Irrational cubic fourfolds with geometric K3 categories

Proof. Resolve the rational maps and use smooth projective models. Compose V\dashrightarrow W with the maximal rationally connected quotient of W. A general fibre of the composite fibres over a rationally connected general fibre of W\dashrightarrow R(W), with rationally connected general fibre. It is rationally connected by the fibration theorem of Graber–Harris–Starr [28, Corollary 1.3]. Since R(W) is not uniruled, this composite is a maximal rationally connected quotient of V. Indeed, a family of rational curves through general points of V that is not vertical would induce a covering family of rational curves on R(W). Uniqueness of the quotient proves the first assertion. A projective bundle has a projective-space generic fibre. The remaining statements follow from the dimension of the quotient, and from the fact that a non-uniruled surface is its own quotient.□

Proof. The field k is rational. If u(\ell)=1, the surface field \ell is the function field of a member of \mathcal S, by Lemma 2.2. A nonidentity field automorphism would give a nonidentity birational, hence regular, automorphism of that K3 surface. This contradicts the choice of \mathcal S. A quadratic extension in characteristic zero has its nonidentity involution.□

Proof. Choose dense open subsets on which f is an isomorphism. Every divisor whose generic point belongs to these opens has a corresponding divisor on the other model with exactly the same valuation. Any valuation in the symmetric difference must therefore be represented by a divisorial component of one of the two closed complements. There are finitely many such components. This also proves that an isomorphism in codimension one has value zero.

For the composition use the common field F=\C(X), transporting \C(Y) by f^* and \C(Z) by f^*g^*. Put D_X=\mathcal D_F(X), D_Y=\mathcal D_F(Y) and D_Z=\mathcal D_F(Z). Their pairwise symmetric differences are finite. On the set of all prime divisorial valuations of F,

All three differences have finite support. Multiplying by u(\kappa(\nu)) and summing proves additivity. Transport by a \C-field isomorphism identifies residue fields, so the last term is precisely c_u(g). Reversing the two valuation sets gives the inverse formula.

For the blow-up, the target of the extraction \pi^{-1}:X\dashrightarrow\widetilde X retains every original divisorial valuation and adds exactly one exceptional divisor E_i=\mathbb P_{Z_i}(N_{Z_i/X}) for each Z_i. Its residue field is purely transcendental over \C(Z_i), of transcendence degree \operatorname{codim}_X(Z_i)-1. Lemma 2.2 gives u(E_i)=u(Z_i). The stated signs now follow from the definition and the inverse formula.□

Proof. Rational connectedness gives H^{p,0}(M)=0 for p>0 [6, Theorem 30]. We use the Hodge decomposition, functoriality and Lefschetz polarizations in [63, Sections 2.1–2.2]; the Hodge-complement statement is Lemma 1 there. Thus H^2(M,\Q) is of type (1,1); hard Lefschetz gives the analogous statement in degree six. The remaining assertion about types follows from Hodge symmetry. In degree four all nonprimitive Lefschetz summands are rational Hodge classes. Hodge–Riemann therefore makes their orthogonal primitive transcendental part nondegenerate. The same argument in degree two applies to a surface, using an ample class to split off its nonprimitive part. Alternatively, the nondegeneracy on the Hodge-class space follows on each primitive Lefschetz summand from definiteness.

By semisimplicity of polarizable rational Hodge structures, the substructure generated by the stated off-diagonal types has a Hodge complement in the transcendental structure. That complement would have only diagonal type, and hence would consist of rational Hodge classes. It is zero by the definition and nondegeneracy of the transcendental part. Finally, i^* takes the off-diagonal part of H^4(M) into H^4(Z), which has only type (2,2); and i_* takes the off-diagonal part of H^2(Z) into H^6(M), which has only type (3,3). Both maps vanish on the generating types and therefore on the whole corresponding rational Hodge structure.□

Proof. There is nothing to prove if W=0. Otherwise p_g(Z)>0, so Z is not uniruled. A fixed-degree connected genus-zero stable map with positive projected degree has projected image a connected union of rational curves. For each irreducible component of its finite-type moduli space the evaluation image in Z has dimension at most one: a dominating evaluation image would give a covering family of rational curves on Z. There are finitely many such components for the fixed discrete data. A holomorphic two-form pulls back to zero through each of their evaluation maps, since their images lie in curves or points. The bilinear tensor, with all other data fixed, is rational and respects Hodge structures. Its associated Hodge endomorphism therefore annihilates the (2,0) generating subspace, and its conjugate, and hence all of W by Lemma 3.1. This proves vanishing as a tensor, independently of the values of the two variables. The argument also applies after projecting an expanded target to Z.

If the projected degree is zero, all evaluations in Z agree. Factoring out base classes by the projection formula, the two variable classes occur as u\smile v. They already have top degree on the surface. Every additional positive-degree base factor kills their product; the remaining fibre integral is a rational number multiplying \int_Zuv. This reasoning is independent of u,v and allows the specified descendants.□

Proof. Cut an edge outside the path. The detached connected subtree has only rational diagonal input classes. Its single output is rational and of Hodge difference zero, because its virtual class and all structure maps respect Hodge difference. At the retained vertex the attachment remains its existing relative root, with the same contact and the resulting diagonal evaluation weight; it is not an additional ordinary marking. Repeating this operation removes all side branches from the tensor contraction. Conservation of Hodge difference at the remaining vertices forces difference two along the path. Lemma 3.1 and the projective bundle formula identify the indicated variable spaces. These path classes and the diagonal branch outputs have even cohomological degree, so their permutations introduce no odd-degree signs. For the first row, contraction of consecutive tensors written in path order as q_M(Au,x) and q_M(By,v) pairs x and y with the inverse cup form and gives q_M(BAu,v). Thus contraction composes operators without taking adjoints. In the second row it composes scalar maps between copies of W. A path changing between the two sorts has a mixed vertex, whose tensor vanishes by the stated vertex hypothesis. The diagonal of a projective bundle gives the same identities between its copies of W, with the relevant nonzero pairing constants. The conclusion follows from closure under composition and rational linear combination. If the two variables meet on one local piece, Lemma 3.4 supplies the second row directly.□

Proof. Replace a relative weight (k,\xi^bp^*\delta) by the ordinary insertion

to form the absolute tensor B_{\mathrm{abs}}(u,v) on M in numerical class \pi_*\beta, retaining all original ordinary insertions and their descendant powers. We apply degeneration to the normal cone using the following specific lifts. In

the strict transform \mathcal Z of Z\times\mathbb A^1 is still Z\times\mathbb A^1, because the blowup ideal restricts to the Cartier ideal (t). Lift i_*\delta by (i_{\mathcal Z})_*(\delta\times1). On the special fibre its restriction to Y is zero and its restriction to P=\mathbb P_Z(N_{Z/M}\oplus\mathcal O_Z) is the zero-section Gysin class. These are transverse fibre restrictions. Original ordinary classes use pullbacks from M.

This lift is valid even when i_*\delta=0. A class on \mathcal W may restrict to zero on the smooth fibre and nontrivially to its special components. For a relative variable in W, Lemma 3.1 says that the associated absolute insertion is zero, so the absolute bilinear tensor is zero. For two ordinary T_M variables the associated absolute tensor is a generator allowed in the first row. Thus the absolute starting tensor has the required restriction in all three cases.

We prove the relative restriction by simultaneous strong induction, keeping the two variables throughout. Fix an integral ample divisor H on M. The outer order is first H\cdot\pi_*\beta, then the number of original ordinary markings. Within fixed outer data, use, in succession, the length of the relative list, the sum of its contacts, and the sum of its hyperplane exponents, with smaller values first. At every smaller outer index the assertion is assumed simultaneously for all numerical curve classes at that index, relative lists, evaluation labels, placements of the two formal variable slots, and ordinary descendant exponents. At equal outer indices the inner induction is likewise simultaneous in the curve classes, labels, variable placements, and descendant exponents. Cutting a side branch only replaces the weight at an existing relative root, so it does not increase the number of original ordinary markings.

A connected relative vertex on Y of zero projected degree cannot have joining markings. Indeed every nonzero effective curve contracted by \pi has class a positive multiple of an exceptional line, and has negative D-degree; relative admissibility would require its D-degree to equal a sum of positive contacts. A zero-class vertex likewise has no contacts. Thus zero projected degree contributes only a constant invariant with no joining marks. Its two middle-degree ordinary variable classes already have total top degree; the invariant is a scalar cup pairing, or zero. This supplies the initial case.

For a noninitial term of the degeneration formula, cut away the branches with no variables, as in Lemma 3.5. Every Y-vertex in a nontrivial connected tree has a joining edge, so the zero-degree argument above makes its projected ample degree positive. The projected classes of all components are effective and add to the original class. Hence, if there is more than one Y-vertex, every one has smaller projected degree and its path tensor is controlled by induction. A unique Y-vertex of smaller projected degree is controlled in the same way. If a Y-vertex has the full projected degree, it is the unique Y-vertex; its tensor is still an earlier case if it retains fewer original ordinary markings. Local path vertices are controlled by Lemma 3.4. In these cases every path vertex therefore satisfies its row, so the conditional contraction rule of Lemma 3.5 controls the term. A path containing no Y-vertex is controlled entirely by the local lemma.

It remains to consider a unique Y-vertex of the full projected class carrying all original ordinary markings. Every P-vertex is then a fibre tail with one joining marking, since the graph is a tree. Let such a tail have joining contact k' and receive the translated markings indexed by a set J. All its evaluations in Z agree. Its pushforward to the joining divisor therefore has the form

where \eta is a rational diagonal class independent of the base labels. The projection formula gives this identity also when a label is one of the two variables. The complex codimension of \eta is

Indeed the relative fibre virtual dimension is r-2+rk'+|J|; each translated zero-section insertion has fibre codimension r+r(k_j-1)+b_j=rk_j+b_j, and pushing to the relative fibre divisor of dimension r-1 gives the displayed difference.

If one tail contains both variable markings, both are relative W variables, since all original ordinary markings remain on the main vertex. Their product is q_Z(u,v) times the point class on Z, so the tail output is q_Z(u,v) times a fixed class. The remaining vertices then have only fixed insertions, so the graph is a rational multiple of q_Z. It already lies in the allowed W,W row, without using induction on its main tensor. For every other nonzero term, expand the tail outputs in the projective bundle basis. A tail with one W variable multiplies it only by fixed diagonal classes, which act by scalars between the allowed copies or annihilate it. Thus each remaining summand retains two separate variable slots in the prescribed spaces, and the following induction applies to it.

A negative c gives zero. An empty tail has c=1-rk'<0, so every tail receives a translated marking. The new relative list therefore has no more entries than the old one. A strict decrease is an earlier induction case. If the lengths agree, each tail has exactly one translated marking. Then c\geq0 and b_j<r imply k'\leq k_j. A strict decrease of the total contact is again earlier. At equal contacts, expand \eta in the projective bundle basis. Its fibre exponent is at most b_j, so either the total hyperplane exponent decreases or all the original indices agree. A decrease of a fibre exponent only multiplies its base label by a rational diagonal class, and preserves the stated variable spaces. Thus all smaller terms with separate variable slots retain the required tensor restriction.

For one translated mark and equal indices, the coefficient of the highest fibre power is obtained by restricting to a fibre. The relative invariant computed in [35, Theorem 7.1] is

The proof of that theorem identifies the descendant line with the cotangent line of the source at the ordinary marked point, so its convention is the one fixed above. Taking j=r-1-b gives exactly the descendant in (3.3). Gluing multiplies these numbers by positive contact and labeling factors. Consequently the remaining term is a nonzero rational scalar times the original relative tensor. At fixed projection, the total contact fixes the numerical class on the blowup, since the kernel on curve classes is generated by the exceptional line.

We have an identity B_{\mathrm{abs}}=C B_{\mathrm{rel}}+R, with C\in\Q^* independent of the two variable vectors. The starting tensor and the remainder belong to the relevant row of Lemma 3.5; hence so does B_{\mathrm{rel}}. The induction is simultaneous in the three rows. It is a tensor identity, not a separate choice of a scalar for each pair of vectors. Testing on opposite off-diagonal types and using Lemma 3.1 extends it to the full rational transcendental structures. This completes the induction.□

Proof. Fix a relative tensor B_{\mathrm{rel}}(u,v) in numerical curve class \beta on Y, with relative list \mu=((k_j,\delta_j))_{j=1}^{L}. Thus D\cdot\beta=\sum_{j=1}^{L}k_j. Use degeneration of Y along D, with the same source cotangent convention, and translate a relative weight (k,\delta) into \tau_{k-1}(j_*\delta) to form an absolute tensor B_{\mathrm{abs}}(u,v) on Y in the same numerical class \beta, retaining the original ordinary insertions and descendants. Use the strict divisor-family lift for translated classes and pullbacks for original ordinary classes. This is the classical divisor correspondence of [53, Section 2.4, Lemma 4]; we give the ordering needed for the restricted tensors.

The ordinary T_M part restricts to zero on D. The rational Hodge structures generated by boundary classes of Hodge difference \pm2 are copies of W: for a surface center they are p^*W\subset H^2(D) and \xi p^*W\subset H^4(D). Gysin carries the first into the exceptional W summand of H^4(Y) and annihilates the second, since H^6(Y) is diagonal. Restriction of the ordinary exceptional class is j^*j_*p^*u=-\xi p^*u. It follows from (3.6) that every associated absolute tensor has the claimed restriction. Local tensors have it by Lemma 3.4. This also treats a translated variable with zero Gysin class; the absolute tensor is then zero.

Fix an integral ample divisor on Y and perform outer strong induction on its curve degree, then on the number of original ordinary markings. As before, these are the ordinary markings before translation, and the induction is simultaneous for all numerical classes, relative data, labels, variable placements, and descendant exponents at the given index. Pruning replaces existing relative weights as in Lemma 3.5.

Every main Y-vertex in a nontrivial tree has a joining root. Its curve class cannot be zero, since a zero class has total contact zero; its ample degree is therefore positive. Under collapse of the degeneration to Y, the projected component classes are effective and add to the original class \beta. If there are several main vertices, all have smaller ample degree. A unique main vertex of smaller degree is also an earlier case. A main vertex with full degree is unique, and is an earlier case unless it retains all original ordinary markings. Thus in these cases induction controls every main path vertex and Lemma 3.4 controls every local one, as required by the conditional path rule. A path with no main vertex is controlled by the local lemma after pruning.

At equal outer data the unique main vertex has curve class \beta: all other components project to points of Y. They are fibre tails of P_D\to D, since the graph is a tree. This argument uses the ample degree on Y, without requiring D to be nef. Hence K=D\cdot\beta, the total contact, is fixed. If K<0 there is no admissible relative datum. A zero curve class has K=0 and no relative marks; its constant tensors are scalar cup pairings, zero on mixed summands. More generally, for K=0 the following inner induction has just the empty-list case.

Let the new main relative list have length t, with tail contacts k'_a. If J_a is the set of translated markings on tail a, the tail calculation (3.4) now applies with r=1: the center of this normal-cone degeneration is the divisor D\subset Y. Thus

Before applying the list order, use the tail-output factorization from the first reconstruction. A tail containing both variable slots gives q_Z(u,v) times fixed classes, so its whole graph contribution is already an allowed scalar pairing. In every other nonzero summand the variable slots remain separate: fixed diagonal classes act by scalars between the copies of W. The following inner induction applies to these summands.

Each nonzero tail has c_a\geq0. Therefore t\geq L. Use decreasing length as the next order; it terminates because t\leq K. This is an inner induction for each fixed \beta, so no unbounded reversal of an order is being used.

At equal lengths every c_a is zero. Each tail has

Take all fixed labels homogeneous. Give (k_j,\delta_j) weight k_j-1+\operatorname{codim}\delta_j. These weights are nonnegative. Each formal variable slot has a fixed cohomological degree, unchanged when its vector is specialized to zero or to the other vector. A boundary W slot has positive codimension, so it always has positive weight. Combining two positive-weight labels on a tail decreases their number, so use that number, with smaller values first, as the final order. A strict decrease of this count is an earlier case with two separately labelled variable slots. If the number of positive-weight labels is unchanged, each tail has at most one positive-weight label. All other labels are (1,1), so every nontrivial contact and label is unchanged. Empty tails have degree one and identity output; redistribution of the trivial labels, at the fixed total length, consequently gives the original weighted partition, with each formal variable slot still distinguished.

These unchanged-data terms cannot cancel. A tail carrying its one positive-weight label has the leading fibre invariant in (3.5), now for r=1, with its positive gluing contact factor. An additional (1,1) label translates to an ordinary point-divisor insertion on \mathbb P^1. By the divisor equation it multiplies the leading number by k'_a. The descendant correction contains the square of the fibre point class and vanishes. For the source cotangent convention this is the ordinary relative divisor equation with the divisor supported in the interior, disjoint from the relative boundary. A degree-one tail without translated markings contributes one. Thus every unchanged-data coefficient is a product of the same positive leading numbers and positive contact, divisor, and labeling factors. All path labels are even, so no permutation signs occur. The graph with one leading fibre tail for each relative entry is among these terms. Their sum is therefore a nonzero rational coefficient C, depending only on the fixed discrete data and the formal labels, not on the two variable vectors. For the empty list the coefficient is one. This is the nonvanishing part of the triangular coefficient calculation in [53, Section 2.4, Lemma 4]; its exact normalization is not needed here.

As before, the formula is an identity of bilinear tensors. All earlier terms are allowed by the simultaneous induction and the path lemma; division by the nonzero, vector-independent coefficient proves the three stated restrictions. Ordinary descendants at retained markings have not changed, and those at markings moved to a fibre occur only in the smaller ordinary-mark cases. Hence no bound or extra hypothesis on original descendants or ordinary diagonal insertions has been used.□

Proof. At \mathbb P^4 the transcendental lattice is zero. A blowup with a nonzero surface contribution adds exactly one summand by Theorem 3.3; a blowup with no transcendental contribution does nothing. A blowdown with zero transcendental contribution also leaves the lattice and algebra unchanged by the same theorem. Suppose instead that the next step is a blowdown with nonzero surface contribution. The theorem, read in reverse, exhibits the removed center as a factor \Q of the operator algebra of the current variety. Its identity is a primitive central idempotent. The primitive central idempotents of the already identified algebra \Q^m are precisely the projectors onto its existing whole summands. Thus the removed rational subspace is one of those summands, even if some summands are mutually Hodge-isomorphic or a surface transcendental structure is reducible.

The lattice of each summand is its intersection with the ambient integral lattice: the historical decomposition and the geometric blowup decomposition are integral direct sums. Equality of their rational subspaces therefore identifies their saturated integral lattices and cup forms. The residual summands are unchanged. This proves the induction.□

Proof. Geometric genus and the transcendental lattice are unchanged by point blowups. Pass to a minimal model using [12, Proposition II.16]. In the following numerical calculation, K, b_2, and q=h^{1,0} refer to that minimal surface. Since p_g=1, it is not ruled and has nonnegative Kodaira dimension [12, Propositions III.20–III.21 and Example VII.3]. Its canonical class is nef [12, Corollary VI.18], so K^2\geq0. Noether’s formula [12, Sections I.14 and III.19] gives c_2=12(1-q+p_g)-K^2=24-12q-K^2. On the other hand its topological Euler characteristic is c_2=2-4q+b_2. Hence

Projectivity gives Picard rank at least one, while the rank assumption gives b_2\geq22. Thus q=0, K^2=0, and the Picard rank is one. Kodaira dimension two is excluded by K^2=0 [12, Proposition X.1]. In Kodaira dimension one the elliptic fibration [12, Proposition IX.2] supplies a nonzero isotropic divisor class, independent of an ample class, contradicting Picard rank one. The classification in Kodaira dimension zero leaves a K3 surface, since p_g=1 and q=0 [12, Theorem VIII.2].

The K3 lattice is unimodular, and its Néron–Severi lattice is primitive by the integral Lefschetz (1,1) theorem [63, Theorem 2]. For a primitive sublattice of a unimodular lattice, its discriminant group and that of its orthogonal have equal orders. The Néron–Severi lattice here is generated by a primitive ample class L, so its discriminant is L^2. This proves the degree assertion.

Any automorphism fixes L and hence acts trivially on this rank-one Néron–Severi lattice. Its action on T_{Z,\Q} is a rational scalar by the endomorphism hypothesis, and preservation of the nondegenerate form makes that scalar +1 or -1. The unimodular gluing identifies the cyclic discriminant group of T_Z with that of \Z L, of order d; see [54, Proposition 1.6.1] for the compatible gluing of primitive orthogonal sublattices of an even unimodular lattice. The action -1 on the first and +1 on the second is compatible only if 2 annihilates this cyclic group, contrary to d>2. Thus the action is +1 on both lattices, and hence on the whole integral second cohomology. The faithful cohomological action for K3 automorphisms, a consequence of global Torelli [37, Proposition 15.2.1], makes the automorphism the identity.□

Proof. Under the rationality assumption all varieties in the smooth factorization of Theorem 2.5 are rationally connected. Lemma 3.8 tracks integral whole blocks through the factorization. A nonzero transcendental rational Hodge structure of this kind has a nonzero off-diagonal part by Lemma 3.1. Since the final h^{3,1} is one, its transcendental structure is simple: a nontrivial semisimple decomposition would require at least two nonzero (3,1) parts. Thus the final lattice is one whole block. The number of additions minus removals of blocks in its integral polarized Hodge-isometry class is one.

Every center in this class has geometric genus one, transcendental rank 21, determinant of absolute value d, and rational Hodge endomorphism ring \Q. Lemma 3.9 makes it birational to a member of \mathcal S, and proves the degree and automorphism assertions. Conversely, a smooth surface center counted by u is birational to a member of \mathcal S: in dimension two an MRC base which is a K3 has the same function field as the surface. Its entire transcendental lattice therefore has exactly the required class. Point and curve centers are not counted.

The exceptional divisor of a smooth blowup is a projective bundle over its center and has the same MRC base, by Lemma 2.2. The sign convention and additivity in Lemma 2.4 therefore identify c_u(f) with the above number of block additions minus removals. This number is one.□

Proof. Resolve the map by point blow-ups \pi:Y\to S. Since h^1(Y,\mathcal O_Y)=0, pullback of differentials shows that the base of its Stein factorization has genus zero. Thus, after Stein factorization, we have a morphism f:Y\to\mathbb P^1 with smooth connected general fibre F. The images on S of these fibres form a pencil without fixed components, of class nL for some integer n\geq1. In the total-transform basis of the successive exceptional curves write

The exceptional basis is orthogonal, with (E_i^*)^2=-1. Consequently adjunction and F^2=0 give

It follows that 2g-2\geq(\sum_i m_i^2)^{1/2}=n\sqrt d. Stein factorization accounts for the geometric components in the last assertion.□

Proof. First consider a bounded smooth projective surface Y birational to S. Its minimal-model morphism \mu:Y\to S is a succession of point blow-downs. Let A be a very ample divisor from the bounded family. In the total-transform exceptional basis,

The pushforward B is ample: at each blow-down its square increases and its intersection with every curve is positive, so the surface Nakai–Moishezon criterion applies [19, Theorem 1]. It is an integral multiple of L. Hence

The right-hand side takes only finitely many values in a bounded smooth projective family after a finite stratification. This proves the assertion in dimension two; in smaller dimensions a K3 MRC base is impossible.

It remains to bound the birational types of the K3 bases of bounded smooth projective threefolds. Put these threefolds into finitely many smooth projective families \mathcal X\to T, with relatively very ample divisor H. We use the fixed-polynomial Hilbert space and its open graph locus [61, Tags 0DPH, 0D1A and 0D1B], together with semicontinuity and base change [61, Tags 0BDN and 0B91]. For e\geq1, the free locus in the relative scheme of degree-e maps \mathbb P^1\to\mathcal X/T is open. Its morphism to T is smooth: for a free map a the obstruction space H^1(\mathbb P^1,a^*T_{\mathcal X_t}) vanishes. Its image is therefore open. If U_D is the union of these images for 1\leq e\leq D, then the U_D are ascending open subsets of the noetherian space T. Their union is precisely the locus of uniruled geometric fibres, since in characteristic zero a smooth projective variety is uniruled if and only if it has a nonconstant free rational curve. The union is quasi-compact, so U_D equals that union for some D. These deformation statements follow, for example, from [6, Remark 9, Proposition 10 and Theorem 15]; this argument proves the uniformity rather than assuming that the uniruled fibres themselves form a separately specified family.

Suppose now that the MRC quotient of X=\mathcal X_t is a surface birational to S. Use its almost-holomorphic presentation r:U\to S^\circ with proper smooth rationally connected fibres; see [44, Chapter IV, Section 5]. Its fibres are smooth copies of \mathbb P^1. A degree-at-most-D free map on X deforms in a covering family. A very general member meets a very general fibre of r and is contained in it, by the MRC property. Since that fibre is a complete integral curve, the image of the map is the whole fibre. The H-degree e of the fibre is thus at most D.

For such a fibre F, smoothness of r gives N_{F/X}\simeq\mathcal O_{\mathbb P^1}^{\oplus2}. Its Hilbert scheme is therefore smooth of dimension two at [F]. The family of fibres defines an injective map from S^\circ to this Hilbert scheme, with isomorphic tangent spaces; its closure is an irreducible component of \operatorname{Hilb}^{en+1}(X) birational to S. The components so obtained are components of fibres of the finite-type projective scheme

They have bounded smooth projective birational models by the preceding bounded-family reduction. Apply (4.1) to these surfaces. This proves a uniform bound independent of the threefold and of its MRC quotient map.□

Proof. Part (i) is Lemma 4.2. For (ii), pass to bounded smooth projective models of the pairs, including the exceptional locus in the boundary. The normalized base-changed covers are still étale on the complementary open. For a smooth curve A of genus g_A with at most r marked points, and a connected degree-e cover A'\to A unramified elsewhere, Riemann–Hurwitz gives

Indeed, over any one branch point the ramification contribution is e minus the number of points above it, and is at most e-1. This proves the curve assertion uniformly for e\leq N.

For a bounded smooth surface pair choose a pencil of hyperplane sections for a relatively very ample divisor H. A general section A is smooth, has uniformly bounded genus by adjunction, and meets the divisorial part of D in uniformly boundedly many points. Choose it to avoid the zero-dimensional part of D. For any one cover, general members can also avoid the images of the singular points of its normalization; normal surfaces have only finitely many singular points. Each normalized component over A has degree at most N and is ramified only over A\cap D, so (4.2) gives a uniform genus bound G. Resolve the pulled-back pencil on a smooth model of the covering surface and take its Stein factorization. Its smooth connected general fibres are precisely these normalized components. If the surface field were that of S, Lemma 4.1 would give \sqrt d\leq2G-2. This proves the surface assertion. The pencil can depend on the individual cover; only its genus and boundary-intersection bounds need to be uniform.

The same curve estimate over \overline{\C(C)} bounds the genera of the geometric generic components in (iii). If T were birational to S, the given map to C, after resolution and Stein factorization, would give a pencil on S. Lemma 4.1 again bounds d. No genus bound for the parameter curve C is needed.

For (iv), suppose the MRC base of T is birational to S, and write r:T\dashrightarrow S and f:T\dashrightarrow C. Put F=\C(C) and choose an algebraic closure \overline F. Resolve the maps when necessary. If f were nonconstant on a general fibre of r, the map (r,f):T\dashrightarrow S\times C would be dominant and generically finite. A component of a geometric generic surface fibre of f would then dominate S_{\overline F} generically finitely. Such a component is uniruled, whereas a generically finite dominant image of an uniruled surface is uniruled. This contradicts the fact that a K3 surface is not uniruled in characteristic zero.

Consequently f=h\circ r for a dominant rational map h:S\dashrightarrow C. This is descent along the geometrically integral general fibres of the MRC quotient, or equivalently the relative algebraic closedness of \C(S) in \C(T). Resolve h and take its Stein factorization. Over \overline F, let \Gamma be a smooth geometric generic fibre of the resulting connected-fibre pencil, of genus g. A geometric generic component P of f dominates the corresponding \Gamma. For a smooth projective model \widetilde P over \overline F this implies

To see the inequality, resolve the rational map to \Gamma and pull back regular one-forms; pullback is injective for a dominant map in characteristic zero, and irregularity is a birational invariant of smooth projective surfaces. The irregularities of \widetilde P are uniformly bounded by bounded smooth projective birational models and upper semicontinuity. Lemma 4.1 therefore bounds d.

All genus, intersection, and coherent-cohomology bounds used here are preserved by extension of algebraically closed fields. The same proofs therefore apply to the geometric generic data specified in (iii) and (iv). Taking the maximum of the finitely many bounds proves all the assertions.□

Proof. Take N=N_1 in the displayed wall diagram, let A=\Theta-\Phi be the ample difference on W, and write A_N,\Theta_N,\Phi_N for transforms. We first transfer the discrepancy bound from W to N. We then make the original log-anticanonical class nef and pass to its ample model. That last map must be small: a divisorial contraction would lose one of the prime valuations we need to retain. On a graph resolution with projections p to W and q to N,

For this comparison at the wall, approach \Theta by interior parameters \Theta_i in the chamber defining N. By [31, Theorem 3.3(3)], W\dashrightarrow N is a log terminal model for K_W+\Theta_i, so the exceptional difference on the same graph resolution is effective. Its finitely many coefficients depend linearly on the boundary. Taking their limits gives E_\Theta\geq0; [31, Theorem 3.3(2)] supplies the contraction N\to Q. Thus the first difference is exceptional and nonnegative at the wall. The second is also q-exceptional, since the map does not extract. On a q-contracted curve its degree equals that of p^*A, so it is q-nef; the negativity lemma proves (4.5). Subtracting gives

It follows valuation by valuation that the original \epsilon-lc bound is retained on N.

There is a useful extension of this argument. The divisor K_N+\Theta_N is pulled back from Q. Any small \Q-factorial modification N'\to Q has the same codimension-one divisor and the same pullback relation. Thus N\dashrightarrow N' is crepant for K_N+\Theta_N. The comparison (4.4) remains nonnegative for W\dashrightarrow N', and the ample subtraction proves (4.6) for N' as well. In particular, the bound holds on any required relative small chamber; there is no need to locate all those chambers in one chosen two-dimensional slice.

The pair (N,\Theta_N) is klt, its boundary is big over Q, and its log-canonical divisor is trivial over Q. Writing that boundary, up to relative rational linear equivalence, as an ample divisor plus an effective divisor, a sufficiently small perturbation makes a klt log-Fano boundary. Therefore N/Q is of Fano type. Furthermore,

The transform of an ample class by a birational contraction is in the interior of the movable cone: choose divisor classes on W mapping onto a basis on N, perturb A slightly in both directions in this basis, and push their ample systems forward. No prime divisor is fixed, because none is extracted. The same argument works after passing to the relative numerical space over Q.

Finite generation now makes this movable class nef on a small model N^{\mathrm c}/Q. Only small steps are needed: a negative divisorial contraction would force its exceptional divisor to be fixed in the corresponding system. The class is big, and its semiample contraction cannot be divisorial because it lies in the interior of the movable cone. Indeed a nef class vanishing on curves covering a contracted divisor lies on the boundary of the movable cone, as follows by intersecting those curves with effective divisors avoiding that divisor. Its ample-model map N^{\mathrm c}\to Z^{\mathrm c} is therefore small. The log-canonical divisor is crepant for this last contraction, so the discrepancy bound passes to Z^{\mathrm c}. Its small \Q-factorializations have precisely the common prime valuations of the link. In particular, N^{\mathrm c}\to Z^{\mathrm c} is one such small \Q-factorialization, and N^{\mathrm c}\dashrightarrow N_i is small for i=1,2. Composing with the contraction towers N_i\to V_i\to B_i\to Q in the wall diagram gives exactly the rational contractions asserted in the statement.

When \Phi=0, exceptional valuations above W already have log discrepancy greater than one. A divisor of W that is contracted has a strictly negative coefficient in (4.5): intersect with a general curve in its positive-dimensional contracted fibre, on which A has positive degree. Its coefficient in (4.6) is consequently strictly positive. This gives terminality also for these remaining exceptional valuations. The same reasoning applies to each small model as above.□

Proof. Write (W',\Phi') for the higher smooth start with its new marking. On the original smooth SNC pair, every exceptional valuation has log discrepancy greater than one: (4.7) gives the bound 2(1-c)>1 for rational c. For irrational c, apply it to rational coefficients decreasing to c and use the affine dependence of log discrepancy on the boundary coefficients. The non-exceptional divisors have discrepancy at least 1-c. These facts also hold on the new smooth SNC start directly, without comparing the number of its exceptional divisors.

Fix one endpoint V and write f:W'\to V for the resolved map. For a divisor of W contracted by its negative program, discrepancy strictly improves. If its retained marking is b\in\{0,c\}, then on the higher start its coefficient in the difference from the pullback of K_V+\Phi_V is

For a divisor exceptional over W, monotonicity gives a(F,V,\Phi_V)\geq a(F,W,\Phi)>1, and adding a new mark b\geq0 again makes this coefficient positive. Thus the entire difference over V is effective with strictly positive coefficients on every endpoint-exceptional divisor.

Run the relative program over V. Existence follows from relative bigness for a birational morphism [15, Theorem 1.2]. At a relative minimal model the exceptional difference is nef over V and hence nonpositive by negativity. Since all the above positive exceptional coefficients must have been contracted, the resulting map to V is small. A projective small birational morphism to a \Q-factorial variety is an isomorphism. This realizes the same endpoint, after which its Mori contraction can be used. The argument works for both endpoints.

We spell out the endpoint perturbations from [31, Lemma 4.1]. Write f_i:W'\to V_i for the two resolved maps and \pi_i:V_i\to B_i for their Mori contractions. Choose small general effective ample rational divisors A,H_1,\ldots,H_s, with the classes of the H_j spanning N^1(W'), and put H=A+\sum_j H_j. If c is irrational, also choose a rational boundary \Phi_0\leq\Phi' sufficiently close to \Phi'; for rational c take \Phi_0=\Phi'. Make these choices so that A+\Phi_0-\Phi' is ample, both f_i are (K_{W'}+\Phi_0+H)-negative, and -(K_{V_i}+(f_i)_*\Phi_0+(f_i)_*H) is \pi_i-ample. The exceptional differences have finitely many strictly positive coefficients, which stay positive under these small changes; the ampleness conditions are open as well. For a sufficiently positive ample rational divisor C_i on B_i, the divisor

is ample. Choose a general effective rational representative F_i^{V_i}\sim_{\Q}L_i and put F_i=f_i^*F_i^{V_i}. Then

Here \Theta_i-\Phi' is ample: it is the sum of the ample divisor A+\Phi_0-\Phi' and the nef divisors H_j,F_i. Simultaneous general representatives of sufficiently divisible multiples make the dominating pair (W',\Phi'+H+F_1+F_2) klt. The negativity of f_i therefore makes B_i the required endpoint ample model. The spanning classes H_j and general two-dimensional slice in [31, Lemma 4.1] complete these two perturbations to one geography, retaining \Phi' as the fixed original boundary. Lemma 4.4 now uses the unchanged lower bound 1-c. With zero boundary the same proof starts from the strictly positive discrepancies of terminal endpoints.□

Proof. We give the generic argument, and then apply noetherian induction. Take the closure of the parameter points under consideration and an irreducible component of this closure. After shrinking and a generically finite base change, we have flat normal geometric fibres \mathcal Y_t, a family \pi:\mathcal R\to\mathcal Y of resolutions, and smooth projective fibres \mathcal R_t on which the marked strict transforms and exceptional divisors form a relative SNC divisor. The individual exceptional components can also be separated by this base change. These are the usual spreading operations on a resolution of the geometric generic fibre.

For the members in question, rational singularities and vanishing give

For log-Fano pairs, the downstairs vanishing follows from Kawamata–Viehweg vanishing and the equality follows from rational singularities; see [26, Theorem 3.13.1 and Corollary 5.7.7]. The parameter points of these members are Zariski dense in the closure chosen above. Thus they satisfy the dense-subset hypothesis of the relative Picard-space theorem [21, Theorem 1.3]: rational singularities and H^1(\mathcal O)=H^2(\mathcal O)=0 are required on that dense subset. Upper semicontinuity also gives the displayed vanishings on an open set. The theorem permits a further generically finite base change and shrinking such that the rational numerical divisor spaces of \mathcal Y_t and \mathcal R_t, their pullback map, and the exceptional divisor classes are identified with fixed rational linear data. In particular, choose relative Cartier divisors whose classes span the downstairs space on every fibre.

Here is the required \Q-Cartier criterion. For a rational Weil divisor D on a fibre, let \widetilde D be its strict transform. Then

The forward implication follows by pullback. For the converse, numerical equivalence of rational divisors on \mathcal R_t is rational linear equivalence: its \operatorname{Pic}^0 vanishes, and the numerically trivial part of the Néron–Severi group is torsion. Thus the asserted relation, after clearing denominators, is a linear-equivalence relation with a Cartier divisor pulled back from \mathcal Y_t and exceptional divisors. Pushing down proves that a multiple of D is linearly equivalent to a Cartier divisor. This is precisely the converse in (4.8).

Apply the criterion to the fixed rational classes on the resolution. For the log-canonical class one may use K_{\mathcal R_t}+\widetilde B_t modulo exceptional classes; this does not presuppose that K_{\mathcal Y_t}+B_t is \Q-Cartier. The membership in (4.8) holds on the dense set of members under consideration, and the linear data are fixed. It therefore holds at the geometric generic point, with fixed rational coefficients in a basis of relative Cartier classes. Rational linear equivalence upstairs, followed by pushdown, proves that the generic log-canonical divisor is \Q-Cartier. Clear denominators and spread this linear equivalence. Its possible vertical errors disappear after shrinking, giving a relative Cartier multiple. This argument applies simultaneously to each additional specified class.

Exceptional coefficients in a pullback are unique. Indeed, an exceptional rational divisor numerically trivial over its target is zero, by applying the negativity lemma to it and its negative. They are thus the solutions of fixed rational linear equations on the resolution, and are constant after shrinking. The klt condition is now the finite set of strict inequalities saying that the crepant SNC coefficients are less than one. Relative log-anticanonical ampleness is open. Thus an open set consists entirely of the required log-Fano pairs, with all the asserted properties.

Finally, all Weil divisors downstairs are \Q-Cartier exactly when the right side of (4.8) exhausts N^1(\mathcal R_t)_{\Q}: every divisor upstairs pushes to a Weil divisor, and the same argument applies. This is again a condition on the fixed linear data. Applying the construction to the remaining closed parameter sets proves all the assertions by noetherian induction.□

Proof. Lemma 4.6 supplies families of Fano-type fibres. Take smooth parameter strata and remove the images of any vertical exceptional components of the simultaneous resolution. The relative \Q-Cartier equality and the crepant coefficients less than one then make the total-space pair klt as well. This verifies the total-space hypothesis of [21, Theorem 1.4]. After a generically finite base change and shrinking, [21, Theorems 1.4–1.6] gives fibrewise small \Q-factorializations and compatible numerical spaces, nef cones, Mori chambers, and MMP steps. Its hypotheses hold: the fibres have rational singularities and the stated vanishings, and every fibre is of Fano type. Thus the very-general-fibre alternative in [21, Theorem 1.2] applies.

For clarity, the finiteness-to-boundedness step is as follows. On the geometric generic fibre of one stratum choose a small \Q-factorialization. Finite generation supplies finitely many small models, and finitely many faces and semiample contractions on each of them. All this finite data spreads after a finite extension and shrinking. The two inclusions needed for constancy of a nef cone can be seen directly: spread its finitely many semiample generators, which remain semiample on an open set, and spread effective curves generating the dual polyhedral cone. The divisor generators give one inclusion; the curve generators give the reverse. The fibrewise \Q-factorializations and compatible numerical spaces ensure that these comparisons take place in the same space. The relative chamber/MMP compatibility cited above then rules out further chambers on a fibre. Here the source has first been made fibrewise \Q-factorial, so the partial-MMP chamber description agrees with that from small modifications; the converse direction in [21, Theorem 1.6(3)] applies. The generic finite list of graphs consequently supplies the complete list on an open set. Apply noetherian induction to the remaining strata. This proves boundedness of the diagrams, not just of their vertices.

We also check the last assertion. If f:Y\dashrightarrow Z is a non-extracting birational map, take a sufficiently ample Cartier divisor H on Z and a general member avoiding the generic points of the finitely many centers of divisors contracted by f. Write D for its strict transform on Y. The corresponding linear system has no fixed prime component. Moreover,

At a prime divisor retained by the map this is the same valuation test on both sides. At a prime divisor contracted on Y, a regular section of \mathcal O_Z(mH) pulls back regularly, and the chosen H has order zero at that valuation. Normality then gives equality of the section spaces. Hence Z is the ample model of a movable system on Y. A subsequent contraction is handled by taking the pullback of an ample divisor from its base. The Mori dream space chamber description therefore includes these maps too. If an intermediate target is not \Q-factorial, apply this description to the composite from the initial \Q-factorial source; the diagram also records the intermediate maps.

Finally take closures of the marked transforms on the finitely many graph families, and resolve their union with the exceptional loci. Proper images, inverse images, and irreducible components of these finite-type families remain bounded after stratification. This gives the stated marked and exceptional data.□

Proof. Apply the geography above to a common smooth resolution of the specified map in (i), and to the given common-start maps in (ii), retaining their function-field identifications. Lemmas 4.4 and 4.5 give a central ample model Z^{\mathrm c}/Q, reached by a small map from N^{\mathrm c}/Q, with \epsilon=1 in the first case and \epsilon=1-c in the second. Put Z=Z^{\mathrm c}_{\overline F}. Its dimension is at most four, the negative log-canonical divisor is ample, and the same discrepancy bound holds. Birkar’s boundedness theorem [14, Theorem 1.1] applies: projective varieties admitting an \epsilon-lc boundary with nef and big negative log-canonical divisor form a bounded family. In particular it bounds the underlying varieties Z just constructed.

The marked boundary is bounded as well. In case (ii), write its restriction to Z as cD_Z, and let H be one of the bounded very ample polarizations supplied by boundedness. In fibre dimension r,

The right side is bounded by the coefficient of the Hilbert polynomial of the bounded polarized varieties. Thus D_Z has bounded degree. Reduced cycles of bounded degree in a fixed projective space form a bounded family, so the marked central pairs are bounded. Marks vertical over Q have empty restriction to the generic fibre. Horizontal components may split over \overline F, but their nonzero coefficients remain c and their total degree is still bounded.

All the geometric generic link vertices and bases are rational contractions of small \Q-factorializations of these central pairs. Lemma 4.7 supplies bounded diagrams, and Remark 4.8 supplies the closed bad loci. Small loci do not acquire divisorial components under the algebraic extension to the geometric generic field. The same constructions apply over algebraically closed fields of characteristic zero. For the family results stated over \C, we use their finite-data spreading arguments and algebraically closed base change to obtain the corresponding geometric generic families here. Every bound was taken from one of these fixed bounded families, not by accumulating constants along a path.

For the assertion about base contractions, the central rank-two model is of Fano type over Q by Lemma 4.4. This property survives small modifications and descends under contractions, including fibre-type contractions [55, Lemma 2.8, with the relative convention after Lemma–Definition 2.6]. It therefore holds for every end and its base B\to Q. These are contractions of normal varieties, so their generic fibres are geometrically integral in characteristic zero. The rational connectedness theorem for klt log-Fano contractions [30], applied also to a resolution, gives the asserted rationally connected smooth proper models.□

Proof. At the generic point of the center of F, of codimension r, use the regular parameters and boundary coefficients in (4.7). If the center is not a boundary stratum, at least one b_i is zero; since an exceptional center has r\geq2, the right side is strictly greater than one. Thus a valuation in question centers at a boundary stratum. Write its orders there as w_i. They satisfy

There are finitely many possibilities because every 1-b_i is positive.

Blow up the stratum. In a chart containing the center of the valuation, choose an index with minimal order; the new orders are that minimum and the differences from it. Zero differences correspond to units unless the center lies in a further proper subvariety of the exceptional stratum. The latter would be a non-stratum center and is excluded by (4.7) applied to the crepant SNC transform. This transform is still klt, so its coefficients remain strictly less than one and the same non-stratum exclusion applies at every step. Thus, until the valuation itself becomes a divisor, its center continues to be a stratum and the sum of its positive orders strictly decreases. This is the usual monomial subtraction algorithm, and proves both that the valuation is a stratum valuation and that finitely many blow-up sequences suffice. In a fixed family there are finitely many strata and bounded orders; separate their geometric components and perform the same finite sequences relatively.□

Proof. Use Lemma 4.6 to obtain simultaneous crepant log resolutions. Their coefficients are in a fixed finite set strictly below one on each stratum. The exceptional divisors already on the resolution are a finite list; all additional valuations of log discrepancy at most one are in the uniformly finite list of Lemma 4.10. Extract all of them on a common resolution.

The extraction theorem [15, Corollary 1.4.3] produces a \Q-factorial model \tau:T^{\mathrm t}\to T extracting exactly this set. The extra restriction concerning non-klt centers in that theorem is empty here. Its crepant boundary \Lambda^{\mathrm t} has coefficient 1-a(F,T,\Lambda) on an extracted divisor, and hence is effective. By construction the pair is terminal. Also

is nef and big, so T^{\mathrm t} is of Fano type. Apply this construction on a geometric generic member and spread it; the fixed extraction list, crepant equality, and the discrepancy inequalities persist after shrinking. Thus the terminalizations and all their marks form bounded families. This construction does not resolve an unbounded set of arbitrarily chosen higher models.

For a model T' extracting a subset, every prime divisor on T' is already a valuation appearing on T^{\mathrm t}. Consequently T^{\mathrm t}\dashrightarrow T' is a birational contraction. Lemma 4.7, applied to the bounded Fano-type terminalization families, supplies its diagram and every indicated subsequent contraction. In applying that lemma one can either use the displayed weak log-Fano pair and a small ample perturbation of its boundary on each stratum, or spread a Fano-type boundary chosen on its generic member. Both give families of klt log-Fano pairs on an open set. Include all the extracted divisors as marks, even those with crepant coefficient zero. The marked support on every output is then among the bounded transforms already considered.□

Proof. Zero-dimensional members have D=0 and impose no restriction on c; we treat the positive-dimensional members below. Apply Lemma 4.6 to the log-Fano pairs (T,0) with additional marked divisor D. On finitely many simultaneous log resolutions the crepant coefficients for (T,0) and the coefficients of \pi^*D are fixed. Write a_E=a(E,T,0)>0 and m_E for the coefficient of \pi^*D on an exceptional component. Choosing c<1/2 and cm_E<a_E/2 for every m_E>0 makes every coefficient of the crepant pullback of cD strictly less than one. These finitely many inequalities give a uniform positive klt bound for c.

For ampleness use a bounded very ample divisor H and let r=\dim T. A common multiple mK_T is Cartier on each of the finitely many strata. Thus

The degree e=D\cdot H^{r-1} is a bounded nonnegative integer. Numerical proportionality says D\equiv(e/k)(-K_T). Since e/k\leq me, choosing cme<1/2 for all these data gives

which is ample. Taking the minimum of the finitely many rational choices gives c_*.

Finally numerical equivalence between divisors defined over the original field persists geometrically. Any geometric curve is defined over a finite extension; all its conjugates have the same intersection with such a divisor. Its orbit descends to a curve cycle over the original field, so a zero intersection downstairs forces zero on every conjugate. The preceding argument therefore applies after the geometric base change.□

Proof. First use only the ordinary part of Theorem 4.9. At the indicated end the tower is N_i\xrightarrow{e_i}V\to T\to Q. Since \dim T=3 and \dim Q\leq1, the last contraction has positive relative dimension and \rho(T/Q)\geq1. Equation (4.3) gives

Consequently \rho(N_i/V)=0 and \rho(T/Q)=1: the arrow e_i is the identity and T\to Q is an elementary contraction of fibre type. This calculation takes place over Q before geometric generic base change. The geometric generic model T_{\overline F} and its conic projection occur in the bounded diagram. The restriction of the original visible ramification is contained in the bounded bad locus of this projection, and its divisorial support is a union of the finitely many divisorial components there. These marked supports are therefore bounded.

Apply [23, Lemma 4.6] to the elementary conic Mori contraction V\to T. Its source is terminal and \Q-factorial, so its base T is \Q-factorial. Divisors on T_F extend by closure to Weil divisors on T, which are \Q-Cartier. Restriction therefore surjects onto the generic numerical divisor space. The relative rank-one condition and the nonzero restriction of an ample class give \rho(T_F/F)=1. Fano type descends under contractions [55, Lemma 2.8]; hence T_F and T_{\overline F} are of Fano type. Over the original generic field F, rank one implies that -K_{T_F} is ample: it is the sum of an ample log-anticanonical class and an effective class. It also implies numerical proportionality of (D_T)_F and -K_{T_F}. These two divisors are \Q-Cartier before, and hence after, base change. Their numerical proportionality persists over \overline F by Lemma 4.12, even if the geometric Picard rank increases. That lemma provides a uniform c_* from these ordinary diagrams.

Choose a rational c<\min\{1/4,c_*\} once. With this fixed value the marked log links have the uniform bound of Theorem 4.9, and the bounded (T_{\overline F},c(D_T)_{\overline F}) data have the extraction bound of Theorem 4.11. Only finitely many bounded families, dimensions, and bounded-degree cover constructions are used subsequently: the exceptional and bad-locus components in Section 5, and the ramification and del Pezzo class covers in Section 6. Their branch and degree bounds are checked there. Take the maximum of their thresholds in Theorem 4.3. Every bound depends only on those families and this chosen c, proving the asserted independence.□

Proof. Choose a rational boundary \Theta making (Y,\Theta) klt and -(K_Y+\Theta) relatively ample. Then -K_Y=-(K_Y+\Theta)+\Theta is relatively big, and a sufficiently divisible multiple of -(K_Y+\Theta) is generated over Z. Theorem 1.2 and Corollary 1.4 of [30] give rational chain connectedness of every fibre; the non-klt locus is empty. The statement after passage to a characteristic-zero geometric generic field follows by descent of the finite data and base extension.

Put T=f(E) and choose a general point e of E outside the other components of f^{-1}(T). The fibre of E\to T through e has positive dimension. A chain of rational curves in the fibre of f joining e to another point starts with a nonconstant rational curve through e. That curve is contained in E: its image is in f^{-1}(T), and an irreducible curve through a point lying on only the component E must lie in that component. Thus rational curves cover E, proving uniruledness. Equivalently, one can select a dominating component of the parameter space of these rational curves. The argument can be made after an uncountable algebraically closed extension and descended, so it does not require an uncountability assumption on the original field.

For (i), f(E) is proper and f^{-1}(f(E)) is a proper closed subset of the integral variety Y. Its component containing E must equal E, and \dim E-\dim f(E)\ge\dim Y-\dim Z>0. The same component argument works in (ii), where exceptionalness gives \dim E-\dim f(E)\ge1. Finally uniruledness is birationally invariant.□

Proof. The assertion is immediate for an isomorphism. For a birational contraction, nonexceptional prime divisors cancel with their birational transforms on Q. The remaining divisors are uniruled by Lemma 5.1. Since \dim B\le3, these divisors have dimension at most two, and their MRC quotients have dimension at most one; they have test value zero.

If the contraction has positive relative dimension, then \dim Q\le2. Every prime divisor of Q has test value zero for dimension reasons. Every divisor of B vertical over Q is uniruled by Lemma 5.1, and again has dimension at most two, so its test value is zero. This includes Q a point, when there are no vertical divisors and no base divisors to sum.□

Proof. In the difference v(V'/Q)-v(V/Q) the base-divisor sums cancel. Common vertical prime valuations cancel as well, leaving precisely the signed exceptional divisors vertical over Q. This proves (5.4). Substitute (5.3) and use Lemma 5.2 on both ends to obtain (5.5).□

Proof. Suppose first that \dim Q=3. Each Mori base has dimension at most three and dominates Q, so its morphism to Q is birational. No divisor of either Mori base, and no divisor of V vertical over that base, dominates Q. Therefore both h_Q terms vanish. Normal proper generic curves over \C(Q) are regular and hence smooth in characteristic zero. A birational map between them is an isomorphism. Their closed points are exactly the prime divisors of the fourfolds dominating Q, so there are no horizontal divisorial contributions either.

For \dim Q=0, the entire marked contraction diagram is bounded by Theorem 4.9. The exceptional prime divisors contributing to c_u(f) therefore have bounded birational models of dimension three. Choose a marked good open for each contraction V\to B. After cancellation, the divisors contributing to h_Q(V/B)=v(V/B) are components of the marked bad locus on B or its preimage on V. These components have bounded birational models by the marked-locus assertion of Theorem 4.9. The bounded-variety exclusion in Theorem 4.3(i) gives a single threshold beyond which every term has test value zero.

Now suppose \dim Q=1, and put F=\C(Q). We use the bounded diagrams after extension to \overline F. If E is a horizontal exceptional prime divisor of the link, the geometric generic fibre components of E\to Q are surfaces. On the side where the divisor is extracted, each is exceptional for a divisorial contraction of the generic threefold. They are uniruled by Lemma 5.1; any intervening small modifications preserve their birational types. Their smooth birational models are bounded by Theorem 4.9. The threefold-over-a-curve exclusion in Theorem 4.3(iv) therefore makes u(E)=0 uniformly for large d. Here u is evaluated on the original complex threefold field \C(E); only its geometric generic surface components, rather than E itself, are being placed in bounded families.

It remains to treat h_Q(V/B), the other end being identical. Work with a good open for the generic contraction V_F\to B_F and spread it over an open of Q. Shrinking Q changes no horizontal prime field. The cancellations defining h_Q remove all divisors meeting that good open. Consequently the geometric generic components of the remaining prime divisors lie in the bounded marked bad locus or its bounded preimage.

A remaining positive term is a divisor P\subset V vertical over B and dominant over Q. Its geometric generic fibre components are vertical divisors of V_{\overline F}\to B_{\overline F}. They are uniruled surfaces by Lemma 5.1, with bounded birational models. Theorem 4.3(iv) therefore gives u(P)=0 for large d. A remaining negative term is a divisor D\subset B dominant over Q. If \dim D\le1, its test value is zero automatically. If \dim D=2, its geometric generic fibre components over Q are bounded curves. Were u(D)=1, the surface D would be birational to a member of \mathcal S; the induced pencil, after Stein factorization, would have general fibre genus at least 1+\sqrt d/2. This contradicts the uniform genus bound for those components when d is large.

Only finitely many bounded families and types of marked components have been used. Take the maximum of their exclusion thresholds. All exclusions concern the full collection of degree-d Picard-number-one K3 surfaces and therefore hold for every subcollection \mathcal S. This proves the stated uniformity.□

Proof. A smooth projective model of Q is a dominant rational image of the rationally connected fourfold, hence is rationally connected. A smooth projective rationally connected complex surface is rational: it is uniruled, its minimal model is ruled over a curve, and rational connectedness forces that curve to have genus zero; see [6, Theorem 30] and [12, Theorem V.1]. Consequently k\simeq\C(x,y).

A terminal fourfold is nonsingular in codimension two [43, Exercise 4.9.1.1], so its singular locus has dimension at most one and cannot dominate Q. The generic surface is regular, as a localization of its regular total space near the generic fibre, and is smooth because k is perfect. The contractions have connected fibres and rationally connected geometric generic fibres as above, giving geometric integrality. Projectivity is inherited by base change. A small birational map between smooth proper surfaces is an isomorphism: the exceptional locus of a non-isomorphic birational morphism of smooth surfaces contains a curve, and resolving a birational map gives the same conclusion if it is non-isomorphic in either direction. Such a curve would spread to a divisor exceptional for the original small map.

If \dim B=2, the contraction B\to Q is birational. The generic end V_k has ample anticanonical divisor and Picard number one over k, hence is a del Pezzo surface. If \dim B=3, the generic base B_k is a normal proper curve with geometrically integral rationally connected generic model, so it is a smooth genus-zero curve. The generic morphism V_k\to B_k has relatively ample anticanonical divisor and relative Picard number one, hence is a smooth-surface Mori conic bundle.

Here Picard numbers mean dimensions of numerical divisor spaces over k, not over its algebraic closure. To justify their restriction from the fourfold, a divisor on V_k closes up to a Weil divisor on V, which is \Q-Cartier. This gives surjectivity onto the generic numerical divisor space. A class numerically zero over the indicated original base remains numerically zero on the generic fibre: a generic curve and its intersection degrees spread over an open of Q. Thus the relative rank is at most the original rank one. The anticanonical class is nonzero, so it is exactly one. The same argument bounds the numerical rank of the generic middle surface by two. No passage to an algebraic closure is made in this rank calculation. In the conic case the base curve has numerical Picard number one, its pullback to the surface is nonzero, and the relative numerical divisor space has dimension one. Thus the total numerical divisor space has dimension two.□

Proof. By Lemma 5.5, \rho(W/k)\leq2. If a divisorial contraction W\to S survives on the generic fibre, then \rho(W/k)=\rho(S/k)+1. The end S therefore has rank one and is a del Pezzo surface. Extraction from a conic end, which has rank two, would force \rho(W/k)\geq3. Thus any surviving extraction forces \rho(W/k)=2. If the opposite end has no surviving extraction, its surface is W itself and cannot be a rank-one del Pezzo. It must instead carry its conic contraction. If both ends have surviving extractions, both are divisorial contractions from W to rank-one del Pezzo surfaces.

If neither side has a surviving extraction, both generic end surfaces are W. Rank one then gives two structures over a point, and the restricted map is a structured isomorphism. Rank two gives two conic contractions. These may agree or may be distinct; the latter case is retained as a two-conic move.

In all the rank-two cases, compare the two end contractions on W. If they contract the same extremal ray, uniqueness of the contraction identifies their normal targets. Indeed, each is constant on the connected fibres of the other, hence factors through it, and the two factorizations are inverse. For conic contractions this identifies their pullback curve fields inside k(W), not merely the abstract isomorphism classes of the curves. For identical birational contractions the common exceptional orbit cancels from the horizontal cocycle. Thus the restricted birational map is a structured isomorphism in either case, with equal vertical divisor data and H_Q=0.

Otherwise the two contractions give distinct K_W-negative extremal rays. These are the two boundary rays of the two-dimensional cone of curves of the projective surface W. The anticanonical divisor is positive on both and is therefore ample. This proves (ii). The argument uses numerical classes and contractions over k and does not require the generic surfaces or conic bases to split.□

Proof. The morphism from the smooth surface to the smooth curve is flat, so its fibres have the arithmetic genus of the generic conic. Over \overline k, every fibre is connected, has arithmetic genus zero, and has anticanonical degree two. The anticanonical degree of each irreducible component is a positive integer. An irreducible reduced fibre therefore has arithmetic genus zero and is a smooth rational curve. A double fibre 2R would have R^2=0 and K_S\cdot R=-1, contradicting adjunction parity. The only remaining possibility is a reduced pair R_1+R_2 with -K_S\cdot R_i=1. Put m=R_1\cdot R_2. Since (R_1+R_2)\cdot R_i=0 and the fibre is connected, adjunction gives

Thus m=1, both curves are smooth rational curves of self-intersection -1, and they meet transversely.

If the component cover over an orbit were split, the sum of one chosen component over each point of that orbit would descend to a divisor on S. It has zero intersection with a general fibre and negative intersection with every component in that chosen orbit. This is impossible when the relative numerical divisor space is one-dimensional: the general fibre is a nonzero relative curve class and spans its dual. Consequently every component orbit has field L quadratic over the corresponding discriminant field K.

The relative anticanonical system embeds the fibres as plane conics: on a smooth fibre it has degree two, and on a reducible fibre it has degree one on each component, with vanishing first cohomology. Cohomology and base change give the local rank-three system. At a singular fibre the total surface is regular, so after diagonalizing the quadratic form over the local discrete valuation ring, its nonsingular binary part has unit determinant and the remaining coefficient has valuation one. More explicitly, with uniformizer t the equation is

An exponent of t greater than one would make the total space singular at the geometric node. We use the Kummer identification and quaternion residue formula of [9, §2, equations (1)–(4)]. The associated quaternion can be written (-ab,-act); its tame residue is the square class of -ab in the residue field. This is exactly the extension separating the two lines ax^2+by^2=0. At a smooth fibre all three diagonal coefficients are units, so the residue vanishes.

The corresponding vertical divisor has function field L(t), since a line is split over its component field. The involution of L/K excludes u(L)=1 by the automorphism clause of Lemma 2.3; the degree of L over k need not be two. All smooth fibres cancel against their base points by the MRC rule. This proves (5.8). On a del Pezzo side the generic base is a point, so there are no vertical primes dominating Q.□

Proof. Use a plane blowup basis L,E_1,\ldots,E_n with n=9-K_S^2. A class xL-\sum y_iE_i belongs to A(S) exactly when

Cauchy’s inequality gives (3x-2)^2\leq nx^2. For n=4 this restricts the integer x to 1,2. At x=1 there is one coefficient y_i=1 and all others vanish; at x=2 all four coefficients are one. These are the four line pencils and the conic pencil. For n=3 the inequality forces x=1, giving the three line pencils. The same calculation for A_2(S) gives x=1,2 and the two classes

The three pencil classes L-E_i and these two plane classes span the degree-six Picard space. Their only linear relation is the equality of the two class sums, each equal to -K_S. The relation is invariant under G_k; semisimplicity yields (5.9). Taking invariants gives

The Brauer obstruction to descending an invariant line bundle is torsion [7, Remark 2.7]. Thus invariant rational Picard classes descend, and the invariant Picard space has dimension one. Both finite sets are nonempty, proving transitivity.□

Proof. By Lemma 5.6, the common smooth surface W is del Pezzo with a two-dimensional invariant numerical plane and the two end contractions generate its two rays. Each is either divisorial to a smooth surface or a conic contraction. For a divisorial contraction, pass first to \overline k. Its exceptional intersection matrix is negative definite. An exceptional component R has R^2<0 and K_W\cdot R<0; adjunction 2p_a(R)-2=R^2+K_W\cdot R forces both intersections to equal -1 and R to be a smooth rational curve. Two such curves must be disjoint, since otherwise their two-by-two intersection matrix would not be negative definite. Thus the contraction is the blowdown of disjoint (-1)-curves to distinct smooth points. Each Galois orbit of these curves contributes an independent invariant relative divisor class. Relative Picard number one therefore gives a single orbit. The contraction is consequently the blowup of one closed point over k, and the equivariant correspondence between the geometric points and their exceptional curves identifies their orbit fields. Put D=K_W^2>0.

Every fibre class used below is the integral class of a geometric fibre in \operatorname{Pic}(W_{\overline k}). It is G_k-invariant, has square zero and anticanonical degree two, and lies in the rational numerical plane over k by the torsion descent argument used in Lemma 5.9. A closed fibre over a point of degree r on the original genus-zero curve has class r times this geometric class. Thus none of the following calculations requires a k-point on the conic base.

For a divisorial end let E denote its orbit of exceptional curves, write e=|E|, and put E_\Sigma=\sum_{R\in E}R on W_{\overline k}. Thus u(E)=u(k(E)) counts one orbit field, whereas E_\Sigma^2=-e counts geometric curves in an intersection calculation. When both ends are divisorial, use F, f=|F|, and F_\Sigma for the corresponding data at the primed end. In the diagram

E is the exceptional set of \pi and F that of \pi'. The factorization \varphi_k=\pi'\circ\pi^{-1} therefore contributes u(E)-u(F) by Lemma 2.4: the extraction has positive sign and the contraction negative sign. Neither term is multiplied by its orbit size.

One divisorial end. Let \pi:W\to S be the contraction to the rank-one del Pezzo end, of degree a, and let W\to C be the other, conic contraction. Then D=a-e>0. Write H=\pi^*(-K_S). The plane spanned by H and E_\Sigma has intersection form H^2=a, H\cdot E_\Sigma=0, and E_\Sigma^2=-e. A nonzero isotropic fibre class in this rational plane therefore gives ax^2=ey^2 with nonzero rational x,y, so ae is a square. Since 1\leq e<a\leq9, the complete list is

Indeed, two positive integers with square product have the same squarefree part. Below ten the only unequal such pairs with smaller entry less than the larger are 1,4, 2,8, 1,9, and 4,9.

Over \overline k, contracting one line in every singular conic fibre gives a ruled surface over \mathbf P^1. Its canonical square is eight, so

The first three cases of (5.11) have extraction degree at most two and discriminant size different from three. Their two w values and their cocycle contributions vanish.

In the remaining case (a,e)=(9,4), work geometrically on the plane blown up at four points. Galois fixes L and permutes the four exceptional classes transitively. Thus the invariant fibre class has the form F=xL-y\sum_{i=1}^4E_i with x,y\in\Z. The equations F^2=0 and -K_W\cdot F=2 give x=2, y=1. The conic pencil is therefore the pencil through the four points. Its three reducible members correspond to the partitions of these points into two pairs. For each point p, select in every partition the pair containing p. This gives a star of three lines; selecting the other pair gives its complementary triangle. The four stars and four triangles exhaust the eight possible selections. The construction commutes with every permutation of the four points, so there is an isomorphism of G_k-sets

Consequently w(W\to C)=u(E), which is the extraction contribution. Reversing the link reverses both signs.

Two conic ends. The same surface has two fibre classes F_1,F_2. Put m=F_1\cdot F_2. They are distinct isotropic rays, so m>0, and their anticanonical degrees are two. In their numerical plane,

Since m is a positive integer, the equality D=8/m excludes degree five. Thus neither conic bundle has discriminant size three, by (5.12). Their w values vanish and there is no divisorial contribution.

Two divisorial ends: the numerical list. Now let e,f be the orbit sizes, so that the end degrees are D+e and D+f, each at most nine. If D=1 or 2, the Bertini or Geiser involution \iota fixes K_W and acts as -1 on K_W^\perp [22, Sections 6.6 and 6.7]. It is defined over k: the corresponding complete anticanonical or bi-anticanonical system is defined over k, and the nontrivial involution of its quadratic function-field extension extends to W. If \iota preserved an exceptional ray with orbit sum E_\Sigma, then \iota_*E_\Sigma=\lambda E_\Sigma for some \lambda>0; intersection with -K_W forces \lambda=1. The fixed-space formula would then give E_\Sigma\in\Q K_W, contrary to E_\Sigma^2<0<K_W^2. Thus \iota exchanges the two contractions and identifies their entire exceptional G_k-sets and output surfaces over k. Both sides of (5.10) vanish.

For the remaining degrees we first restrict the possible orbit sizes by intersection theory, then identify the finite G_k-sets. The target is

Small orbits contribute zero; the other terms will be matched through equivariant bijections of exceptional or pencil sets.

Suppose D\geq3 and put m=E_\Sigma\cdot F_\Sigma. The Gram determinant of -K_W,E_\Sigma,F_\Sigma, which lie in a plane, gives

Since the two effective exceptional sums meet nonnegatively, the solution is

Each curve in an orbit has the same total incidence with the opposite orbit. Thus m/e and m/f are integers. The anticanonical embedding realizes (-1) curves as lines; two different such curves meet at most once. In particular m\leq ef.

These conditions give precisely the following necessary possibilities, with e\leq f:

Here is a direct exhaustion. For e=f, formula (5.14) gives m/e=1+2e/D, so D\mid2e; impose also 1\leq e\leq9-D and m\leq e^2. This gives exactly the diagonal entries displayed. For e<f, the values i(i+D) for 1\leq i\leq9-D must have matching squarefree parts. The lists for D=3,4,5,6,7 are respectively

They give only (2,5),(1,5),(1,3),(1,2) in their respective rows. For D\geq8 the permitted range has at most one entry and supplies neither a diagonal nor an off-diagonal case.

Identifying the finite sets. In the cases (D,e,f)=(3,2,5) and (3,3,3), every curve in F meets every curve in E once. After contracting E, its image class C on S has

respectively. Therefore -K_S-C is in A(S). This construction is injective: equal image classes have equal pullbacks, and all the multiplicities at the extracted points equal one, so their strict transforms have the same (-1) class. A (-1) class has a unique effective representative. The cardinalities in Lemma 5.9 then show that

as actual G_k-sets. For (3,3,3) the construction on the other side also gives E\simeq A(S'). For (3,2,5) the other side has degree eight and u(E)=0. Thus the desired difference is u(E)-u(F) in both cases.

For (D,e,f)=(4,1,5) and (5,1,3) every opposing curve meets the single extracted curve once. Its image already has square zero and anticanonical degree two, so the image itself identifies F with A(S). The injectivity argument is unchanged. The opposite end has degree nine or eight, respectively, and the one-point orbit contributes zero. These cases again give (5.10).

In the diagonal cases (D,e)=(3,6),(4,4),(6,3), the value of m/e is e-1. Each curve therefore misses exactly one curve in the opposite orbit. This rule gives an equivariant bijection E\simeq F, since the same statement holds in both directions. The output degrees are nine or eight, so no pencil terms occur. In degree seven both orbit sizes are at most two and all terms vanish by (5.7).

It remains to treat (D,e,f)=(4,2,2), whose two outputs have degree six. Here we must identify the actual pencil G_k-sets, since an equality of rational permutation representations alone need not identify their orbit fields. The Picard blowup decompositions imply

Insert (5.9) on both sides. The permutation representations of E,F,A_2(S),A_2(S') have only one-dimensional rational constituents, because all four sets have two elements. By transitivity, the reduced representation of each triple A(S) or A(S') is irreducible over \Q: its image is either C_3, acting through \Q(\zeta_3), or S_3, acting through its standard representation. Semisimplicity consequently identifies these two reduced representations. Both are faithful on their image, so their kernels in G_k agree. The quotient is C_3 or S_3; its transitive action on three points is unique up to isomorphism (the stabilizer is trivial or an order-two subgroup, and all such subgroups of S_3 are conjugate). Hence

as G_k-sets. The two extraction orbits contribute zero by (5.7), completing the final case.

Finally, positivity of D in every nontrivial conic case and (5.12) give |\Delta|\leq7. In the structured-isomorphism case of Lemma 5.6, the horizontal cocycle is zero and the h_Q values agree; this case includes cancellation of identical generic end contractions. Thus H_Q=0 even when the conic discriminant is arbitrarily large.□

Proof. Resolve the diagram and marked support in bounded families, marking all resolution exceptional divisors as well. Write D for the resulting SNC support on the smooth total space. It is enough that D contain the ramification; its unramified components do no harm. Every prime on the resolution is either the strict transform of a prime on the old model or an exceptional prime. Marking all exceptional primes therefore includes any newly visible ramification, even over the singular locus of the old model.

We form residues and their quadratic covers before extending constants: the ground field F is \C, or \C(Q) on the ordinary generic fibre over Q. After finding their unramified open, we will base change these same covers to \overline{\C(Q)} to use the geometric bounds. The Kummer isomorphism identifies the Brauer 2-torsion of the total function field with its H^2 with coefficients in \mu_2 [9, §2, equations (1)–(4)]. On the smooth total model, the first two residue maps form a complex

Here Y^{(j)} denotes the codimension-j points. These are the coniveau differentials of [16, Example 2.1 and Proposition 3.9]; the same complex applies over the characteristic-zero function field of Q. Consider a ramification component D_i and a codimension-one point z of D_i outside its intersections with the other components of D. Every term except that from D_i is zero in the displayed secondary residue at z. Since D_i is smooth, that term is its ordinary DVR residue, so \partial_z(\partial_{D_i}\alpha)=0. If the quadratic residue is represented by a\in F(D_i)^*/F(D_i)^{*2}, this says that \operatorname{ord}_z(a) is even. After multiplication by a square, a is a unit at the DVR, and adjoining its square root is unramified there. Normalize the indicated open of D_i in the quadratic algebra, componentwise in the split case. This normalization is finite, is unramified at every codimension-one point, and has regular target. Purity of the branch locus makes it étale throughout that open [59, Lemma 58.21.4, Tag 0BMB].

The horizontal ramification components are generically finite over S. Delete their nonfinite and branch loci on S, the images of their pairwise intersections, the images of their intersections with the remaining marked support, and the singular and resolution bad loci of the diagram. These images are proper closed subsets: over the generic point of S the distinct horizontal components are distinct points of the smooth proper curve. The deleted sets form a bounded embedded boundary after stratifying the family. Over the resulting smooth open U\subset S, the horizontal union is finite étale, its components are smooth and disjoint from the other marked divisors, and its residue double cover is finite étale. Their degrees over U are r_k and 2r_k. The scheme of sections defining P_k is then a finite étale scheme of degree eight over U. Thus all the fields in the statement have the asserted degree and branch bounds.

These degree and branch bounds exclude the large-degree K3 fields. First consider a bounded diagram over \C. Fix an orbit field E/k in the statement, and let S_E\to S be the normalization in E. Take a general pencil in a uniformly bounded very ample system on the resolved base surface S. Let h bound the genus of its normalized general member and let b bound its intersections with the deleted boundary. Each normalized component of the pullback of that member to S_E has degree n\leq14 and, by Hurwitz [60, Lemmas 53.12.2 and 53.12.4],

If u(E)=1, a smooth projective model of S_E is birational to a test K3 surface. Resolve the pulled-back pencil on this model and take its Stein factorization. Its connected general fibre is one of these normalized components. This contradicts the bound 2g-2\geq\sqrt d in Lemma 4.1 for large d.

When the family is given only on the geometric generic fibre of Q, the same construction is performed over \overline{\C(Q)} on the base curve there. Finite étale covers remain finite étale after this extension, although they may split. The same degree and Hurwitz bounds therefore hold on every geometric component. We retain the base changes of the original ramification support and residue covers; we do not recompute the support from the extended Brauer class, whose residues may have become zero. For any orbit field E/k, the inclusions \C(Q)\subset k\subset E give a rational map from its surface model to Q. If u(E)=1, resolve that map on the corresponding K3 surface and take its Stein factorization. Then the geometric general fibres of that map are precisely the components whose genera were bounded. Lemma 4.1 again gives the contradiction. No assertion about the complexity of an unmarked Brauer class is used.□

Proof. Let k'\subset L be any other eligible field; its candidate need not be nonzero. Resolve the two presentations on a common start (6.5). By (6.6), its log MMP has an output with base field k and another with base field k'. Connect these outputs by the log Sarkisov links of Theorem 4.9, and write their successive nodes as Z_i/S_i for 0\leq i\leq N. The birational maps from the common start identify every \C(Z_i) with L; set K_i=\C(S_i)\subset L. Thus K_0=k and K_N=k'. The class \alpha\in\operatorname{Br}(L)[2] is fixed throughout this path, and the support of each boundary is its full visible ramification on the corresponding model.

Every Mori base in this three-dimensional programme has dimension at most two. If a common base of a link has dimension two, the endpoint bases also have dimension two and their contractions to the common base are birational: they are connected-fibre contractions between normal varieties of the same dimension. The induced identifications are the ones coming from L. Such a link therefore preserves the embedded surface field.

If k'\ne k, let m be the first index for which S_{m+1} is not a surface with K_{m+1}=k. Its common base Q has dimension at most one, by the preceding paragraph. At each preceding node the data (k\subset L,\alpha) are unchanged. Hence its proper generic curve, residue covers, and candidate are the original C_k, the original covers, and \sigma_k(L,\alpha). The relevant part of the path is

By Theorem 4.9, on the geometric generic fibre of Q its whole marked diagram is uniformly bounded, including the projection to the endpoint surface base. All ramification points on the proper generic curve are visible as divisors on this total model: their closures dominate the surface base and have codimension one. They are among the marked transforms by the full visible convention. If Q is a curve, its function field is contained in k because the endpoint base contracts to Q. Lemma 6.1 applies and gives \sigma_k(L,\alpha)=0, a contradiction. Thus k'=k.□

Proof. The relative generic diagram over Q is bounded by Theorem 4.9. On the smooth del Pezzo locus, let a\in\{5,6\} be the fibre degree. The vanishings H^1(\mathcal O)=H^2(\mathcal O)=0 make the relative Picard scheme étale [42, Theorem 4.8, Corollary 5.13 and Proposition 5.19]. We identify its finite part that parametrizes conic-pencil classes. For a line-bundle class F on a geometric fibre S, Riemann–Roch gives

Consequently the equations F^2=0 and -K_S\cdot F=2 are equivalent to having the fixed anticanonical Hilbert polynomial \tfrac a2m(m+1)+2m+2. This fixed-polynomial locus in the relative Picard scheme is open and closed and of finite type; in the smooth projective family it is also projective [42, Theorem 6.20, Exercise 5.7 and its answer]. It is therefore proper and quasi-finite, hence finite étale. Lemma 5.9 identifies its geometric fibres with A(S), of cardinality five or three according as a=5 or 6. These are statements about the Picard scheme and require no universal line bundle on the original family. The complement of this smooth family is bounded as an embedded locus of the diagram. Each field factor of the finite scheme is therefore a cover of bounded degree unramified outside this bounded locus. Apply the bounded-cover exclusion of Theorem 4.3, or the Hurwitz calculation (6.3). When Q is a curve its field is contained in each tested surface field. Thus no factor is counted by u and w=0.□

Proof. Put L=\C(T) and suppose an eligible k\subset L has nonzero candidate. The bounded diagram on T does not yet include the presentation over k; in particular, k need not contain \C(Q). We will construct an output with bounded marked geometric generic diagram over Q and compare it with the output over k. Choose a common smooth SNC start Y mapping to T and to a projective surface model of k, with the full visible boundary (6.5). One log MMP output has base field k by (6.6).

A bounded output over Q. First run the log MMP over T to a relative minimal model b:Y^*\to T. This is the birational case: relative bigness holds, so [15, Theorem 1.2] applies. Let D_* be the transform of D_Y on Y^*; its pushforward under b is D_T.

We first study the geometric generic fibre over Q, retaining the same symbols for these restrictions. The marked supports are the base changes of the original supports, regardless of whether the extended Brauer class still ramifies on them. We have

The divisor E is b-exceptional and b-nef; the latter follows from relative nefness of the left side. The negativity lemma gives E\leq0. For an exceptional prime F of b, its coefficient is

Consequently every valuation extracted by b on this geometric generic fibre satisfies a(F,T,cD_T)\leq1; an extracted valuation occurring in D_* even satisfies a(F,T,cD_T)\leq1-c.

Theorem 4.11 now bounds the geometric generic model and all its subsequent non-extracting contraction diagrams, including the marked support. Indeed the only extracted divisors are among the bounded terminalization data for the bounded klt log Fano pairs (T,cD_T). On the geometric generic fibre over Q, every component of D_* is either the strict transform of a component of D_T, or one of these extracted prime divisors. The markings here are retained under extension of constants. The terminalization theorem marks the entire extraction list, including primes with crepant coefficient zero, so it bounds this full support and all of its subsequent transforms on that geometric generic fibre. Primes of the resolution exceptional over T there and absent from this list cannot survive on Y^* there by (6.8). This bounds the geometric generic model Y^* over Q and its subsequent non-extracting diagrams, not the arbitrary resolution Y. To obtain a Mori output over Q, we verify that the log canonical divisor on this geometric generic fibre is not pseudo-effective. To see this from (6.7), choose a general complete-intersection curve in T avoiding the centres of the exceptional divisors of b. Its strict transform has negative intersection with K_{Y^*}+cD_*, because K_T+cD_T is anti-ample. Such curves form a movable covering family, so a pseudo-effective divisor could not have this negative intersection.

Return to the models over \C. We may therefore run the log MMP from Y^* over Q to a Mori fibre space Z/B. Its geometric generic diagram over Q is bounded by the terminalization theorem, since these additional steps do not extract divisors.

Comparison with the output over k. Connect the output with field k to Z/B by log links from the same initial pair. These links are taken in the absolute common-start geography; the arbitrary eligible field k need not contain \C(Q). If B is not a surface with embedded field k, consider the first departure from k along this path, as in Proposition 6.2. Call the common base of that link Q'. It has dimension at most one, its incoming endpoint has the original candidate \sigma_k(L,\alpha), and, when Q' is a curve, its morphism to Q' gives \C(Q')\subset k. The bounded diagram over Q' therefore forces that candidate to vanish by Lemma 6.1.

If instead B is a surface with \C(B)=k, the bounded geometric generic diagram of Z/B over the original Q gives the same contradiction directly. Here B\to Q gives \C(Q)\subset k when Q is a curve, so the other application of Lemma 6.1 has its required field inclusion as well. Both alternatives contradict the nonzero candidate.□

Proof. If there is a nonzero candidate, Proposition 6.2 says that its eligible field is the only eligible field. If there is no nonzero candidate, all eligible candidates are zero and agree with the definition. In particular a zero candidate cannot be ignored in favour of a nonzero candidate from another field. The last assertion follows from Proposition 6.4, Lemma 6.3, and the definition when \dim B\leq1.□

Proof. Choose an ordinary Sarkisov factorization supplied by Theorem 4.9, and orient each link from V/B to V'/B'. Write \Delta for final minus initial value of a function on its nodes. Proposition 5.3 gives

We check H_Q=\Delta s in every dimension of the common base Q.

If \dim Q\leq1, Proposition 5.4 gives H_Q=0, while Theorem 6.5 gives s=0 at both ends.

If \dim Q=3, both endpoint bases have dimension three and are birational to Q. On the generic fibre the link is an isomorphism of smooth projective conics. Thus the endpoint base fields and quaternion classes are identified, including all eligible embedded surface fields and their candidates. Therefore \Delta s=0, which equals H_Q by Proposition 5.4.

Suppose \dim Q=2 and put k=\C(Q). In case (ii) of Lemma 5.6, the calculation of Proposition 5.10 gives c_{u,\mathrm{hor}/Q}=\Delta w. Here w is evaluated on the structured generic surface over k, including its conic structure when \dim B=3, as in Definition 5.8. On a del Pezzo side, h_Q=0 and s=w. On a conic side, its generic curve base over k is geometrically integral of genus zero. Lemma 5.7 identifies its singular-fibre set and component cover with \Delta_k and the residue cover. The common del Pezzo surface W has positive degree, so (5.12) gives r_k=\deg_k\Delta_k=8-K_W^2\leq7. Hence k is eligible. The same lemma gives h_Q=-u(\Delta_k), and by (6.2) and Theorem 6.5,

This remains valid if the candidate is zero. Consequently H_Q=\Delta(w-h_Q)=\Delta s.

In case (i) of Lemma 5.6, the structured generic surfaces are isomorphic after cancelling any identical generic contractions. Their exceptional horizontal contribution and H_Q are zero by that lemma. At a del Pezzo node the Picard-class sets defining w identify. At a conic node the isomorphism identifies the embedded base field and its quaternion class, so the intrinsic corrections identify. Thus \Delta s=0 here as well. This last argument requires no bound on the discriminant degree of an isomorphic conic structure.

It follows from (6.10) that every link satisfies

Add over the factorization and use the cocycle property of Lemma 2.4. At a node over a point there are no vertical divisors in the defining difference for v, and s=0 by definition. The sum therefore vanishes. All lower bounds on d arose from the finitely many fixed families specified after (6.4); none depends on the length or starting resolution of the factorization.□

Proof. Choose d_0>2 at least as large as the uniform bound in Theorem 6.6. Its uniformity allows the collection of K3 surfaces to be restricted to any specified integral transcendental Hodge-isometry class.

Suppose that f\colon\mathbb P^4\dashrightarrow X is birational. Take as test collection the Picard-number-one K3 surfaces of degree d whose transcendental Hodge lattice, after the shift of Hodge types and reversal of the cup pairing, is isometric to T_X. These surfaces have no nonidentity automorphisms by Lemma 3.9. The assumed rationality makes X rationally connected, and the cubic Hodge calculation gives h^{3,1}(X)=1 [33, Section 2.1]. Thus all hypotheses of Proposition 3.10 hold, and it gives

Both varieties are Mori fibre spaces over a point. Indeed, they are smooth, and integral weak Lefschetz and the exponential sequence give \operatorname{Pic}(X)=\Z\mathcal O_X(1), while adjunction gives -K_X=3H [38, Chapter 1, Section 1.1, Lemma 1.6 and Corollary 1.9]. The ordinary Sarkisov program therefore factors f into the links used in Theorem 6.6. Applying that theorem to this factorization yields c_u(f)=0, contradicting (7.1).□

Proof. We spell out the exceptional sets, including their algebraicity. Pass to Hassett’s marked space and then add finite level structure. Fix a smooth irreducible component that is finite and surjective over a dense open U\subset\mathcal C_d, and denote it by U'. The rank-two marking makes K constant. Its orthogonal \mathbb T=K^\perp is an integral polarized variation on U'; we trivialize this complementary local system only on period charts. All exceptional loci below are constructed on this one fixed cover.

The cubic Hodge numbers and middle lattice are

see [33, Section 2.1 and Proposition 2.1.2]. Thus \mathbb T has rank 21 and signature (19,2) for the cubic cup pairing. In a marked period chart its period line varies in an open subset of

Here T denotes the underlying fixed lattice in that chart. The openness follows equivalently from the associated K3 period description (7.2). The domain \mathcal D_T is analytically open in a smooth irreducible projective quadric, and this quadric spans \mathbb P(T\otimes\C).

For 0\ne t\in T\otimes\Q, consider the locus in U' where some flat translate of t is a Hodge class. After clearing denominators and Tate twisting, [20, Theorem 1.1 and Corollary 1.3] makes this locus closed algebraic, with finitely many local branches. Here one fixed denominator works for the whole monodromy orbit, and the flat polarization has the same value on all its translates. The cited corollary obtains this orbit locus from images of connected components of the bounded-norm space of Hodge classes, which the theorem makes finite over the base. In a period chart each branch lies in (t')^\perp\cap\mathcal D_T for a nonzero flat translate t'. Each such section is proper because the period domain spans T\otimes\C; hence the global locus is proper as well. There are countably many rational t. Since H^4(X,\Q)=K_\Q\oplus K_\Q^\perp, excluding these loci removes every Hodge class outside K_\Q. Thus A_X\otimes\Q=K\otimes\Q; saturation gives A_X=K and consequently \operatorname{rank}T_X=21.

The middle lattice is unimodular, and both A_X and T_X are primitive. To recall the determinant argument, write L=H^4(X,\Z). Every homomorphism A_X\to\Z extends to L, since L/A_X is torsion-free. Unimodularity therefore makes the restriction map L\to A_X^* surjective with kernel T_X. Reducing modulo A_X gives

Interchanging A_X and T_X gives the same quotient as T_X^*/T_X. The two discriminant groups have the same order; hence |\det T_X|=\det A_X=d.

Next fix a nonscalar a\in\operatorname{End}_{\Q}(T\otimes\Q). If a is a Hodge endomorphism, then \C\omega is an eigenline for a. Hence its Hodge locus is contained in

This is a finite union of proper linear sections of the period quadric. Indeed, an eigenspace equal to T\otimes\C would make a scalar; and an irreducible quadric spanning the ambient vector space cannot be contained in a finite union of proper linear subspaces. There are countably many rational endomorphisms a. Apply the same argument to the weight-zero polarized variation \operatorname{End}(\mathbb T), clearing denominators. The locus where some flat translate of a is Hodge is closed algebraic with finitely many local branches. Monodromy acts by conjugation and preserves nonscalarity, so the eigenline argument makes every local branch proper. These global loci are therefore proper. Excluding them gives \operatorname{End}_{\mathrm{Hdg}}(T_X\otimes\Q)=\Q.

For completeness, simplicity also follows directly from the Hodge numbers. By polarizability, a rational Hodge substructure has a Hodge complement. In any nontrivial direct-sum decomposition of T_X\otimes\Q, only one summand could contain the one-dimensional (3,1) part, and that summand would also contain its conjugate. Every other summand would consist entirely of rational (2,2) classes, contrary to the definition of T_X.

Finally take the finite images in U of the exceptional loci in U'. These images are closed and proper, since U' is irreducible and finite over U. Their closures in \mathcal C_d are still proper. Adding \mathcal C_d\setminus U gives a countable union of proper closed algebraic subsets, whose complement is nonempty over \C. The categorical assertion holds throughout \mathcal C_d by (7.3); if one uses only the Addington–Thomas theorem, one additionally removes the complement of its dense open subset. This proves the simultaneous assertion.□

Proof of Theorem 1.1. Let d_0 be as in Proposition 7.1, and fix any admissible d>d_0. The divisor \mathcal C_d is nonempty. By Lemma 7.2, outside a countable union of proper closed algebraic subsets of this divisor, the lattice T_X has rank 21, absolute determinant d, and rational Hodge endomorphism ring \Q. The same lemma supplies an exact \C-linear equivalence

for a smooth projective K3 surface S. Proposition 7.1 applies to each such X and proves that it is irrational.

The integer d_0 was fixed before choosing d, so the conclusion holds very generally in every admissible \mathcal C_d with d>d_0. Its ineffectiveness is inherited from the uniform bound in Theorem 6.6. Such a cubic satisfies the categorical condition of Kuznetsov’s conjecture and fails its rationality condition; hence the categorical-to-rational implication is false.□

Proof of Corollary 1.2. For admissible d, the classical input (7.2) supplies the indicated polarized K3 surface for every labelled cubic in \mathcal C_d. Theorem 1.1 supplies irrationality on the stated very-general set for d>d_0. Since K contains h^2, the isometry gives

The inclusion is primitive: if a nonzero integer multiple of an integral class in (h^2)^\perp is orthogonal to K, then the class itself is orthogonal to K. This is the primitive polarized K3 Hodge embedding in the cited formulation, with the stated Tate twist and pairing reversal.□

Proof of Corollary 1.3. For n\ge2, the integer d=2n^2+2n+2 is greater than six and is congruent to zero or two modulo six. Taking a=1 realizes Addington’s condition (***), namely d=(2n^2+2n+2)/a^2 for integers n,a with a\ne0. That condition implies his condition (**), the remaining divisibility restrictions in (1.1); hence d is admissible. Addington’s Theorem 2 then supplies the stated birationality for every X\in\mathcal C_d, with a projective K3 surface by his convention [4, condition (***), Theorem 2, and the convention in the introduction]. Theorem 1.1 supplies irrationality for a very general such X when d>d_0. Finally, the values 2n^2+2n+2 are strictly increasing and unbounded for n\ge2, so infinitely many exceed d_0.□

Proof of Corollary 1.4. First let X\in\mathcal C_d be any cubic with a labelling K of admissible discriminant d. The intersection form satisfies (h^2,h^2)=\int_Xh^4=3, so h^2 is primitive in the integral middle lattice: if h^2=m\eta for an integral class \eta, then m^2 divides 3. It is therefore primitive in K and extends to a basis (h^2,\sigma) of this rank-two lattice.

Voisin’s Theorem 5.6 is stated for the lattice P generated by the independent integral Hodge classes h^2 and \sigma, which are algebraic on a smooth cubic fourfold as recalled in its proof [64, Theorem 5.6 and its proof]. Here P=K, and its discriminant in that theorem is exactly

Admissibility includes 4\nmid d. Voisin’s theorem therefore gives universal triviality of \operatorname{CH}_0(X) for every such labelled cubic, without requiring K to be the entire lattice of Hodge classes or X to be very general. The field-extension scope and the equivalence with the displayed integral Chow-theoretic decomposition are given in the introduction and equation (1) of the same paper [64]. Theorem 1.1 supplies irrationality for the stated very general cubics when d>d_0.□