Minimal models in numerical dimension one

Proof. Apply the log abundance theorem [Ope26, Theorem 1.1] to the projective lc pair (Y,0) over \mathbb C, whose \mathbb Q-Cartier canonical divisor is nef.□

Proof. Take a common smooth projective resolution carrying F:

Write g=h\circ p=h^+\circ q:W\to R and set D=p^*K_U-q^*K_{U^+}. This divisor is q-exceptional, since the step extracts no divisors. On every curve contracted by g, the divisors -p^*K_U and q^*K_{U^+} have nonnegative degree; in the divisorial case the second has degree zero. Thus -D is g-nef, and hence q-nef. The negativity lemma gives D\geq 0; see [KM98, Lemmas 3.38–3.39].

To verify the support assertion, let r\in R be a closed point with a positive-dimensional fiber on either side. There is a curve in g^{-1}(r) mapping onto a curve in that fiber: take a component dominating that curve and cut it by sufficiently general ample divisors. The relative ampleness signs give D\cdot C<0 for this lifted curve. Consequently g^{-1}(r) meets \operatorname{Supp}D.

We use the fiber-support form of the negativity lemma [KM98, Lemma 3.39]: the support of an effective divisor antinef over a proper birational morphism to a normal variety either contains or is disjoint from each fiber. In the present setting it can also be seen directly on the reduced irreducible components of the fiber. If such a component T is not contained in \operatorname{Supp}D but meets it, a Cartier multiple of D restricts to a nonzero effective Cartier divisor on the integral projective variety T. Its degree against sufficiently many ample hyperplanes is positive, producing a curve contracted by g with positive D-degree. This contradicts antinefness. This argument does not require T to be normal or the fiber to be reduced: only its support is at issue. Components of dimension zero already lie in the support if they meet it. The fibers of g are connected because R is normal and g is proper and birational [Har77, Corollary III.11.4]. Containment therefore propagates through their reduced irreducible components.

It follows that every fiber in question is contained in \operatorname{Supp}D. If the center of F on R is contained in this locus, then F\subset\operatorname{Supp}D on our chosen resolution. Finally,

which proves both assertions.□

Proof. The usual MMP properties give preservation of \mathbb Q-factoriality, invariance of the Picard number under a flip, and a drop of one under an elementary divisorial contraction [KM98, Propositions 3.36–3.37]. There are therefore only finitely many divisorial contractions. Since the steps extract no divisors, only finitely many prime divisors of X can disappear; denote this finite set by \mathcal D.

All models are terminal. This follows from [KM98, Corollaries 3.42–3.43], or directly from Lemma 2.2: previously exceptional valuations retain log discrepancy greater than one, and a newly contracted divisor increases strictly from log discrepancy one. Moreover, every valuation exceptional over the smooth X has integral log discrepancy at least two. Discrepancies never decrease along the sequence.

The flipped loci. Consider a flip whose target has a codimension-two component T of its flipped locus. A terminal variety is smooth in codimension two [KM98, Corollary 5.18]. Blowing up T at its generic smooth point therefore defines a prime valuation F with

Its center on the contraction base is contained in the locus of positive-dimensional fibers of the flipped contraction. Lemma 2.2 implies

Thus F is not exceptional over X: it is an original prime divisor. It is exceptional over U_{i+1}, so it belongs to \mathcal D. After this occurrence its discrepancy is at least two forever, and it cannot satisfy (2.2) at a later step. Each flip with such a target component therefore uses a distinct member of a finite set. After a finite prefix, all steps are flips and every flipped locus has codimension at least three.

The flipping loci. We have removed codimension-two components on the target side. It remains to remove them on the source side; only then can a general surface be transported unchanged through every later finite stage. The argument adapts the homological decrease used in the proof of [AHK07, Lemma 3.1], keeping only classes of algebraic cycles. Put d=n-2 and define the finite-dimensional real vector space

Cycle classes, Borel–Moore localization, and their compatibility with restriction to an open subset are as in [Ful98, Section 19.1]. The homology of the compact complex algebraic varieties here is finite-dimensional and vanishes above their real dimension.

For a remaining flip U\dashrightarrow U^+ over R, remove the images in R of both exceptional loci and take inverse images. This gives a common open subset \mathcal U. Indeed, a proper birational morphism to a normal variety is an isomorphism over its quasi-finite locus. Each exceptional locus consists of positive-dimensional fibers; its image has dimension at most one less than its own. Since the source contraction is small and the target exceptional locus has dimension at most n-3, the closed complements Z\subset U and Z^+\subset U^+ satisfy

The additional points removed outside the exceptional loci lie in an isomorphism locus and have dimension at most n-3.

Restriction gives maps

with the same image: intersect an integral d-fold with \mathcal U and take its closure on the other model, while a d-fold contained in the complement restricts to zero. The right-hand map is injective, by the localization sequence and H_{2d}(Z^+,\mathbb R)=0. Hence

If the flipping locus has a d-dimensional component T, its class belongs to the kernel of the left-hand restriction and is nonzero: for an ample Cartier divisor A,

Then (2.3) is strict. A nonnegative integer can drop only finitely many times, so eventually the flipping loci also have codimension at least three. The same construction of the common open then gives this codimension bound on both complements.□

Proof of Proposition 2.1. Choose an effective ample rational divisor B for which (X,B) is klt and K_X+B is ample. For example, take a sufficiently positive general smooth very ample divisor and multiply it by a rational number strictly between zero and one. Since K_X is pseudo-effective, K_X+tB is big for every rational t>0.

We shall repeatedly use a simple consequence of normality. If a birational map X\dashrightarrow U extracts no divisors, every prime divisor of U corresponds to a prime divisor of X. For compatible transformed divisors, the divisorial criterion for regularity of a rational section consequently gives

whenever the multiples are Cartier. Thus effectiveness and bigness of B_U, and bigness of K_U+tB_U, persist on every model constructed below. In particular, we do not need B_U to be ample.

Finite stages. Set t_j=2^{-j} for every integer j\geq0. Starting with V_0=X, construct V_{j+1} from V_j by running the MMP for

with scaling of C_j=(t_j-t_{j+1})B_{V_j}. The precise input is [BCHM10, Corollary 1.4.2]: for a projective \mathbb Q-factorial klt pair (V,\Delta) with big boundary \Delta, and C\geq0 such that (V,\Delta+C) is klt and K_V+\Delta+C is nef, the MMP with scaling of C terminates. Here the inductive hypothesis is that (V_j,t_jB_{V_j}) is klt and its adjoint is nef. The smaller pair is klt because B_{V_j} is effective; its boundary is big by (2.4); and the larger pair is exactly the one in the inductive hypothesis.

The adjoint remains big by (2.4), so a Mori fiber space cannot occur: an effective representative of a positive multiple cannot have negative degree on curves covering general fibers. Each stage is consequently a finite, possibly empty, sequence ending at a nef adjoint. Standard MMP preservation gives the next klt pair and a projective \mathbb Q-factorial model. The rational divisor K_{V_j}+t_jB_{V_j} is nef and big, hence semiample by the klt base point free theorem [KM98, Theorem 3.3].

Canonical steps. To apply Lemma 2.3, we must identify every adjoint step as a canonical MMP step. This requires canonical negativity on the contracted ray and, for a flip, canonical ampleness on the flipped side. On its source U, let D=K_U+t_{j+1}B_U. The contracted ray R_0 satisfies

where C=(t_j-t_{j+1})B_U is the current scaling divisor. Therefore B_U\cdot R_0>0 and K_U\cdot R_0<0. This is the usual ray calculation for a directed program [BCHM10, proof of Corollary 1.3.3], and it already identifies each divisorial step as a K-negative divisorial contraction.

For a small contraction h:U\to R, the adjoint flip makes D^+ relatively ample. We show that K_{U^+} is relatively ample as well. Choose the positive rational number

The divisor K_U-bD is numerically trivial over R and descends up to rational linear equivalence. To see this, a Cartier multiple is relatively semiample by the relative klt base point free theorem for the pair (U,t_{j+1}B_U) [KM98, Theorem 3.24]: subtracting its adjoint D gives relative ampleness, because -D is relatively ample. Its semiample morphism is constant on each connected projective fiber, as its degree on every fiber curve is zero. The resulting image is proper and quasi-finite over R, because each connected fiber maps to one point. It is therefore finite and birational over the normal R, hence equals R. Thus K_U-bD\sim_{\mathbb Q}h^*A for a rational Cartier divisor A on R.

Both morphisms in the adjoint flip are small, so this relation transforms to

Since D^+ is relatively ample and b>0, so is K_{U^+}. The adjoint flip is therefore a K-flip. Thus all steps of the finite stages satisfy the hypotheses of Lemma 2.3.

The fixed model. Concatenate the finite stages and apply Lemma 2.3. Choose an endpoint Y=V_s after the resulting finite prefix. If only finitely many nonempty stages occur, choose s after the last one. For every j>s, the finite composition Y\dashrightarrow V_j is an isomorphism outside codimension-at-least-three closed subsets on both models. This property is preserved under finite composition: on a common open, remove the next bad set and take its closure on the preceding model; its dimension does not increase. Let Z_j be the resulting closed subset of Y.

The divisor M_j=K_Y+t_jB_Y agrees on this common open with K_{V_j}+t_jB_{V_j}. Choose a multiple Cartier on both varieties and base point free on V_j. A line bundle on a normal variety has the same global sections after deleting a subset of codimension at least two, since it is reflexive. The two spaces of sections are therefore identified, and the complete system on Y is generated outside Z_j. This proves (ii) for each j.

Finally, Y is projective, \mathbb Q-factorial and terminal by the preceding construction, and a finite composition of these steps extracts no divisors. On a common smooth projective resolution of the finite chain, the effective differences in Lemma 2.2 telescope to (2.1). A divisor mapping onto a prime divisor of Y also corresponds to a prime divisor of X, so its coefficient in E is zero. Thus E is q-exceptional. For an original prime divisor contracted on Y, terminality gives log discrepancy greater than one on Y, compared with one on X. Its strict transform has positive coefficient in E, proving (iii).□

Proof. Put r=n-2. Very general members of |H| have a smooth complete intersection S avoiding \operatorname{Sing}Y and every Z_j. Indeed, each of these countably many closed sets has dimension at most n-3, so the condition that r hyperplanes meet any one of them is a proper closed incidence condition. Over \mathbb C these can be avoided simultaneously, together with the closed conditions excluded by Bertini’s theorem. The inverse image of S is therefore exactly the smooth complete intersection cut out by the r pulled-back equations; there are no additional components over the singular locus. Each M_j|_S is semiample, hence nef. Passing to the limit on every curve on S shows that L|_S is nef and proves (3.2).

Write A=\pi^*H, choose an ample Cartier divisor A_0 on \widehat Y, and make the fixed choice

Since A is nef and ampleness is open [Laz04a, Corollary 1.4.10 and Theorem 1.4.23], there exists \eta>0 such that

For a given positive m with d\mid m, first choose j=j(m) so that mt_j<\eta, and then choose a multiple k of d_j with k>m. The pullback of |kM_j| is generated near S. A general member D_m is smooth there, by Bertini, and

Here \mathcal J(R_m) is the multiplier ideal; its asserted local triviality follows from m/k<1 and the smoothness of D_m there.

The divisors m\pi^*L+P-iA are integral Cartier divisors, and

is ample for 0\leq i\leq r. Nadel vanishing [Laz04b, Theorem 9.4.8], equivalently [Laz10, Theorem 2.4], gives

Set \mathcal F= \mathcal O_{\widehat Y}(m\pi^*L+P)\otimes\mathcal J(R_m). Tensoring the Koszul complex of the r equations with \mathcal F yields the exact complex

At points of S, the equations form a regular sequence and \mathcal F is locally free. Outside S, one equation is a unit, so the Koszul complex is contractible even after tensoring with an arbitrary sheaf. This proves exactness everywhere. Splitting the complex into short exact sequences and using (3.7) gives surjectivity on global sections onto its last term. The inclusion \mathcal F\subset\mathcal O_{\widehat Y}(m\pi^*L+P) then proves (3.3).

Adjunction and (3.5) give P|_S-K_S=A_0|_S. Hence mL|_S+P|_S-K_S is ample. Kodaira vanishing, the zero-boundary ample case of [Laz10, Theorem 2.4], and surface Riemann–Roch [Har77, Chapter V, Section 1] give

If S is disconnected, the formula is read componentwise and summed. All its coefficients are fixed before m,j,k,D_m vary. Together with (3.3), this proves (3.4).□

Proof. Choose a line bundle G of positive degree on the fixed smooth curve N such that \iota^*\Omega^1_V embeds as a subbundle of G^{\oplus a} for some a. Such a choice follows by global generation of (\iota^*\Omega^1_V)^*\otimes G for a sufficiently ample G, followed by dualization. Set g=\deg G>0.

For a nonzero s, the integer

is finite and attained. The leading ambient jet gives a nonzero section of

Here is a justification valid also at singular points of \Gamma. For b\geq1, the bundles of principal parts on the smooth variety V have the locally split exact sequence

The lower jet of s has zero value at every closed point of the reduced curve \Gamma. A section of a vector bundle on a reduced variety with this property is zero, so that lower jet restricts to the zero section. The principal-parts sequence is locally split as a sequence of \mathcal O_V-modules, and therefore remains exact after restriction to \Gamma and pullback to N. The restricted order-b jet consequently belongs to the left-hand bundle. Its value is nonzero at a point where the minimum b is attained; the induced map on its fiber at any point of N above that point is an isomorphism. Its pullback is thus regular and nonzero. For b=0, use s|_\Gamma directly. This proves (3.9) without assuming smoothness of \Gamma.

Taking symmetric powers of the subbundle inclusion above embeds (3.9) into a direct sum of copies of G^{\otimes b}\otimes\iota^*\mathcal D. A nonzero component is a nonzero section of this line bundle on N, so

Since \operatorname{ord}_z(s)\geq b for every z\in\Gamma, (3.8) follows.□

Proof. Every curve in \operatorname{Supp}F_j maps to a point of T, by the generation assumption. Fix a very ample Cartier divisor H_T on T and an ample Cartier divisor J on T_1, and put Q=f^*H_T. The projection formula gives

The Hodge index theorem on the fixed smooth surface T_1 [Har77, Chapter V, Section 1] makes the negative intersection form positive definite on Q^\perp\subset N^1(T_1)_{\mathbb R}. Consequently there is one constant K>0 such that

Effectiveness of F_j gives J\cdot F_j\geq ck_j(J\cdot e). Thus, with the fixed positive number \epsilon=c^2(J\cdot e)^2/K,

After removal of F_j, the residual linear system has no fixed curve. Its square is nonnegative: choose two members with no common curve and take their effective intersection, or use a nowhere vanishing member if the system is trivial. By (3.11) and the projection formula,

Now (3.12) proves (3.10).□

Proof. Suppose that an integral curve C_0\subset Y satisfies L\cdot C_0<0. Fix a projective resolution \pi:\widehat Y\to Y which is an isomorphism over the smooth locus. Choose a closed point x\in C_0 outside the images of those irreducible components of \pi^{-1}(C_0) which do not dominate C_0. There are only finitely many such images, and each is a point. Consequently every point of \pi^{-1}(x) lies on a component of \pi^{-1}(C_0) dominating C_0.

A surface through x. Choose n-2 very general members H_1,\ldots,H_{n-2}\in|H|, all passing through x. They meet properly, and their intersection meets each Z_j and \operatorname{Sing}Y in at most finitely many points. Indeed, the hyperplanes through x have no other base point; successive general cuts avoid all positive-dimensional components of the relevant intersections. The countably many conditions can again be imposed simultaneously over \mathbb C. Write their intersection cycle as

where each T^{(i)} is an integral surface. The dimension theorem guarantees a component through x; fix one and denote it by T.

On every T^{(i)}, the system induced by |d_jM_j| has at most a finite base locus. Its square is nonnegative, including when T^{(i)} is nonnormal. To check this directly, choose a section nonzero at the generic point, and then choose a second section not vanishing identically on any curve of the first zero divisor. The resulting Cartier intersections with [T^{(i)}] form an effective zero-cycle. If the first section has no zero, the restricted line bundle is trivial and its square is zero. These are the usual Cartier intersection and projection formulas on cycles [Ful98, Chapter 2]. It follows that

by (3.13). In particular,

Fixed geometry above x. The surface T meets \operatorname{Sing}Y only in finitely many points. Its strict transform T_{\widehat Y} is thus birational to T and maps onto it. Choose any z\in T_{\widehat Y} above x. There is an integral curve \Gamma\subset\widehat Y through z dominating C_0. To see this, take an irreducible component A\subset\pi^{-1}(C_0) through z. By the choice of x, it dominates C_0. If \dim A=d, its fiber A_x is a proper closed subset of dimension at most d-1. General very ample cuts through z, in number d-1, cut A_x to dimension at most zero and leave a curve component through z on A. That component cannot lie in A_x, so it dominates C_0. For d=1, take \Gamma=A.

Let \iota:N\to\widehat Y denote the normalization map of \Gamma, followed by inclusion. Resolve T_{\widehat Y} by a smooth projective surface T_0\to T_{\widehat Y}, choose a point w\in T_0 above z, and blow it up. Write T_1 for the resulting surface, e\subset T_1 for the exceptional curve of this blowup, and f:T_1\to T for the composite map:

The images of T_0 and N in \widehat Y meet at z. The exceptional curve e, all these varieties, and all these maps are fixed throughout the remaining argument.

The contradiction on the fixed surface. The projection formula and M_j\to L give, for one fixed \delta>0,

In fact this degree equals the positive degree of N\to C_0 times M_j\cdot C_0. For each such j, choose any positive multiple k_j of d_j. Lemma 3.2, applied on the fixed \widehat Y and \Gamma, shows that every nonzero pullback section s'\in H^0(\widehat Y,k_j\pi^*M_j) of a section on Y satisfies

The constant c depends only on the fixed normalized curve and the negative-degree margin in (3.15).

The sections on Y induce a nonzero linear system of k_jf^*(M_j|_T), generated outside f^{-1}(T\cap Z_j). Nonzeroness follows from generation at points of T\setminus Z_j.

Every nonzero section in this induced system has order at least ck_j at w before the last blowup. Indeed, the local ring map \mathcal O_{\widehat Y,z}\to\mathcal O_{T_0,w} sends \mathfrak m_z^b into \mathfrak m_w^b; apply (3.16) to a section inducing it. Sections whose restriction is identically zero do not contribute to the induced system. Thus its fixed divisor F_j satisfies \operatorname{coeff}_eF_j\geq ck_j. All the hypotheses of Lemma 3.3 hold with D_j=M_j|_T. Lemma 3.3 gives a fixed \epsilon>0 such that

contrary to (3.14). Therefore no curve C_0 of negative L-degree exists, and L is nef.□

Proof of Theorem 1.1. Apply Proposition 2.1 to obtain a finite canonical MMP

and the divisors L=K_Y, B_Y, and M_j=L+t_jB_Y, with generated multiples outside closed subsets Z_j of codimension at least three. The variety Y is terminal and therefore smooth in codimension two. These are the standing hypotheses of the numerical results.

Fix a very ample Cartier divisor H on Y and a projective resolution \pi:\widehat Y\to Y that is an isomorphism over the smooth locus. Lemma 3.1 gives a fixed Cartier divisor P on \widehat Y. Choosing one positive integer r_0 for which r_0L is Cartier, we obtain

It also gives L^2\cdot H^{n-2}\ge0.

Suppose that this intersection number is positive. Choose a smooth projective common resolution W dominating \widehat Y and all the models in the finite MMP prefix, with maps

Proposition 2.1 gives

For every positive multiple m of r_0, this effective difference gives an injection

Set P_X=u_*r^*P. This is a fixed integral Weil divisor, hence Cartier on the smooth X. A rational section f in the right-hand space satisfies

Pushing this inequality forward gives

Thus there is an injective linear map

Choose a single ample Cartier divisor A_X for which A_X-P_X has a nonzero section. Multiplication by that fixed section gives

By (4.1), the left-hand side is bounded below by cm^2, for a fixed c>0, along all sufficiently large multiples of r_0. Consequently

contradicting \kappa_\sigma(X,K_X)=1 and the convention (1.1). It follows that

Proposition 3.4 now implies that L=K_Y is nef, including on curves contained in the singular locus. All other conclusions of Theorem 1.1, including the effective exceptional comparison (1.2) and its strict support condition, are supplied by the finite MMP in Proposition 2.1.□