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 (3) 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 \(q\,\mathop{\mathrm{Sym}}^{b-\deg q}U\) gives the constant-coefficient differential equation \[ q(\partial)w=0. \tag{6}\] 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 [19].

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 \[r(r-1)p^{r-2}h\,q(\nabla p) \pmod{p^{r-1}}.\] For a linear \(q\), it is \(rp^{r-1}h\,q(\nabla p)\pmod{p^r}\). Equation (6) 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 \(\mathcal O(\deg p-1)\). 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 \[w=c\prod_{\nu=1}^r p_\nu^{m_\nu}, \qquad m_\nu\ge2,\] 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 \[\frac{q(\partial)w}{w} = q\left(\sum_{\nu=1}^r\frac{m_\nu v_\nu}{p_\nu}\right) -\sum_{\nu=1}^r\frac{m_\nu q(v_\nu)}{p_\nu^2}.\] 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 \(\mathbb P(U)\), 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 \(\mathcal O_{\mathbb P^1}(b-1)\); 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 \(\mathop{\mathrm{Sym}}^b U\) 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 [20], 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 \((\mathbb P^{n-1})^\vee\times(\mathbb P^{n-1})^\vee\). 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 \(\mathbb C\), 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 \(L_1,\ldots,L_{v_0}\in\mathbb C[x]_1\) whose squares span \(\mathbb C[x]_2\). 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 \[y_{\rho j}^{\pm1}, \qquad 1\le\rho\le m,\quad 1\le j\le v_0,\] 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 \[k=\mathbb C(y_{\rho j}^2:1\le\rho\le m,\ 1\le j\le v_0).\] 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 \[s_\rho=\sum_{j=1}^{v_0}y_{\rho j}L_j(x).\] Forms from different rows commute, and their squares are \[s_\rho^2=q_\rho(x) :=\sum_{j=1}^{v_0}y_{\rho j}^2L_j(x)^2.\] 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 \[\mathcal C(Q)_\Omega\longrightarrow\mathbb P^{n-1}_\Omega, \qquad [x:s]\longmapsto[x],\] is defined everywhere and is finite. Its pullback of \(\mathcal O(1)\) is the ambient \(\mathcal O(1)\). 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 \[\deg T=d\deg Z.\] In particular \(d\le D\) and \(\deg Z\le D\). Choose a field of definition \(F_0\subseteq\Omega\) for \(Z\) as in Lemma 8, 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 \(\mathbb C\), 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 \[d=[\Omega(T):\Omega(Z)]\ge2^{m-d_*}\] 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. Inside \(\mathbb P^{b-1}\), with coordinates \(p_0,\ldots,p_{b-1}\), take the inverse image under coordinate-wise \(b\)-th powering of the line parameterized by \[[u:z]\longmapsto[z-u:z-\zeta u:\cdots:z-\zeta^{b-1}u].\] 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 \[(p_j/p_0)^b=\frac{1-\zeta^j s}{1-s},\qquad 1\le j<b.\] These \(b-1\) functions generate a free \((\mathbb Z/b)\)-submodule of \(\mathbb C(s)^\times/\mathbb C(s)^{\times b}\): 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 \(\widetilde G=\mu_b^b\rtimes(\mathbb Z/b)\). 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 \[t=s^b\] defines a degree-\(b^b\) map \(C'\to\mathbb P^1_t\). 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 \(\mathbb C[[x]]\) has a \(b\)-th root, so all local extensions are contained in \(\mathbb C((x^{1/b}))\). 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 \[ d^b=\frac{t(t-1)}{t-2}. \tag{17}\] 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 \(C'\to\mathbb P^1_t\) 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 \[2g_T-2=-2b+4(b-1)=2b-4,\qquad g(C)-1=|G|(g_T-1).\] For \(b=2\), this construction starts with \(C'=\mathbb P^1\) and produces a degree-four étale \(C_2\times C_2\)-cover between curves of genus one.

Pull \(\mathcal O(b)|_{C'}\) 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 \(\mathbb P^1_s\); hence \(\deg\mathcal O(b)|_{C'}=b^{b-1}\). Since \(C\to C'\) has degree \(b\), we have \(\deg L=b^b=e\).

The standard \(\widetilde G\)-linearization on \(\mathcal O(b)\) 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 (14), and their product gives \[v^b=\prod_{j=0}^{b-1}(z-\zeta^ju)=z^b-u^b.\] If \(u=v=0\), this identity gives \(z=0\), which was excluded. Thus \(u,v\) define a base-point-free pencil. Its morphism \(\pi:C\to\mathbb P^1\) satisfies \(L=\pi^*\mathcal O(1)\); since \(\deg L=e>0\), the morphism is finite of degree \(e\). ◻

Proof. Let \(g=g(C)\). For generic \(A\in\mathop{\mathrm{Pic}}^0(C)\), the line bundle \(A L^{g_T-1}\) is generic of degree \[e(g_T-1)=g-1.\] The effective locus in \(\mathop{\mathrm{Pic}}^{g-1}(C)\) is the image of \(\mathop{\mathrm{Sym}}^{g-1}C\), 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 \(\mathop{\mathrm{Pic}}^0(C)\), is a nonempty \(G\)-stable open subset, because \(L\) is linearized.

Fix \(A\) in that open set, and put \[\mathcal V=\pi_*(A L^{g_T}).\] This is a rank-\(e\) vector bundle on \(\mathbb P^1\): the map \(\pi\) is finite, and its pushforward is torsion-free on a smooth curve. Projection formula and the preceding vanishing give \(H^*(\mathbb P^1,\mathcal V(-1))=0\). For every geometric point \(q\in\mathbb P^1\), the exact sequence \[0\longrightarrow\mathcal V(-q)\longrightarrow\mathcal V \longrightarrow\mathcal V|_q\longrightarrow0\] therefore identifies \(H^0(\mathbb P^1,\mathcal V)\) with \(\mathcal V|_q\). It follows that the evaluation map \[H^0(\mathbb P^1,\mathcal V)\otimes\mathcal O_{\mathbb P^1} \longrightarrow\mathcal V\] 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 \(\mathcal O(j)\), taking global sections, and using projection formula proves (15), with its indicated multiplication map.

Let \[\mathcal M_A=\bigoplus_{j\ge0}H^0(C,A L^{g_T+j}).\] Equation (15) 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 \[M_a(U,V)=A_aU+B_aV,\qquad A_a,B_a\in\mathop{\mathrm{End}}_\kappa(E_A).\] Multiplication of sections is commutative, so these matrix polynomials commute as elements of \(\mathop{\mathrm{End}}_\kappa(E_A)[U,V]\). 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 (14). 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 \(\mathop{\mathrm{Pic}}^0(C)\times C\). 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 \(A\) [33]. 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 \[m_a:E_A\longrightarrow E_A\otimes\langle u,v\rangle\] and using \(g_*u=\chi(g)u\), \(g_*v=\psi(g)v\) gives (16). On the generic parameter field this is semilinear transport of the coefficients. ◻

Proof. Suppose an element \(g\ne1\), of order \(r\), fixes \([V]\in\mathop{\mathrm{Pic}}^1(C)\). 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 \(\mathbb Z/b\) by exponentiation. Compose the monodromy of the cover with \(\xi\) to obtain \(\bar\alpha\in H^1(T;\mathbb Z/b)\). Since \(H_1(T;\mathbb Z)\) is free abelian, \(\bar\alpha\) lifts to a class \(\alpha\in H^1(T;\mathbb Z)\). Represent \(\alpha\) by a harmonic real one-form with integral periods. The form \[\omega=\frac1b q^*\alpha\] 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 \(\mathbb Z\) 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 \[f_C(gc)=\xi(g)f_C(c).\] Its differential, expressed as a real one-form after local lifting to \(\mathbb R\), is the \(G\)-invariant form \(\omega\).

The Abel map \[a:C\longrightarrow\mathop{\mathrm{Pic}}^1(C),\qquad c\longmapsto\mathcal O(c),\] 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 \(\mathop{\mathrm{Pic}}^1(C)\) with harmonic real one-forms on \(C\), and induces an isomorphism on integral first cohomology; see [27]. 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 \(\mathop{\mathrm{Pic}}^1(C)\). 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 13. Lemma 14 supplies parts (i) and (ii), and Lemma 15 supplies part (iii). Apply Lemmas 16 and 17 to the étale cover \(C\to T_0\) to obtain part (iv). ◻

Proof. Choose finitely many rational sections spanning \(\mathcal F_\eta\), and let \(\mathcal F_0\) be the coherent subsheaf of rational sections generated by them and their multiples by a \(k\)-basis of \(\Delta_0\). For each \(g\), use (22) to view \(J_g\otimes g_*\mathcal F_0\) as a coherent subsheaf of rational sections. Its image is coherent because \(T\) is noetherian. Let \(\mathcal F\) be the sum of these finitely many images. Corrected transport permutes the summands by the composition law in (22); hence it preserves \(\mathcal F\). 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, \[\chi(A)=1-g_C,\qquad \chi(A\otimes V)=\chi(V)=2-g_C.\] A scalar acting on a line inside a determinant of cohomology has exponent equal to its Euler characteristic. Thus the corresponding factor of \(\mathcal D\) has scalar weights \[\bigl(-\chi(A\otimes V)+\chi(A), -\chi(A\otimes V)+\chi(V)\bigr)=(-1,0)\] in \(A,V\). The first weight cancels the weight one of \(\mathcal F\), and the second leaves no dependence on identifications of \(V\). Local choices of line identifications therefore give the same transport on \(\mathcal F\otimes\mathcal D\). 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 \(\mathcal F\) and \(\mathcal D\), 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 \(\mathcal F\) 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 \[c_1\!\left(\bigl(\det Rq_*(A\otimes V)\bigr)^{-1} \otimes\det Rq_*A\otimes\det Rq_*V\right) =-q_*(av).\] 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 \(H^1(\mathop{\mathrm{Pic}}^0(C),\mathbb Z)\) and \(H^1(\mathop{\mathrm{Pic}}^1(C),\mathbb Z)\). Tensoring by \(\mathcal O_C(-c_0)\), for the chosen curve point \(c_0\), identifies the torsor \(\mathop{\mathrm{Pic}}^1(C)\) with \(\mathop{\mathrm{Pic}}^0(C)\); 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 \(H^1(C,\mathbb Z)\) 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 [28].

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 \[\int_{\mathop{\mathrm{Pic}}^0(C)\times\mathop{\mathrm{Pic}}^1(C)}\frac{\theta^d}{d!} =\pm\det\Lambda=\pm1.\] The factors for distinct curves form separate blocks, so the same unit calculation holds on \(Z\). Each Picard tangent bundle is trivial, giving \(\mathop{\mathrm{td}}(Z)=1\). To obtain top degree on \(P\) from \(\exp(c_1(\mathcal D))\), one must already use top degree on \(B\), because every mixed term has degree one on each side. Thus every positive-degree term of \(\mathop{\mathrm{ch}}(\gamma)\) contributes zero. Only its rank remains. ◻

Proof. The vector bundle \(\delta_*\mathcal O_{T'}\) has rank \(a\). Put \(d=[\delta_*\mathcal O_{T'}]-a\in K^0(Q)\). 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 [33]; 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 \(G_0(Q)\), \[(a+d)d=0.\] Over \(\mathbb Q\), the operator \(a+d\) is invertible by a finite nilpotent expansion, hence \(d\) acts by zero. Euler characteristic takes values in \(\mathbb Z\), so it kills torsion and \(\chi(Q,d\alpha)=0\). Finite pushforward and the projection formula now give the claimed equality. ◻

Proof of Proposition 19. The construction and the first identity have been established above. By Proposition 13, \(H\) 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 [33], 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 [33]. The same holds for \(\beta\otimes\mathcal O_X(l)\), since \(\mathcal O_X(1)\) 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 23 multiplies its Euler characteristic by \(|H|=e_1\) on passing upstairs. Applying this term by term to the coherent class gives \[\chi(Z\times X,\beta\otimes\mathcal O_X(l))\in e_0^2e_1\mathbb Z,\] which proves (21). Finally, field extension preserves coherent cohomology dimensions by flat base change [33], even for a complex embedding that does not fix the original constants. Over \(\mathbb C\), 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 \(S^{2h+1}\to\mathbb P^h(\mathbb C)\) 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 \((\mathbb T/H')\times\mathbb P^h(\mathbb C)\) 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 \(\mathbb T\times\mathbb P^h(\mathbb C)\). 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 17 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 \(i:V\hookrightarrow\mathbb P^a(\mathbb C)\), with complex normal bundle \(\nu\). The complex \(K\)-theory Thom class, followed by extension from a tubular neighborhood, defines \[i_!:K^0(V\times\mathbb P^h(\mathbb C)) \longrightarrow K^0(\mathbb P^a(\mathbb C)\times\mathbb P^h(\mathbb C)).\] 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 \[\prod_{\alpha=1}^c\frac{1-e^{-x_\alpha}}{x_\alpha} =\mathop{\mathrm{td}}(\nu)^{-1}.\] For completeness, this identity follows on the zero section from \(\mathop{\mathrm{ch}}(\lambda_{-1}(\nu^\vee))=\prod_\alpha(1-e^{-x_\alpha})\) 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 \[\mathop{\mathrm{ch}}(i_!\xi)=i_*\bigl(\mathop{\mathrm{ch}}(\xi)\mathop{\mathrm{td}}(\nu)^{-1}\bigr).\] The exact tangent-normal sequence now implies \[ \int_{\mathbb P^a}\mathop{\mathrm{ch}}(i_!\xi)\mathop{\mathrm{td}}(T_{\mathbb P^a}) =\int_V\mathop{\mathrm{ch}}(\xi)\mathop{\mathrm{td}}(T_V). \tag{27}\]

The integral projective bundle formula identifies \[K^0(\mathbb P^a(\mathbb C)\times\mathbb P^h(\mathbb C)) =\mathbb Z[v,u]/(v^{a+1},u^{h+1}),\qquad v=1-[\mathcal O_{\mathbb P^a}(-1)].\] Write \(i_!\xi=\sum_{i=0}^a\sum_{j=0}^h c_{ij}v^iu^j\), with \(c_{ij}\in\mathbb Z\). The number \[\int_{\mathbb P^a}\mathop{\mathrm{ch}}(v^i)\mathop{\mathrm{td}}(T_{\mathbb P^a}) =\chi\bigl(\mathbb P^a,(1-[\mathcal O(-1)])^i\bigr)\] is an integer by Hirzebruch–Riemann–Roch, applied additively to the finite sum of line-bundle classes \(v^i\). Thus the left side of (27) 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 (25): if \(H=c_1(\mathcal O(1))\), then \(\mathop{\mathrm{ch}}(u)=1-e^{-H}\) has leading term \(H\), so these \(h+1\) Chern characters are linearly independent over \(\mathbb Q\). The projective bundle and Thom constructions used here are standard complex \(K\)-theory, with the Koszul convention for the Thom class specified above [23]; for the Riemann–Roch formula see [5]. ◻

Proof. Pull back the topological class of \(\beta'|_{Z\times X_{\mathrm{sm}}}\) 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 \((Z\times X_{\mathrm{sm}})/H\). 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 \(\tau\in K^0(V\times\mathbb P^h(\mathbb C))\) 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 25 gives \[ \int_Z\mathop{\mathrm{ch}}(\sigma)\mathop{\mathrm{td}}(T_Z) =e_1\mathop{\mathrm{ch}}(I_V(\tau)) =\mathop{\mathrm{ch}}\!\left(\sum_{j=0}^h q_j u^j\right), \qquad q_j\in e_1\mathbb Z. \tag{29}\]

Nonequivariantly, Lemma 24 makes \(\gamma\) homotopic to \((z,t)\mapsto(z,\Phi_0(t))\). The equality (29) therefore computes the same fiber integral for this latter pullback of \(\beta'\). Evaluate it at any point of \(\mathbb P^h(\mathbb C)\). Only \(q_0\) remains on the right, whereas (20) 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 (21). A smooth linear section of \(X\) retains that divisibility and can also be compared with the projective-space test. Put \(d=\dim X\). Theorem 12 gives \(\operatorname{codim}_X(X\setminus X_{\mathrm{sm}})>h\). 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 \[f:D_0\longrightarrow\mathbb P^h(\mathbb C), \qquad f^*\mathcal O_{\mathbb P^h}(-1)\simeq\mathcal O_{D_0}(-1).\] The inclusion \(j:D_0\hookrightarrow X_{\mathrm{sm}}\) 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 24, 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 (29) back by \(f\) now identifies the fiber integral on \(Z\times D_0\) with \[\mathop{\mathrm{ch}}\!\left( \sum_{j=0}^h q_j(1-[\mathcal O_{D_0}(-1)])^j \right). \tag{\(\ast\)}\] Define \[p_{D_0}(l)= \chi\!\left(Z\times D_0, \beta'|_{Z\times D_0}\otimes\mathcal O_{D_0}(l)\right),\] where restriction of a coherent class is derived. Hirzebruch–Riemann–Roch on the smooth projective product, followed by \((\ast)\), gives \[ p_{D_0}(l)= \sum_{j=0}^h q_j\, \chi\!\left(D_0, (1-[\mathcal O_{D_0}(-1)])^j\otimes\mathcal O_{D_0}(l)\right). \tag{30}\]

It remains to retain the original divisibility when passing to this section. Recall from (21) the integers \[p(l)=\chi(Z\times X,\beta\otimes\mathcal O_X(l))/e_0 \in e_0e_1\mathbb Z.\] 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 \[ p_{D_0}(l) =\sum_{a=0}^{d-h}(-1)^a\binom{d-h}{a}p(l-a) \in e_0e_1\mathbb Z. \tag{31}\] 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 (31) are the original integer Euler characteristics. Combining (30) and (31) proves (28). ◻

Proof. We successively construct integer polynomials \(Q_i\), for \(0\le i\le v\), with \(Q_0=Q\), such that \[Q_i(0)=Q(0)/p^i,\qquad A Q_i=0\pmod{(p^{v-i},u^{h-i\nu+1})} \quad(0\le i<v).\] 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 \(\mathbb Z/p\), so the displayed congruence forces every coefficient of \(Q_i\) through degree \[m_i=h-(i+1)\nu\] 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 \[Q_i-pQ_{i+1}\in u^{m_i+1}\mathbb Z[u],\] restricting the old congruence to degrees at most \(m_i\) gives \[p A Q_{i+1}=0\pmod{(p^{v-i},u^{m_i+1})}.\] 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 18. Use \(D_0\), \(q_0,\ldots,q_h\), and \(Q_*\) just constructed for the chosen module. If \(e_0=1\), then \(q_0=\pm r\in e_1\mathbb Z\) already gives the required divisibility, so assume \(e_0>1\). The Hilbert series of the coordinate ring of the complete intersection \(D_0\) is \[\frac{(1-t^b)^s}{(1-t)^{h+s+1}}.\] Euler characteristics of sufficiently positive twists agree with Hilbert functions. Moreover, multiplication by \((1-[\mathcal O(-1)])^j\) 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 (30) that \[ \sum_{l\ge0}\frac{p_{D_0}(l)}{e_1}t^l =Q_*(1-t)\frac{(1-t^b)^s}{(1-t)^{h+s+1}}+R(t), \qquad R(t)\in\mathbb Z[t]. \tag{32}\] By (31), every coefficient on the left is divisible by \(e_0\).

Multiply (32) 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 \[Q_*(u)\bigl(1-(1-u)^b\bigr)^s =0\pmod{(e_0,u^{h+s+1})}.\] Define \[A_b(u)=\frac{1-(1-u)^b}{u}\in\mathbb Z[u].\] Cancelling the common monomial \(u^s\) yields \[ A_b(u)^sQ_*(u)=0\pmod{(e_0,u^{h+1})}. \tag{33}\] This is coefficientwise cancellation of a monomial, valid over \(\mathbb Z/e_0\) even when \(e_0\) is composite. When \(s=0\), the factor is \(1\), and the constant coefficient of (33) 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 \(\mathbb Z/p\), \[1-(1-u)^b =1-(1-u^{p^a})^c =c\,u^{p^a}+\text{terms of higher degree}.\] 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 18 gives \(h\ge v\nu\). Apply Lemma 27 to (33), 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 \[e_0\mid Q_*(0),\qquad e_0e_1\mid q_0=\pm r.\] 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 of Corollary 3. If \(I=S_F\), take \(J=S_F\). Otherwise, first work over \(\mathbb C\). 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 \(\mathbb C[x_1,\ldots,x_c]\) form a full regular sequence. Corollary 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], 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].

For general \(F\), choose homogeneous generators of \(I\) and let \(F_0\subseteq F\) be the subfield generated over \(\mathbb Q\) 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 \(\mathbb C\) 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 3, applied in \(\overline S\) to every homogeneous ideal containing this sequence, supplies precisely the Artinian EGH premise of Caviglia–Sbarra [10]. 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 \(\mathbb Z\)-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 \(A=S/(I_\Gamma,\ell)\), \[\deg\Gamma=\sum_{j\ge0}\dim_F A_j, \qquad H_\Gamma(m)=\sum_{j=0}^m\dim_F A_j.\] Some \(A_e\) is therefore nonzero with \(e\ge m+1\). Corollary 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 [17].

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 \[\deg\Gamma-H_\Gamma(m) =\dim_F(I_Z/I_\Omega)_k>0.\] The equality is the modern Cayley–Bacharach theorem in [17]; 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 \[\deg(\Omega\cap X)=2^r-\deg\Gamma \le 2^r-2^{r-k}.\] For \(k=r\), the same inequality merely says that a proper closed subscheme of \(\Omega\) has integer degree at most \(2^r-1\). ◻