Commuting Division-Coefficient Forms and the Artinian Eisenbud–Green–Harris Conjecture

Proof. The regular sequence gives

Thus the f_i have no common projective zero, and this remains true after any field extension.

Put N=\Delta[x_1,\ldots,x_n], considered as a right A-module. The inclusions

make A a submodule of the finite free k[x]-module N. Consequently A is finite over k[x], and both A and the finite right A-module N are Noetherian.

We claim that t_1,\ldots,t_n is an N-regular sequence. First, A/(t_1,\ldots,t_n) is zero dimensional. Indeed, at a geometric point where all t_j vanish, the identities (2.1), read in increasing order of i, give f_1=\cdots=f_n=0. Equation (2.2) then forces all x_j=0, and finiteness over k[x] gives the asserted zero-dimensional quotient.

Let \mathfrak m=A_{>0}, the graded maximal ideal. The central variables x_1,\ldots,x_n form an N-regular sequence because N=\Delta[x] is free over k[x]. Localization preserves the successive injections. Their quotient is \Delta, on which every positive-degree element of A acts trivially, so it remains nonzero on localization. Thus the sequence is also N_{\mathfrak m}-regular. The support dimension of N_{\mathfrak m} is at most n by finiteness over k[x]. It therefore equals its depth n. The system-of-parameters theorem for finite Cohen–Macaulay modules now implies that t_1,\ldots,t_n is regular on N_{\mathfrak m}; see [?, Proposition 10.103.4, Tag 00N6]. It is regular on the graded module N as well. To see the localization step explicitly, a nonzero homogeneous element of any graded kernel or quotient cannot be annihilated by an element of A\setminus\mathfrak m: the nonzero constant term of that element would survive in the lowest-degree component.

Since all the t_j have degree one,

Its degree-zero part is already \Delta. Hence, in every positive degree, N_d=\sum_jN_{d-1}t_j. Induction on degree, using commutativity of the t_j, shows that

spans N with coefficients on the left in \Delta. In degree d its cardinality is \binom{n+d-1}{n-1}=\dim_\Delta N_d; it is therefore a left \Delta-basis.□

Proof. Put N=\Delta[x_1,\ldots,x_n], with the full left \Delta-basis of Lemma 2.1.

Extend I to the two-sided ideal \Delta I of N. For each nonzero homogeneous element of this ideal, take the least exponent in its t-basis expansion under lexicographic comparison of the coordinates in the order

Let E\subseteq\Z_{\ge0}^n be the set of all such exponents. Right multiplication by t_j translates every exponent by the same vector \mathbf e_j, without moving any left coefficient. Thus E+\mathbf e_j\subseteq E. In each degree, elimination over the division algebra \Delta shows that the number of exponents in E is the left dimension of (\Delta I)_d.

For each i, Equation (2.1) puts

in \Delta I. Expand the C_{id} in the t-basis with coefficients on the left. Every exponent in a term C_{id}t_d has a positive coordinate of index greater than i, and hence is strictly larger than a_i\mathbf e_i in the chosen order. It follows that a_i\mathbf e_i\in E.

The upward-closed set E defines a single monomial ideal

It contains the required pure powers. Finally,

as graded left \Delta-vector spaces. The complementary exponent count in every degree gives the asserted Hilbert function.□

Proof. Every specified generator of \mathfrak j has degree at most b: the generators of Q have degree at most two, b\ge2, and a_l\le a_i=b for l\le i. At a projective point, a nonzero generator can be multiplied by a monomial of complementary degree nonzero at that point. Thus a base point of \mathfrak j_b would annihilate every generator of \mathfrak j.

At such a point f_l=0 for l\le i and t_d=0 for d>i. The earlier triangular rows, taken in the order i+1,i+2,\ldots,n, imply successively that the remaining f_d also vanish. Equation (2.2) forces all x_j to vanish, which is impossible at a point of \mathcal C(Q) by finiteness over k[x].□

Proof. Fix w\in W\setminus\{0\} over an algebraically closed extension. We first show that every repeated factor of w must be linear. If there were no simple factor, a second gradient-map argument would then force w to be a pure power, contradicting base-point-freeness.

For every linear or quadratic generator q of Q, the annihilation of q\,\Sym^{b-\deg q}U gives the constant-coefficient differential equation

Here a basis of U identifies its elements with the corresponding constant differential operators on polynomial functions on U. This is the differential-operator description of the degreewise annihilator pairing; see [?, pp. 1082–1083, Definitions 1–2 and (5a)].

Suppose an irreducible factor p occurs in w=p^r h with r\ge2 and p\nmid h. For a quadratic q, the term of lowest possible p-adic order in q(\partial)w is

For a linear q, it is rp^{r-1}h\,q(\nabla p)\pmod{p^r}. Equation (3.3) therefore shows that, generically on the hypersurface p=0, its projective gradient belongs to \mathcal C(Q).

If that gradient map is nonconstant, it has nonzero differential at some smooth point a of p=0. Choose a tangent vector v at a detected by this differential, and an ambient tangent vector v_{\perp} transverse to the hypersurface. A projective plane through a with tangent plane spanned by v,v_{\perp} cuts the hypersurface in a curve smooth at a, with tangent line spanned by v. Its component through a has degree at most \deg p, and its gradient map is nonconstant. On the normalization of this component, the partial derivatives are sections of the pullback of \cO(\deg p-1). After removing their common base divisor, they define a morphism whose pulled-back hyperplane bundle has degree at most \deg p(\deg p-1). This degree is the degree of the image curve times the degree of the morphism onto its image. The integral image is therefore a curve in \mathcal C(Q) of degree at most b(b-1), a contradiction. If the gradient map is constant instead, Euler’s identity places the hypersurface p=0 in the hyperplane perpendicular to that constant gradient. Irreducibility then forces p to be linear. Thus every repeated irreducible factor of w is linear.

Assume, seeking a contradiction, that w has no simple factor. We may then write

with distinct linear forms p_\nu, and put v_\nu=\nabla p_\nu\in U^\vee. The preceding argument gives [v_\nu]\in\mathcal C(Q). For a quadratic generator q, logarithmic differentiation yields

The left side and each q(v_\nu) vanish. Together with the linear differential equations, this proves that the ambient projective gradient map of w has image in \mathcal C(Q). Here the map is defined on the ambient \PP(U), not just on one factor hypersurface. If it is nonconstant, restrict it to a line along which it varies. The partial derivatives restrict to sections of \cO_{\PP^1}(b-1); removing their common zeros gives a nonconstant morphism whose image has degree at most b-1. That image is again a forbidden curve in \mathcal C(Q).

Finally, a constant projective gradient means that all partial derivatives of w are proportional to one fixed vector. After a linear change of coordinates, w depends on only one coordinate. Homogeneity gives w=c\,p^b for a nonzero linear form p\in U^\vee. The corresponding functional on \Sym^b U is, up to the nonzero scalar c, evaluation at p. Its annihilation of \mathfrak j_b contradicts base-point freeness.□

Proof. We use the hyperplane-incidence description of the Chow form [?, Section 1, Proposition 1.1], recalling the argument for curves to make the field-of-definition bound explicit. Consider pairs of hyperplanes whose intersection meets Z. The incidence variety over Z, with the two hyperplanes chosen through a point of Z, is irreducible of dimension 2n-3. A general pair in its image meets Z in finitely many points, so its image is an irreducible divisor in (\PP^{n-1})^\vee\times(\PP^{n-1})^\vee. Its defining bihomogeneous equation has degree \deg Z in each block: for a general first hyperplane, the corresponding divisor in the second block is the reduced union of the \deg Z hyperplanes representing the points of intersection.

This equation recovers Z. A point on Z has the property that every pair of hyperplanes through it satisfies the equation. For a point outside Z, choose a first hyperplane through it meeting Z properly, and then a second hyperplane through it avoiding the resulting finite set. That pair does not satisfy the equation. This recovery condition is defined over the field of coefficients. Indeed, on a point chart x_a\ne0, a hyperplane through x has coefficient h_a=-\sum_{j\ne a}h_jx_j/x_a. Substitute this expression and its counterpart for the second hyperplane into the incidence equation, clear denominators, and set every coefficient in the remaining hyperplane variables to zero. These finite equations express the recovery condition on the chart and agree on overlaps. Over the algebraic closure of the coefficient field, take the reduced structure. In characteristic zero it is geometrically reduced, so extension to \Omega gives exactly Z.

An equation of bidegree (d,d), d\le D, uses at most \binom{n-1+D}{D}^2 coefficients. Adjoin those coefficients to \C, then take its algebraic closure inside \Omega. This proves the claim, for instance with d_*(n,D)=\binom{n-1+D}{D}^2.□

Proof. Choose an even integer v_0 and ordinary linear forms L_1,\ldots,L_{v_0}\in\C[x]_1 whose squares span \C[x]_2. Such a choice is possible by polarization, increasing v_0 if necessary. For an integer m to be specified, form the skew Laurent algebra on symbols

in which distinct generators in the same row anticommute and generators in different rows commute. Ordered Laurent monomials give a basis: multiplication is defined by reordering and inserting the prescribed signs. Their leading terms show that this algebra is a domain.

The squares of the generators are central and algebraically independent. Localize their Laurent polynomial algebra to

The resulting algebra \Delta_0 is a finite-dimensional domain over k, with basis the monomials whose exponents are zero or one; it is therefore a division algebra. Its center is exactly k. Indeed, a parity vector \alpha in one row commutes with its j-th generator precisely when \sum_{l\ne j}\alpha_l=0 modulo two. Writing T=\sum_l\alpha_l, these equations give \alpha_j=T for all j. Since v_0 is even, they imply T=0 and then \alpha=0. Applying this in every row proves the claim; linear independence of the parity monomials prevents cancellations.

Evaluate the new formal forms by

Forms from different rows commute, and their squares are

The full-rank linear map from the independent square variables to the coefficients of each quadric shows that the coefficient vectors of the q_\rho are jointly algebraically independent. The indicated ideal Q consists of identities, and its quotient is finite over k[x], since every s_\rho is integral.

This supplies the commuting forms and the quadratic relations. We now choose m so that the resulting finite cover contains no curve of degree at most D. Over any algebraically closed extension \Omega of k, the linear projection

is defined everywhere and is finite. Its pullback of \cO(1) is the ambient \cO(1). If n=1, its target is a point, and there is no curve upstairs. We may then take m=2.

Assume n\ge2, and suppose an integral curve T of degree at most D occurs upstairs. Its image is an integral curve Z; if the map degree is d, then

In particular d\le D and \deg Z\le D. Choose a field of definition F_0\subseteq\Omega for Z as in Lemma 3.3, and write d_*=d_*(n,D). Let r=\binom{n+1}{2}, the size of each coefficient block. Since all mr coefficients are independent over \C, their transcendence degree over F_0 is at least mr-d_*.

At least m-d_* entire coefficient blocks are jointly independent over F_0. For completeness, scan the blocks, retaining a block whenever it contributes all r new independent parameters over the retained blocks and F_0. Each rejected block contributes at most r-1 even after all retained blocks have been adjoined. Indeed, enlarging the field generated by the retained blocks can only decrease the transcendence degree contributed by a rejected block. Adjoin all retained blocks first, then the rejected ones. If j blocks were rejected, the total transcendence degree would be at most mr-j, so j\le d_*.

Restrict the retained generic quadrics to Z. Their zero divisors are nonempty, reduced, supported in the smooth locus, and pairwise disjoint. They can also be taken disjoint from the zeros of a fixed coordinate x_j nonzero on Z. These are nonempty open conditions on the jointly generic full quadrics: the complete linear system of quadrics is base-point-free, separates tangent directions on the smooth curve, and can avoid each specified finite set. At a zero of the \rho-th such divisor, q_\rho/x_j^2 has valuation one, while the other retained functions have valuation zero. Thus their classes are independent in \Omega(Z)^\times/\Omega(Z)^{\times2}. The function field of T contains square roots of all these functions, by the equations s_\rho^2=q_\rho. Consequently

by the elementary degree formula for independent squareclasses. Choose m\ge2 with 2^{m-d_*}>D. This contradicts d\le D and proves the proposition.□

Proof. The quotient of the larger graded algebra by all positive-degree old forms is finite dimensional. Hence no point of its projective scheme has all old coordinates zero. The finite graded algebra map therefore induces the stated finite projective morphism, with the asserted hyperplane bundle. On an integral curve, its degree equals the degree of its image multiplied by the positive degree of the restricted morphism. A forbidden curve upstairs would thus give a forbidden curve downstairs.□

Proof. Projectivization makes w^{-1}(1) a cyclic cover of degree b of \PP(U)\setminus\{w=0\}. This complement is connected. A small meridian around a smooth point of the multiplicity-one factor, away from all other factors, permutes the b choices of a root by a generator of \mu_b. The monodromy is therefore transitive, proving connectedness.

For the join assertion, consider nonzero homogeneous polynomials f and g, of positive degrees a and c, in disjoint sets of variables. Cover the level f(x)+g(y)=1 by

On V_-, a continuous c-th root of 1-f(x) identifies V_- with

The first factor contracts to the origin by x\mapsto tx, 0\le t\le1, because f(tx)=t^a f(x). Similarly, V_+\simeq f^{-1}(1). On the intersection both f(x) and 1-f(x) admit continuous roots, giving a homeomorphism

Under these identifications, the two inclusion maps are homotopic to the projections onto g^{-1}(1) and f^{-1}(1). The homotopy pushout for this open cover is their join. Iteration proves the asserted description of F_w. The integral homology formula for joins, including its Tor terms, now gives the stated vanishing: joining M connected spaces has no reduced homology below degree 2M-1. Smoothness of every nonzero homogeneous level follows from Euler’s identity.□

Proof. All geometric assertions may be checked after extension to \C. The multiplicity-one hypothesis persists under this extension. Indeed forms without a multiplicity-one factor constitute a finite union of images of projective multiplication maps (p_1,\ldots,p_t)\mapsto\prod_\nu p_\nu^{a_\nu}, with a_\nu\ge2 and \sum_\nu a_\nu\deg p_\nu=b. Their intersection with \PP(W) is empty geometrically over k, and remains empty after extension. Write r=\dim U, d=N_*-s, and \kappa=M+2h. If s=0, then X=\PP^{N_*-1} and Y^o=\C^{N_*}\setminus\{0\}\simeq S^{2N_*-1}. The conclusions follow from 2N_*-2\ge\kappa. Henceforth assume s\ge1.

Dimension and the singular locus. Choose a basis w_1,\ldots,w_s of W. A point has defective Jacobian rank for the defining equations of Y only if, for some [w]\in\PP(W),

For any nonzero w, its gradient vanishing set has dimension at most r-1. The incidence with \PP(W) therefore bounds the defective-rank locus in U^{\oplus M} by

Every irreducible component of Y has dimension at least N_*-s. Since M>2s-1, no component is contained in the locus in (4.1). Full Jacobian rank gives local dimension N_*-s, so all components have dimension d and are generically smooth. The s equations thus have height s in the polynomial ring and form a regular sequence. The resulting complete intersection is Cohen–Macaulay, with no embedded associated primes. As it is generically reduced, it is reduced.

Away from the origin, the cone projection is locally a product with \C^\times, so smoothness of Y is equivalent to smoothness of X. Equation (4.1) gives

Here the origin causes no additional restriction: d\ge c. The same bound applies to \operatorname{Sing}X. It remains to prove integrality and connectivity; we have not yet assumed that the smooth locus is connected.

Homology of the complement. A point outside Y makes some linear combination of the defining equations nonzero. Recording that combination replaces the complement by a space with contractible fibers over it, while exposing the nonzero hypersurface levels of Lemma 4.1. Define the open incidence variety

Projection to \C^{N_*}\setminus Y is an affine bundle of rank s-1. For a fixed \ell\notin Y, each line [w] in the fiber has a unique representative satisfying \sum_m w(\ell_m)=1. Thus the fiber is the affine hyperplane on which the nonzero evaluation functional on W equals one. Consequently this projection is a homotopy equivalence.

Evaluation also defines a map

The value at (\ell,[w]) is the nonzero linear functional on the tautological line kw obtained by evaluation at \ell. Euler’s identity shows that p is a smooth submersion. Its fibers have complex dimension N_*-1; after choosing a representative of [w] and rescaling, they are the levels F_w of Lemma 4.1.

The join lemma controls the homology of every fiber of p. To pass from these individual fibers to \cH, we use direct image with proper support; we do not require p to be a locally trivial fibration. Fiber homology becomes compact-support cohomology near the top degree, where the Leray spectral sequence gives the needed comparison. Set q=2N_*-2. Compact-support Poincaré duality on the fibers [?, Theorem 3.35] and Lemma 4.1 give

The top sheaf is constant: a submersion chart over a small base open set supplies an oriented open ball in each fiber. Extension by zero maps its top compact-support orientation class to the positive generator for that fiber. This gives an isomorphism from the constant sheaf to the top direct-image sheaf on the open set. The local isomorphisms agree by the complex orientation.

The compact-support Leray spectral sequence is

The stalk formula and this spectral sequence follow from proper-support base change and derived composition [?, Theorems 5.9 and 5.10]; maps between the locally compact Hausdorff spaces here satisfy their hypotheses. Equivalently, one may compactify p, extend the constant sheaf by zero, and apply proper base change and Leray. The base is a real 2s-dimensional manifold, so its compact-support cohomology with coefficients in any abelian sheaf vanishes above degree 2s [?, Proposition 5.1.2(ii)]. In total degree q+2s-j, with 0\le j\le2M-2, only the top row of (4.3) can contribute. In fact, the next possible row is i=q-(2M-1), which would require

An outgoing differential from the top row to another nonzero row has length at least 2M>2s, and hence has zero target. An incoming differential would come from above the top row. Poincaré duality on \cH and \mathcal B therefore yields

The punctured line bundle \mathcal B retracts to S^{2s-1}.

Homology of the smooth cone. The closed/open compact-support sequence for Y\subseteq\C^{N_*}, followed by duality on its complement, gives

for 0\le j\le\kappa. Both adjacent ambient compact-support groups vanish, since their degrees are below 2N_*. The homology range in (4.4) applies because

Removing Y\setminus Y^o does not change the left side of (4.5). Indeed, a finite filtration of the removed algebraic set by smooth locally closed strata, together with the compact-support closed/open sequence, gives vanishing above its real dimension from the same manifold bound. Equation (4.2) bounds that dimension by 2(d-c), whereas

Both (4.6) and (4.7) follow from M\ge2h+4s+10. Duality on Y^o now gives

Every irreducible component of Y meets Y^o densely. The smooth loci of distinct components are disjoint, so their union could be connected only if there is one component. Thus Y, and hence X, is integral.

From homology to homotopy. The preceding argument leaves only simple connectivity to prove. The quasi-projective Lefschetz theorem [?, Theorem 1.1.3(ii)] applies to the smooth open X_{\sm} in the projective variety X, with deleted algebraic subset \operatorname{Sing}X. It states that the smooth open has the homotopy type of a space obtained from a sufficiently general hyperplane section by attaching cells of dimension at least the complex dimension of the open. Apply the theorem successively, intersecting the deleted subset with each hyperplane as well. Each section down to a surface therefore surjects on the fundamental group of the preceding smooth open.

Since \operatorname{codim}_X\operatorname{Sing}X>h\ge2, a general surface linear section avoids the singular locus. It is a smooth complete intersection S_0\subseteq\PP^{s+2} of s hypersurfaces of degree b. Such a surface is simply connected. To check the usual smooth flag needed here, replace its defining equations by general invertible linear combinations. Off their common base locus, Bertini gives smooth successive intersections. Along S_0, the independent differentials of the defining equations give smoothness of every initial intersection. Smooth projective Lefschetz, applied to these very ample divisors, preserves the fundamental group down to dimension two, starting from projective space. Therefore \pi_1(S_0)=0, and the surjections above give \pi_1(X_{\sm})=0.

The cone Y^o is the complement of the zero section in \cO_{X_{\sm}}(-1). Its unit circle bundle has a homotopy exact sequence in which \pi_1(S^1)=\Z surjects onto \pi_1(Y^o), since the base is simply connected. Hence \pi_1(Y^o) is abelian. Its abelianization is H_1(Y^o;\Z)=0, so it is trivial. Smooth complex algebraic varieties have CW type. Successive applications of the Hurewicz theorem [?, Theorem 4.32] to the homology vanishing already proved now give \pi_j(Y^o)=0 for 1\le j\le\kappa.

Radial retraction gives the same connectivity for the unit cone. Finally, the strict bound on the singular codimension and Bertini give the asserted smooth h-dimensional sections. All conclusions hold after every complex embedding and imply the stated geometric assertions over k.□

Proof. Inside \PP^{b-1}, with coordinates p_0,\ldots,p_{b-1}, take the inverse image under coordinate-wise b-th powering of the line parameterized by

Thus the preimage is defined by the b-2 independent linear relations on the p_j^b; no equation is imposed when b=2. Let C' be its normalization. To establish its irreducibility, put s=u/z. On the generic fiber over the s-line the coordinate ratios satisfy

These b-1 functions generate a free (\Z/b)-submodule of \C(s)^\times/\C(s)^{\times b}: valuation at s=\zeta^{-j} detects the j-th exponent independently. Kummer theory gives a field extension of degree b^{b-1}. This argument uses valuations equal to one and applies also when b is composite. Every component of the projective preimage has dimension at least one, since there are b-2 equations, and the coordinate-power map is finite. Hence every component dominates the line. The integral generic fiber therefore proves that its reduced support, and consequently C', is irreducible.

Phase changes of the p_j, together with their cyclic permutation, act on this curve. The group before projectivization is \widetilde G=\mu_b^b\rtimes(\Z/b). Its scalar subgroup is the diagonal \mu_b, and the cyclic permutation already has order b before taking the quotient. Thus the resulting group is the stated semidirect product G. Its action is faithful: a nontrivial cyclic permutation moves s by a nontrivial b-th root, whereas a phase change fixing a general point with all coordinates nonzero must be diagonal. The invariant coordinate

defines a degree-b^b map C'\to\PP^1_t. Since |G|=b^b, this is a Galois cover with group G.

The Kummer cover of the s-line is ramified precisely at the b points s=\zeta^{-j}, with index b. In particular, although several of the displayed ratios have a pole at s=1, their simultaneous adjunction still has local index b: every unit in \C[[x]] has a b-th root, so all local extensions are contained in \C((x^{1/b})). The map is unramified over s=0,\infty. It follows that the map to the t-line has exactly the branch values 0,1,\infty, with index b above each.

Let T_0 be the smooth projective model of

The valuations of the right side at 0,1,2,\infty are 1,1,-1,-1, respectively. This is a cyclic degree-b cover, totally ramified at those four points and unramified elsewhere. Each nontrivial intermediate extension is ramified at 2, whereas C'\to\PP^1_t is unramified there. The two Galois extensions are consequently linearly disjoint. Normalize their fiber product to obtain C. It is connected, and C\to T_0 remains a G-cover.

This last cover is étale. Above 0,1,\infty, both original ramification indices are b; in completed local coordinates, adjoining the same b-th root of a uniformizer removes the ramification after normalization. Above 2, and away from the four listed points, the base change of the original map is already étale. Riemann–Hurwitz now gives

For b=2, this construction starts with C'=\PP^1 and produces a degree-four étale C_2\times C_2-cover between curves of genus one.

Pull \cO(b)|_{C'} back to C, and call the resulting bundle L. The sections u,z defined in the proposition have no common zero, since their simultaneous vanishing forces every p_j to vanish. On C' their pencil is the degree-b^{b-1} map to \PP^1_s; hence \deg\cO(b)|_{C'}=b^{b-1}. Since C\to C' has degree b, we have \deg L=b^b=e.

The standard \widetilde G-linearization on \cO(b) descends to a genuine G-linearization: a diagonal scalar in \mu_b acts trivially in degree b. Pullback preserves this linearization. Degree-b monomials therefore define the sections a_\alpha and their asserted action. Phase changes fix u,z, cyclic permutation fixes z and multiplies u by a primitive root, and v=\prod_jp_j transforms by the product of phases. This product is invariant under cyclic permutation and trivial on the diagonal subgroup, so it defines a character of G. Denote the resulting transport characters of u,v by \chi,\psi.

The equations of the curve give the first relations in (5.1), and their product gives

If u=v=0, this identity gives z=0, which was excluded. Thus u,v define a base-point-free pencil. Its morphism \pi:C\to\PP^1 satisfies L=\pi^*\cO(1); since \deg L=e>0, the morphism is finite of degree e.□

Proof. Let g=g(C). For generic A\in\Pic^0(C), the line bundle A L^{g_T-1} is generic of degree

The effective locus in \Pic^{g-1}(C) is the image of \Sym^{g-1}C, a projective variety of dimension g-1. It is a proper closed subset of the g-dimensional Picard variety. Thus H^0(C,A L^{g_T-1})=0, and Riemann–Roch gives H^1(C,A L^{g_T-1})=0 as well. Here g\ge1; when g=1, the effective locus in degree zero consists only of the trivial line bundle. The complement of this locus, translated to \Pic^0(C), is a nonempty G-stable open subset, because L is linearized.

Fix A in that open set, and put

This is a rank-e vector bundle on \PP^1: the map \pi is finite, and its pushforward is torsion-free on a smooth curve. Projection formula and the preceding vanishing give H^*(\PP^1,\mathcal V(-1))=0. For every geometric point q\in\PP^1, the exact sequence

therefore identifies H^0(\PP^1,\mathcal V) with \mathcal V|_q. It follows that the evaluation map

is an isomorphism. This also proves the assertion over the field of definition of A, since an isomorphism can be checked after algebraic closure. In particular \dim E_A=e. Tensoring by \cO(j), taking global sections, and using projection formula proves (5.2), with its indicated multiplication map.

Let

Equation (5.2) identifies this graded module canonically with E_A\otimes_\kappa\kappa[U,V]. Multiplication by a section a of L is a homogeneous degree-one endomorphism of this free module over \kappa[U,V]. Such an endomorphism is determined by its value on degree zero and has the form

Multiplication of sections is commutative, so these matrix polynomials commute as elements of \End_\kappa(E_A)[U,V]. Every polynomial identity among sections acts as the zero endomorphism and remains an identity among the matrices. In particular u,v act as U,V, and the identity for M_z follows from (5.1). This does not assert that all the individual matrices A_a,B_a commute.

Choose a point of the constant curve to normalize the universal line bundle on \Pic^0(C)\times C. On the open subset just specified, the dimensions of the section spaces are constant and the higher cohomology vanishes in every degree g_T+j, j\ge0. Cohomology and base change, applied to these line bundles along the proper constant-curve projection, then makes the preceding evaluation and multiplication maps algebraic in A [?, Lemma 30.22.1, Tag 07VK]. The universal lines are flat over the Noetherian parameter base, as required there.

Finally, all multiplication maps are natural under transport. An isomorphism between a transported universal line and the universal line at the transported parameter is unique up to a scalar. It acts by the same scalar on the source and target section spaces, so the induced transport on endomorphisms is independent of that choice and composes as a group action. Transporting

and using g_*u=\chi(g)u, g_*v=\psi(g)v gives (5.3). On the generic parameter field this is semilinear transport of the coefficients.□

Proof. Suppose an element g\ne1, of order r, fixes [V]\in\Pic^1(C). Choose an isomorphism g_*V\to V. Its r-fold composite is an automorphism of V, hence a nonzero scalar. Rescaling the chosen isomorphism by an r-th root of the inverse scalar gives a linearization of V under \langle g\rangle. Since this cyclic subgroup acts freely on C, descent along the degree-r étale cover q:C\to C/\langle g\rangle gives V=q^*V_0. But then 1=\deg V=r\deg V_0, a contradiction.□

Proof. Identify \mu_b with \Z/b by exponentiation. Compose the monodromy of the cover with \xi to obtain \bar\alpha\in H^1(T;\Z/b). Since H_1(T;\Z) is free abelian, \bar\alpha lifts to a class \alpha\in H^1(T;\Z). Represent \alpha by a harmonic real one-form with integral periods. The form

has integral periods on C: the projection of a closed loop in C has trivial covering monodromy, so its \alpha-period is divisible by b. It is also harmonic, since q is holomorphic and unramified. Integrating \omega modulo \Z and exponentiating defines a circle-valued map on C. For a path from c to gc, the projected loop has character \xi(g); choosing the sign of the monodromy convention accordingly, the map satisfies

Its differential, expressed as a real one-form after local lifting to \R, is the G-invariant form \omega.

The Abel map

is equivariant. By the Jacobian description as the quotient of the dual space of holomorphic differentials by the integral period lattice, pullback by a identifies translation-invariant real one-forms on \Pic^1(C) with harmonic real one-forms on C, and induces an isomorphism on integral first cohomology; see [?, Proposition 2.2 and Theorem 2.5]. Consequently \omega=a^*\widetilde\omega, where \widetilde\omega is translation-invariant with integral periods. Integrating \widetilde\omega gives a continuous circle-valued map f_\xi on \Pic^1(C). Adjust its constant so that f_\xi\circ a=f_C.

The action on the Picard torsor is affine. The difference between the pullback of \widetilde\omega under g and \widetilde\omega is translation-invariant and pulls back by a to zero, since \omega is invariant. Injectivity of this pullback makes the difference zero. It follows that f_\xi(g_*A)/f_\xi(A) is constant on the connected Picard variety. On the Abel image it equals \xi(g), so it has that value everywhere.□

Proof of Proposition 5.1. Lemma 5.2 supplies parts (i) and (ii), and Lemma 5.3 supplies part (iii). Apply Lemmas 5.4 and 5.5 to the étale cover C\to T_0 to obtain part (iv).□

Proof. Choose finitely many rational sections spanning \cF_\eta, and let \cF_0 be the coherent subsheaf of rational sections generated by them and their multiples by a k-basis of \Delta_0. For each g, use (6.5) to view J_g\otimes g_*\cF_0 as a coherent subsheaf of rational sections. Its image is coherent because T is noetherian. Let \cF be the sum of these finitely many images. Corrected transport permutes the summands by the composition law in (6.5); hence it preserves \cF. All maps commute with \Delta_0, and their composition identities hold inside rational sections. The generic fiber is unchanged.□

Proof. Write g_C for the genus of C. For one pair A,V of degrees zero and one,

A scalar acting on a line inside a determinant of cohomology has exponent equal to its Euler characteristic. Thus the corresponding factor of \cD has scalar weights

in A,V. The first weight cancels the weight one of \cF, and the second leaves no dependence on identifications of V. Local choices of line identifications therefore give the same transport on \cF\otimes\cD. To check the group law, compare the two successive transports with the composed identifications of the universal lines. Composition in J_g\otimes g_*J_h\longrightarrow J_{gh} is the associative composition from g_*h_*A to g_*A to A. Functoriality of determinants and the genuine transport of the original module identify the two maps for these composed choices. Any scalar change of the A-identification contributes opposite weights on \cF and \cD, and a change of the V-identification has weight zero. Thus no scalar cocycle remains, and the maps glue to genuine equivariance.

The origin of the representation R is H-fixed. Its immersion in the product has the finite equivariant Koszul resolution on the linear coordinates of R, even when X is singular. The alternating coherent Tor class of restriction therefore defines \beta, preserving both actions. Since \cF does not depend on P, the last assertion follows.□

Proof. It suffices to calculate the determinant correction for one curve. Write a=c_1(A), v=c_1(V), and let q project off C. Grothendieck–Riemann–Roch for this smooth proper projection gives

Indeed the signed exponential sum -e^{a+v}+e^a+e^v has degree-two part zero and degree-four part -av; a curve has no degree-four Todd term.

Normalization at the chosen curve point removes pure parameter-space degree-two components from both universal classes. The class a has no pure curve component because \deg A=0. The pure curve component of v cannot contribute to q_*(av). The resulting class is therefore purely mixed between H^1(\Pic^0(C),\Z) and H^1(\Pic^1(C),\Z). Tensoring by \cO_C(-c_0), for the chosen curve point c_0, identifies the torsor \Pic^1(C) with \Pic^0(C); different choices change this identification by translation, which acts trivially on integral first cohomology. The universal classes identify these integral lattices with the curve’s first-cohomology lattice: pullback by an Abel map gives the mixed component of the diagonal class on C\times C, and the Abel map induces an integral H^1-isomorphism. The intersection form on H^1(C,\Z) is unimodular. Hence the mixed class has a unimodular matrix \Lambda. This is the cohomological Poincaré pairing; the rank extraction below is the numerical counterpart of the Fourier–Mukai calculation in [?, Theorem 2.2 and Example 2.6].

More explicitly, if d=2g_C is the real dimension of either Picard factor and b_i,p_j are integral bases, a mixed class \theta=\sum_{i,j}\Lambda_{ij}b_i\wedge p_j satisfies

The factors for distinct curves form separate blocks, so the same unit calculation holds on Z. Each Picard tangent bundle is trivial, giving \td(Z)=1. To obtain top degree on P from \exp(c_1(\cD)), one must already use top degree on B, because every mixed term has degree one on each side. Thus every positive-degree term of \ch(\gamma) contributes zero. Only its rank remains.□

Proof. The vector bundle \delta_*\cO_{T'} has rank a. Put d=[\delta_*\cO_{T'}]-a\in K^0(Q). Multiplication by d lowers the dimension-of-support filtration on the coherent-sheaf group G_0(Q). To see this, the successive quotients of that filtration are generated by integral closed supports [?, Lemma 42.23.2, Tag 02S9]; on an integral support the two bundles in d have the same rank and are isomorphic on a dense open set. Localization therefore puts their difference in smaller support dimension. In particular this operator is nilpotent.

The torsor becomes the disjoint union of a copies of T' after pullback to T', so \delta^*d=0. The projection formula gives, as operators on G_0(Q),

Over \Q, the operator a+d is invertible by a finite nilpotent expansion, hence d acts by zero. Euler characteristic takes values in \Z, so it kills torsion and \chi(Q,d\alpha)=0. Finite pushforward and the projection formula now give the claimed equality.□

Proof of Proposition 6.2. The construction and the first identity have been established above. By Proposition 5.1, H acts freely on P, hence on Z\times X. Its quotient is a projective scheme, and the quotient morphism is a finite étale H-torsor; existence of the scheme quotient and torsor follows from [?, Tag 07S7], and a product of the translates of an ample line bundle gives projectivity of the quotient. Equivariant coherent sheaves descend along this torsor, together with their commuting \Delta_0-actions [?, Proposition 35.5.2, Tag 023R]. The same holds for \beta\otimes\cO_X(l), since \cO_X(1) is linearized.

Each cohomology group of a descended sheaf is a finite right module over the division algebra \Delta_0/k, so its k-dimension is divisible by e_0^2. Lemma 6.6 multiplies its Euler characteristic by |H|=e_1 on passing upstairs. Applying this term by term to the coherent class gives

which proves (6.4). Finally, field extension preserves coherent cohomology dimensions by flat base change [?, Tag 02KH], even for a complex embedding that does not fix the original constants. Over \C, idempotent reduction divides these dimensions by e_0, proving the last assertion.□

Proof. The product of the principal bundles \mathbb T\to\mathbb T/H' and S^{2h+1}\to\PP^h(\C) is a principal H'\times S^1-bundle. An equivariant lift \widetilde\Phi is a section of its associated bundle with fiber Y_1. The base (\mathbb T/H')\times\PP^h(\C) has a finite CW structure of dimension M+2h. Extend a section cell by cell: over an a-cell the extension obstruction is a map S^{a-1}\to Y_1, which is null-homotopic by the connectivity just established. The initial choice over zero-cells and extension over one-cells are allowed because Y_1 is nonempty and path connected. This constructs \widetilde\Phi, and passage to the scalar-circle quotient gives \Phi.

To compare \Phi with \Phi_0, forget H'-equivariance. The lifts \widetilde\Phi(t,z) and \widetilde\Phi(t_0,z) are sections of the same circle-associated bundle over \mathbb T\times\PP^h(\C). Construct a homotopy relative to its two endpoints by extending over cells of the base times [0,1]. This relative CW pair has dimension M+2h+1, so the largest obstruction belongs to \pi_{M+2h}(Y_1), which also vanishes. Projectivizing this homotopy proves the assertion about \Phi. The circle-equivariant map S^{2h+1}\to Y_1 obtained at t_0 identifies the associated tautological line bundles, giving the stated isomorphism for \Phi_0.

Finally, any character of H=G^{M-1} is a product of characters of its factors. Lemma 5.5 realizes each factor character by an equivariant circle-valued map on the corresponding factor of P. Multiplying these maps realizes each coordinate of \rho. Their product is \vartheta.□

Proof. Choose a projective embedding i:V\hookrightarrow\PP^a(\C), with complex normal bundle \nu. The complex K-theory Thom class, followed by extension from a tubular neighborhood, defines

We use the Koszul convention for this Thom class: its restriction to the zero section of \nu is \lambda_{-1}(\nu^\vee). If x_1,\ldots,x_c are the Chern roots of \nu, its Chern character, divided by the ordinary cohomological Thom class, is

For completeness, this identity follows on the zero section from \ch(\lambda_{-1}(\nu^\vee))=\prod_\alpha(1-e^{-x_\alpha}) and from the Euler class \prod_\alpha x_\alpha. The splitting principle verifies the identity universally for sums of line bundles, where the product of the universal Chern classes is a non-zero-divisor; the Thom isomorphism then gives the asserted identity before restriction as well. Consequently

The exact tangent-normal sequence now implies

The integral projective bundle formula identifies

Write i_!\xi=\sum_{i=0}^a\sum_{j=0}^h c_{ij}v^iu^j, with c_{ij}\in\Z. The number

is an integer by Hirzebruch–Riemann–Roch, applied additively to the finite sum of line-bundle classes v^i. Thus the left side of (7.3) is the Chern character of an integral combination of 1,u,\ldots,u^h, which gives I_V(\xi). Uniqueness follows because the Chern character is injective on the torsion-free group in (7.1): if H=c_1(\cO(1)), then \ch(u)=1-e^{-H} has leading term H, so these h+1 Chern characters are linearly independent over \Q. The projective bundle and Thom constructions used here are standard complex K-theory, with the Koszul convention for the Thom class specified above [?, Proposition 2.24, Theorem 2.25, and pp. 113–114]; for the Riemann–Roch formula see [5, §7].□

Proof. Pull back the topological class of \beta'|_{Z\times X_{\sm}} along \gamma and call the resulting class \sigma. This pullback is permitted because the class is perfect on the smooth open product.

The equivariant coherent class \beta' descends on the free smooth quotient (Z\times X_{\sm})/H. That quotient is smooth quasi-projective, so the descended coherent class has a topological vector-bundle class. Pulling it back along the quotient of \gamma gives \tau\in K^0(V\times\PP^h(\C)) whose pullback is \sigma. Since Z\to V is an étale covering of degree e_1 and T_Z is the pullback of T_V, Lemma 7.2 gives

Nonequivariantly, Lemma 7.1 makes \gamma homotopic to (z,t)\mapsto(z,\Phi_0(t)). The equality (7.5) therefore computes the same fiber integral for this latter pullback of \beta'. Evaluate it at any point of \PP^h(\C). Only q_0 remains on the right, whereas (6.3) gives \pm r on the left. Thus q_0=\pm r.

This already gives e_1\mid r. The stronger divisibility by e_0e_1 is encoded in the algebraic twist indices p(l) of (6.4). A smooth linear section of X retains that divisibility and can also be compared with the projective-space test. Put d=\dim X. Theorem 4.2 gives \operatorname{codim}_X(X\setminus X_{\sm})>h. A general linear section of codimension d-h therefore misses the singular locus; Bertini gives a smooth section D_0. As X is a complete intersection of s degree-b equations, D_0 has the description in the statement.

Choose h+1 linear forms with no common zero on D_0; these exist by general position since \dim D_0=h. They define a morphism

The inclusion j:D_0\hookrightarrow X_{\sm} and \Phi_0\circ f have circle-equivariant lifts from the unit frame bundle of this same line bundle to Y_1. Indeed, choose the indicated line-bundle isomorphisms and normalize them by their positive fiber norms to identify the unit frame bundles. The relative section-extension argument in Lemma 7.1, now over D_0\times[0,1], makes those lifts homotopic: the real dimension of this base is 2h+1, below the available connectivity bound plus one. Thus j and \Phi_0\circ f are homotopic.

Pulling (7.5) back by f now identifies the fiber integral on Z\times D_0 with

Define

where restriction of a coherent class is derived. Hirzebruch–Riemann–Roch on the smooth projective product, followed by (\ast), gives

It remains to retain the original divisibility when passing to this section. Recall from (6.4) the integers

The d-h hyperplanes defining D_0 have a Koszul resolution on X: along D_0 they cut transversely in the smooth locus, and off their common zero locus the complex is exact because one of the equations is locally a unit. Tensoring this resolution derivedly with \beta, and then passing to the complex embedding and the matrix-idempotent reduction, yields

Here cohomology commutes with the field extension, and the matrix-idempotent reduction divides every cohomology dimension by e_0. The particular hyperplanes need only be defined after that extension: the terms on the right of (7.7) are the original integer Euler characteristics. Combining (7.6) and (7.7) proves (7.4).□

Proof. We successively construct integer polynomials Q_i, for 0\le i\le v, with Q_0=Q, such that

Suppose i<v and Q_i has been constructed. Modulo p, write A=u^\nu B, where B(0)\ne0. The power series B is invertible over \Z/p, so the displayed congruence forces every coefficient of Q_i through degree

to be divisible by p. This degree is nonnegative because h\ge v\nu. Let Q_{i+1} be the truncation of Q_i through degree m_i, divided by p. It is an integer polynomial and has the required constant coefficient. Since

restricting the old congruence to degrees at most m_i gives

Coefficientwise division by p gives the next congruence modulo p^{v-i-1}. At i=v-1, the construction shows directly that Q(0)/p^{v-1} is divisible by p. This proves the conclusion. The same induction includes \nu=0, when no degrees are lost.□

Completion of the proof of Theorem 6.1. Use D_0, q_0,\ldots,q_h, and Q_* just constructed for the chosen module. If e_0=1, then q_0=\pm r\in e_1\Z already gives the required divisibility, so assume e_0>1. The Hilbert series of the coordinate ring of the complete intersection D_0 is

Euler characteristics of sufficiently positive twists agree with Hilbert functions. Moreover, multiplication by (1-[\cO(-1)])^j gives the same finite difference as multiplication of generating functions by (1-t)^j, apart from finitely many initial coefficients. The coefficients in these differences are integers. It follows from (7.6) that

By (7.7), every coefficient on the left is divisible by e_0.

Multiply (7.8) by (1-t)^{h+s+1} and reduce coefficients modulo e_0. The resulting right side is a polynomial that vanishes coefficientwise. We may therefore substitute t=1-u, obtaining

Define

Cancelling the common monomial u^s yields

This is coefficientwise cancellation of a monomial, valid over \Z/e_0 even when e_0 is composite. When s=0, the factor is 1, and the constant coefficient of (7.9) immediately gives e_0\mid Q_*(0).

For the general case, fix a prime p\mid e_0 and write v=v_p(e_0), b=p^a c, where p\nmid c. Over \Z/p,

Thus A_b(u)^s has exact order \nu=s(p^a-1) modulo p, with nonzero leading coefficient. The choice of h in Theorem 6.1 gives h\ge v\nu. Apply Lemma 7.4 to (7.9), reduced modulo p^v, to obtain p^v\mid Q_*(0). This also covers primes not dividing b, for which \nu=0; no assumption that b is a prime power has been made. Doing this for every prime divisor of e_0 gives

The module was arbitrary. A nonzero proper right ideal of the central simple algebra \Delta, whose degree is e_0e_1, would have reduced rank strictly between 0 and e_0e_1. The divisibility just proved excludes such an ideal, so \Delta is division.□

Proof. The pair exchanges identify any two products of the same number of symbols whose total exponent vectors agree. To see this directly, regard such a product as an allocation of exponent units among factors, each of total degree b. Choose the exponent vector desired in the first factor. Whenever that factor has an excess in one coordinate and a deficit in another, some other factor has a unit in the deficient coordinate. Exchange this unit for a unit in the excess coordinate. This is one relation of the form (8.1). Once the first factor has its desired vector, leave it fixed and continue with the remaining factors.

Applying this observation to T_\alpha^b proves the second identity in (8.3). In particular,

Thus z is integral over A. Every D_j belongs to A[z], and the second identity in (8.3) makes every T_\alpha integral over A[z]. There are finitely many symbols, so A' is finite over A. The grading assertion follows from the displayed defining relations.□

Proof. Put b=a_i, let e_0 be the degree of \Delta_0, and take \mathfrak j and W as in (3.1) and (3.2). Write s=\dim_k W. Lemma 3.1 gives base-point-freeness of \mathfrak j_b, and Lemma 3.2 shows that every nonzero geometric member of W has a factor of multiplicity one.

Parameters and the invariant sum. Choose integers in the following order:

The prime-indexed set in (8.5) is empty if e_0=1. Theorem 4.2 supplies the geometrically integral projective constraint variety

Use the curve and its two characters \chi,\psi from Proposition 5.1. For M-1 copies of the curve, indexed by m=2,\ldots,M, let

and write \chi_m,\psi_m:H\to\mu_b for the corresponding characters. Define characters recursively by

Let H act on block m of the constraint cone by the point character \eta_m^{-1}. This preserves its equations because their degree is b, and it gives the natural linearization of \cO_X(1).

As in Theorem 6.1, put

where R is an affine space with a faithful linear H-action. Then K/F is Galois with group H. Let E=\bigotimes_{m=2}^M E_{A_m}, of dimension e_1 over K, and let \Delta_1 be the descent of \End_K(E). Theorem 6.1, applied with the block characters just chosen, makes

a division algebra. Its scalar extension to K is \Delta_0\otimes_k\End_K(E), and H fixes \Delta_0 pointwise. Notice that e_0,s,b were fixed before h, and M before e_1; there is no restriction involving the new degree e_1 in (8.5).

The generic fiber of \cO_X(1), pulled back to K, is a one-dimensional semilinear H-space. By Galois descent it has a nonzero invariant generator \tau. Choose a k-basis of U. In each cone block, divide its coordinate sections by \tau and use the resulting coefficients to form \ell_m\in U_K. Transport of a coordinate section is pullback by g^{-1}. The inverse point action \eta_m^{-1} therefore gives

In particular, dividing by the invariant \tau does not reverse the character.

The constraint equations say that

Since every \eta_m^b=1, the element S_b is invariant and belongs to (\mathfrak j_F)_b. We claim that it has a decomposition

For this, let

Then \mathfrak j_b=V_b+kf_i. If f_i\in V_b, any prescribed nonzero coefficient of f_i may be absorbed by changing the term in V_b, proving the claim.

Suppose instead that f_i\notin V_b. Pure b-th powers span \Sym^b U in characteristic zero. Over an algebraic closure, express f_i as a linear combination of at most \dim\Sym^b U such powers, absorb each scalar into a b-th root, and pad with zero forms. The bound (8.6) gives a nonzero cone point with \sum_m\ell_m^b=f_i. Thus the homogeneous coefficient of the class of f_i in \mathfrak j_b/V_b is not identically zero on X. Geometric integrality makes it nonzero at the generic point. Dividing all the coordinates by \tau only multiplies that coefficient by a nonzero scalar to the b-th power. Finally, the spaces V_b and \mathfrak j_b are defined over k. Invariant linear algebra over F gives (8.10), with homogeneous multipliers of the indicated degrees.

The chain of binary additions over K. We next enlarge the formal space over K, keeping the old equations. Start with s'_1=\ell_1. In link m, introduce a fresh symbol T_{m,\alpha} for every degree-b monomial p^\alpha in the curve coordinates. Each symbol has formal degree one. Evaluate it by the multiplication matrix from Proposition 5.1, on the m-th tensor factor of E, with substitution

Let s'_m be the formal linear combination of the diagonal symbols corresponding to z.

These substitutions preserve commutativity of the entire list. Indeed, the coefficient matrices of a new link commute coefficientwise with \Delta_0 and with the earlier tensor factors. The two inputs in (8.11) commute with one another and with all earlier forms. The inputs therefore centralize the new coefficient algebra \End_K(E_{A_m}), embedded in the m-th tensor factor. Consequently substitution defines a K-algebra homomorphism

where the inputs on the right denote their evaluated forms. It preserves the polynomial commutation identities of the new multiplication matrices, and every new evaluated form commutes with every earlier one. This argument uses no coefficientwise commutativity among the matrices within a single link.

For each link impose the pair-exchange quadrics (8.1), the diagonal relations (8.2), and the identifications

They are identities under evaluation because they are identities of sections on the curve. Lemma 8.1 shows that the resulting quotient is finite over the preceding formal relation algebra and gives

After all links, the formal quotient is finite over \Sym_K U_K/Q_K, is defined by the old and new linear and quadratic relations, and satisfies

Equivariance and descent. Give the new symbol space in link m the degree-b action on the p-monomials from Proposition 5.1, multiplied by the character \lambda_m, and extend it semilinearly over K. Keep U fixed. The pair-exchange relations are stable under the monomial phase and permutation action and under the common twist. The diagonal relations are stable as well. The distinguished symbols u_T,v_T,z have respective weights

These are the weights of their prescribed inputs and output: the input identifications are stable by induction, starting from (8.9). Thus the full ideal of linear and quadratic relations is stable.

Evaluation is equivariant, not merely the relation ideal. For a section a before the twist, write M_a(U_1,U_2) for its linear multiplication matrix. The transport identity (5.3) reads

where g_{\mathrm{coeff}} includes the field action and transport on the corresponding endomorphism factor. Scalar choices in transporting the universal line cancel on these endomorphisms. Using the weights of the two inputs and the degree-one homogeneity of the matrix polynomial, we obtain

This is exactly the action assigned to the twisted symbol. It proves equivariance successively through all the links.

Take invariant vector spaces under semilinear Galois descent. The enlarged formal space descends to an F-space containing U_F, and the full degree-one and degree-two pieces of the stable relation ideal descend to finite-dimensional spaces over F. Let Q_{\mathrm{new}} be the ideal generated by these descended spaces. Scalar extension commutes with the multiplication maps generating this ideal. Since the original relation ideal was generated in degrees one and two, extension to K recovers exactly that ideal. Equivariant evaluation takes these invariant forms into \Delta[x], with \Delta as in (8.8). They are mutually commuting.

The last output s'_M is invariant because \lambda_M=1; define t_i=s'_M. Combining (8.10) with (8.12) proves (8.4). All earlier equations are retained, so every old triangular identity continues to hold modulo Q_{\mathrm{new}}.

It remains to check the other induction conditions. Finiteness of the new relation algebra over the old one can be checked after the faithfully flat extension F\subset K, where it was proved link by link. In particular it remains finite over F[x]. Lemma 3.5 now preserves the geometric no-small-curve condition: the linear projection of the new characteristic set is finite and pulls back the old hyperplane bundle. The central variables still evaluate to themselves, and the formal space still contains U_F. Finally K is finitely generated over k, and so is its finite-group invariant field F. Thus the center is still finitely generated over the original \C, and all induction conditions have been verified.□

Proof of Theorem 1.1. Proposition 3.4 supplies the initial data with no triangular rows yet constructed. Apply Proposition 8.2 in the order i=n,n-1,\ldots,1. Every step retains all preceding identities as consequences of an ideal generated in degrees at most two. Thus a later, smaller value of a_i does not lose an earlier power identity. After finitely many steps we have the division algebra and commuting linear forms satisfying (2.1) for all i. Let A be the commutative graded algebra generated by all the evaluated formal forms. The center is finitely generated over \C, all multipliers in the triangular identities belong to A in the stated degrees, and all c_i are nonzero. Lemma 2.1 gives finiteness over k[x] and the full ordered left-\Delta basis. This proves every assertion of Theorem 1.1.□

Proof of Corollary 1.2. Apply Theorem 1.1 to the regular sequence contained in I. Proposition 2.2 gives one monomial ideal J containing all x_i^{a_i} and having the same Hilbert function as I in every degree.□

Proof of Corollary 1.3. If I=S_F, take J=S_F. Otherwise, first work over \C. Choose a linear coordinate change for which f_1,\ldots,f_c,x_{c+1},\ldots,x_n is regular. The images \overline f_1,\ldots,\overline f_c in \C[x_1,\ldots,x_c] form a full regular sequence. Corollary 1.2 applies to these images and, more generally, to every full regular sequence of the prescribed degrees in c variables and every homogeneous ideal containing it. This universal assertion is the premise of the Artinian reduction of Caviglia–Maclagan [9, Proposition 10], which gives the desired Hilbert-function comparison for every ideal containing the original sequence in n variables. The lexicographic embedding modulo P_F gives the stated lex-plus-powers choice; this reduction and its lex formulation are also recalled in [8, Section 4].

For general F, choose homogeneous generators of I and let F_0\subseteq F be the subfield generated over \Q by their coefficients and those of the f_i. Let I_0\subseteq F_0[x] be generated by those generators together with the f_i. Then I_0F[x]=I, and faithful flatness of F/F_0 descends regularity of the sequence to F_0[x]. Embed the finitely generated field F_0 into \C and apply the complex case after scalar extension. Define a monomial ideal over F_0, and then over F, using the same monomial exponents as the resulting complex ideal. Dimensions of graded pieces commute with both field extensions from F_0, so the Hilbert equalities descend to F. The same observation identifies the degreewise lex-plus-powers choice.□

Proof. If I=S_F, both sides vanish. Otherwise, since F is infinite, complete f_1,\ldots,f_c to a regular sequence by n-c linear forms. After a linear change of coordinates, the images \overline f_1,\ldots,\overline f_c in \overline S=F[x_1,\ldots,x_c] form a full regular sequence of the same degrees. The list of added linear forms is empty when c=n.

Corollary 1.3, applied in \overline S to every homogeneous ideal containing this sequence, supplies precisely the Artinian EGH premise of Caviglia–Sbarra [?, Theorem 4.4]. Their Clements–Lindström image of I/(f_1,\ldots,f_c) in S_F/P_F has preimage J in S_F. The base-independence identification used there gives local cohomology with support in \mathfrak m, and their Hilbert-series comparison is coefficientwise in the full internal \Z-grading for every cohomological degree i\ge0. This gives the displayed inequalities directly over F.□

Proof. First let m\ge0 and let \Gamma\subseteq\Omega fail to impose independent conditions on degree-m forms, so that H_\Gamma(m)<\deg\Gamma, where H_\Gamma(d)=\dim_F(S/I_\Gamma)_d, S=F[x_0,\ldots,x_r], and I_\Gamma is the saturated homogeneous ideal. Choose a linear form \ell nonvanishing on \Omega. It is a nonzerodivisor on the homogeneous coordinate rings of both \Omega and \Gamma. Thus the images of the defining quadrics form a regular sequence in S/(\ell)\cong F[x_1,\ldots,x_r], and, for A=S/(I_\Gamma,\ell),

Some A_e is therefore nonzero with e\ge m+1. Corollary 1.3 replaces A by a quotient by a monomial ideal containing every variable square, with the same Hilbert function. A surviving degree-e monomial is squarefree, and all its 2^e divisors survive. Consequently \deg\Gamma\ge2^e\ge2^{m+1}. This proves, in characteristic zero, the numerical assertion \mathrm{IV}_m of [?, Section 3].

Now suppose X does not contain \Omega and k\le r-1. Put Z=\Omega\cap X, and let \Gamma be residual to Z in \Omega. If I_Z and I_\Omega are their saturated homogeneous ideals, the residual scheme is defined by I_\Gamma=(I_\Omega:I_Z). Set m=r-k-1. Residual duality for the complete intersection of r quadrics gives

The equality is the modern Cayley–Bacharach theorem in [?, Section 3, p. 195]; its complementary degrees sum to \sum_{i=1}^r2-r-1=r-1. The strict inequality holds because the defining equation of X belongs to I_Z but not to I_\Omega. This theorem may be applied over an algebraic closure: flat field extension preserves lengths, Hilbert functions, residual schemes, and scheme containment. Thus \Gamma fails to impose independent degree-m conditions. The bound just proved and the residual degree identity give

For k=r, the same inequality merely says that a proper closed subscheme of \Omega has integer degree at most 2^r-1.□