Proof. First take the locus where both the source and its cone are sheaves in degree zero on classical fibers, with the prescribed Hilbert polynomials. We explain the family conditions in this description. For a perfect complex on \(X\times Y\), the locus where it has relative Tor-amplitude zero over \(Y\) is open. Locally one represents the complex by vector bundles. At a fiber on which it is exact away from degree zero, the positive end can be split off. At the negative end, fiberwise injectivity between modules flat over the base implies injectivity with base-flat cokernel, by the local criterion of flatness. Successively replacing the negative end by these cokernels gives a sheaf flat over the base near that fiber. Openness of flatness gives an open neighborhood in \(X\times Y\); properness of \(X\times Y\to Y\) then gives the required open neighborhood in \(Y\). Applying this to the source and cone proves the asserted openness.

Conversely, let \(V_Y\twoheadrightarrow Q\) be a flat quotient family over a noetherian base \(Y\). Its kernel \(S'\) is flat over \(Y\). Both \(Q\) and \(S'\) are perfect on \(X\times Y\), even if \(Y\) is singular. Indeed, resolve either sheaf locally by finite-rank free modules. Every successive syzygy remains flat over \(Y\). On a surface fiber the local rings are regular of dimension at most two, so after two steps the syzygy is locally free on that fiber. A finitely presented, base-flat module with locally free fibers is locally free near the point, again by the local flatness criterion. This truncates the resolution. This argument suffices over noetherian bases, and finite presentation extends the moduli identification to arbitrary classical test schemes. Flat approximation and the perfection criteria in [33] also give this last passage: a finitely presented base-flat module is pseudo-coherent, and the regular surface fibers give absolute Tor-amplitude contained in \([-2,0]\).

On our open locus the cone sequence is consequently an exact sequence of sheaves with flat quotient. An automorphism \(a\) of its source satisfying \(i\circ a=i\) is the identity, since \(i\) is injective. Negative Ext groups between these sheaves vanish as well, so no higher automorphisms remain in the classical moduli problem. The truncation is therefore the Quot scheme, including its functor on nonreduced bases.

Here is a tangent calculation that also records the relevant maps. Allow both vertices of an arrow \(F\to G\) to vary. Its derived endomorphism complex is \[\mathop{\mathrm{fib}}\left\{ R\!\mathop{\mathrm{Hom}}(F,F)\oplus R\!\mathop{\mathrm{Hom}}(G,G) \xrightarrow{(a,b)\mapsto f a-b f}R\!\mathop{\mathrm{Hom}}(F,G) \right\}.\] One can compute this complex with the two-term resolution of a one-arrow diagram by diagrams induced from its vertices. More explicitly, if \(P_1=(\mathcal O_X\xrightarrow{\mathop{\mathrm{id}}}\mathcal O_X)\) and \(P_2=(0\to\mathcal O_X)\), the resolution has terms \(P_2\otimes F\) and \((P_1\otimes F)\oplus(P_2\otimes G)\). Its map at the target vertex sends \(s\) to \((s,-f(s))\). Thus perfect vertices give a perfect diagram. The one-arrow path algebra is finite dimensional and has this finite projective resolution; the usual perfect-object tangent formula applies to its diagrams on \(X\). The tangent of their moduli is the displayed endomorphism complex shifted by \([1]\).

Fixing the target takes the tangent fiber over \(R\!\mathop{\mathrm{Hom}}(V,V)[1]\). The result is the first cone in (4); applying \(R\!\mathop{\mathrm{Hom}}(S',-)\) to (3) gives the second identification. In particular, forgetting the arrow and remembering its source has differential the boundary \[ R\!\mathop{\mathrm{Hom}}(S',Q)\longrightarrow R\!\mathop{\mathrm{Hom}}(S',S')[1]. \tag{5}\]

These are identifications of perfect complexes in families. For completeness, perfection and functoriality can also be checked before fixing a vertex by taking pointed loops. The relative loop stack at a perfect family is its self-equivalence stack, open in the linear mapping stack of its perfect endomorphism complex. Its cotangent at the identity is the dual endomorphism complex. The cotangent triangle for the loop fiber product identifies this with the pulled-back moduli cotangent shifted by \([1]\). Forgetting vertices and taking direct sums act on these loop tangents by projection and block-diagonal sum. Taking the fixed-target fiber preserves these identifications. This proves the claimed functoriality, including the specified boundary.

Finally, at a geometric point \(S'\) is torsion-free and \(Q\) is torsion. Surface Serre duality gives \[\mathop{\mathrm{Ext}}^2(S',Q)^\vee=\mathop{\mathrm{Hom}}(Q,S'(K))=0.\] Negative Ext groups vanish for sheaves; groups above degree two vanish on a smooth surface. Consequently the derived tangent has geometric fibers in degrees \([0,1]\). For a perfect complex the geometric-fiber criterion is precisely the criterion for Tor-amplitude in this range. ◻

Proof. The cotangent map is induced by \(j\), so its functoriality is that of cotangent complexes. The comparison statement is [31]; the following argument gives it directly in the present scheme-truncation situation. Work on an affine open \(x:\operatorname{Spec}R\to Z\), and write \(E=x^*E_Z\). For every ordinary \(R\)-module \(N\), split square-zero extensions and agreement of \(Z\) and \(\mathfrak Z\) on classical rings give an isomorphism \[\mathop{\mathrm{Hom}}(L_R,N)\xrightarrow{\sim}\mathop{\mathrm{Hom}}(E,N).\] There is also an injection \[\mathop{\mathrm{Hom}}(L_R,N[1])\longrightarrow\mathop{\mathrm{Hom}}(E,N[1]).\] Indeed, let a derivation class \(\delta:R\to R\oplus N[1]\) map to zero on the right. The two points of \(\mathfrak Z(R\oplus N[1])\) obtained from \(x\) by \(\delta\) and by the zero derivation are then homotopic, over \(x\). Infinitesimal cartesianness of a locally geometric derived stack produces a lift of \(x\) over the homotopy pullback \[R'=R\mathbin{\times^h_{R\oplus N[1]}}R,\] whose two maps are \(\delta\) and zero. This is the classical square-zero extension of \(R\) by \(N\) classified by \(\delta\). The lift is a classical point of \(Z\) and factors through the same open \(\operatorname{Spec}R\); it therefore gives a splitting of \(R'\to R\). The extension class \(\delta\) is zero, proving injectivity. Put \(C=\mathop{\mathrm{Cone}}(E\to L_R)\). The two tests, together with connectivity, give \(\mathop{\mathrm{Hom}}(C,N)=\mathop{\mathrm{Hom}}(C,N[1])=0\) for every \(N\). Since \(C\in D^{\leq0}\), the first equality implies \(h^0(C)=0\), and the second then implies \(h^{-1}(C)=0\). Thus \(C\in D^{\leq-2}\). Together with the given perfect amplitude, this is the definition of a perfect obstruction theory in [5]. ◻

Proof. There is nothing to prove when \(H^2(\mathcal O_X)=0\). Otherwise Serre duality and the exact sequence show that a section \(\omega\in H^0(K)\) annihilates the image precisely when \(\omega\mathop{\mathrm{id}}_F\in\mathop{\mathrm{Hom}}(F,F(K))\) lifts to a map \[a:V_0\longrightarrow F(K),\qquad a|_F=\omega\mathop{\mathrm{id}}_F.\] Writing \(i:F\to V_0\), the maps \(i(K)a\) and \(\omega\mathop{\mathrm{id}}_{V_0}\) agree where \(F=V_0\), a dense open because \(Q_0\) has rank zero. They are maps between vector bundles, so they agree everywhere. Composing with the quotient to \(Q_0(K)\) gives \(\omega Q_0=0\).

Suppose now that \(\omega\ne0\), and write its effective divisor as \(C=(\omega)\). The quotient \(Q_0\) factors through \(V_0|_C\). At the generic point of any integral curve \(C_j\subset X\) the local ring is a DVR. If \(\omega\) has order \(a_j\) there, a quotient of \((\mathcal O_{X,C_j}/(\omega))^r\) has length at most \(ra_j\). The divisorial cycle of a torsion sheaf is the sum of these generic lengths, and represents its first Chern class: the elementary-divisor decomposition over a DVR identifies this length with the order of the determinant. Therefore \[[Q_0]_{\mathrm{div}}\leq rC \quad\hbox{componentwise},\qquad Kc_1(Q_0)\leq rK^2.\] The latter inequality uses ampleness of \(K\). It contradicts the hypothesis. The argument allows embedded points and nonreduced curve support, since it uses only generic lengths. Thus no nonzero \(\omega\) annihilates the image, proving surjectivity. ◻

Proof. The cotangent triangle of the derived fiber gives (8) as an identity of perfect complexes. Both its input and target have Tor-amplitude \([0,1]\); the input has this amplitude by Lemma 6. Their fiber has no negative cohomology and has no cohomology above degree two. Its degree-two cohomology vanishes exactly when \[\mathop{\mathrm{Ext}}^1(S',Q)\longrightarrow H^2(\mathcal O_X)\] is surjective at every geometric point. In setting (i) its target is zero. In setting (ii), Lemma 8 applies with \(r=2\), since \(KD=3K^2>2K^2\). The geometric-fiber criterion for perfect complexes now gives amplitude \([0,1]\). Lemma 7 therefore supplies the specified obstruction morphism.

Postcomposition with the given \(G\)-action on \(V\) defines an action on the arrow stack, leaving the source determinant map invariant. It thus acts on the fiber (7), and naturality of the cotangent map makes its obstruction theory equivariant. Equivalently, one can construct the truncation map on the quotient stacks relative to \(BG\), obtaining exactly this equivariant complex and morphism.

The Euler characteristic of the determinant target is \(q-p_g=0\). If \(q_2=\int_X\mathop{\mathrm{ch}}_2(Q)\), Riemann–Roch gives \[\chi(V,Q)=2q_2-(K+L_1+L_2)D,\qquad \chi(Q,Q)=-D^2.\] For the second identity, the only degree-four term in \(\mathop{\mathrm{ch}}(Q^\vee)\mathop{\mathrm{ch}}(Q)\mathop{\mathrm{td}}(X)\) is \(-D^2\), since \(Q\) has rank zero; here \(Q^\vee\) is the derived dual. Consequently \[\begin{align*} v&=\chi(S',Q)-(q-p_g) =\chi(V,Q)-\chi(Q,Q)\\ &=2\left(n+L_1D-\frac{D^2}{2}\right) -(K+L_1+L_2)D+D^2 =2n+(A-K)D. \end{align*}\] ◻

Proof. The Grassmannian construction of the projective Quot scheme is equivariant for the action on \(V\). The determinant fiber is closed and invariant, so restriction and a Plücker embedding give an equivariant embedding \(B% % \mathrel{% \BeginAccSupp{method=hex,unicode,space=false,pdfliteral=direct,ActualText=21AA}% \lhook\joinrel\rightarrow% \EndAccSupp{pdfliteral=direct}% }% % M\) with \(M\) smooth projective. In particular \(B\) has a linearized ample bundle \(\mathcal O_B(1)\).

Here is the global resolution argument for \(E=E_B\). For \(m\) sufficiently large, \(h^0(E)(m)\) is generated by sections and \(H^a(h^{-1}(E)(m))=0\) for every \(a>0\). The hypercohomology sequence then lifts its sections to \(\mathbb H^0(E(m))\). These are finite-dimensional torus representations; linear reductivity allows an equivariant choice of lifts. Evaluation gives an equivariant bundle \(V_0=\mathcal O_B(-m)\otimes W_0\) and a map \(V_0\to E\) surjective on \(h^0\). Its derived fiber \(V_1\) is perfect. At every geometric point \(b\), the same map is surjective on \(h^0\), since \(h^0(E\otimes^L k(b))=h^0(E)\otimes k(b)\). Its fiber has cohomology only in degree zero. Thus \(V_1\) is a vector bundle, and \(E\simeq[V_1\to V_0]\) in degrees \([-1,0]\).

Let \(J\) be the conormal sheaf of \(B% % \mathrel{% \BeginAccSupp{method=hex,unicode,space=false,pdfliteral=direct,ActualText=21AA}% \lhook\joinrel\rightarrow% \EndAccSupp{pdfliteral=direct}% }% % M\), and set \(\Omega=\Omega_M|_B\). The truncated cotangent complex \(\tau_{\geq-1}L_B\) is represented by \(C=[J\to\Omega]\) in degrees \([-1,0]\). Increase \(m\) above so that \(\mathop{\mathrm{Ext}}^1(V_0,J)=0\) as well; this is Serre vanishing for \(J(m)\). The triangle \(J\to\Omega\to C\to J[1]\) then lifts \(V_0\to E\to C\) to \(V_0\to\Omega\), equivariantly. The map \(E\to C\) includes a nullhomotopy on its restriction from \(V_1\). Together with the chosen lift, that nullhomotopy gives a map \(V_1\to J=\mathop{\mathrm{fib}}(\Omega\to C)\). These are ordinary sheaf maps and give a commuting square \[\begin{array}{ccc} V_1&\longrightarrow&V_0\\ \downarrow&&\downarrow\\ J&\longrightarrow&\Omega \end{array}\] representing the original map to \(C\). All choices can be made in the equivariant category, since taking torus invariants is exact. ◻

Proof. Direct sum is an actual morphism of perfect-arrow moduli problems. The determinant of a direct sum is the tensor product of determinants, compatibly with normalization. Consequently direct sum induces a derived map \(\mathfrak k:\mathfrak W\to\mathfrak B\) lifting \(k\). This map is coherently equivariant. Over the classifying stack \(BG\), tensor the source and target of each summand arrow with the same character line of weight \(+1\) or \(-1\). Normalizing the determinant cancels each base character line against its restriction at \(o\). These canonical cancellations respect tensor products and the chosen homotopy in the determinant fiber. Thus direct sum defines \(BG\times\mathfrak W\to[\mathfrak B/G]\) over \(BG\), with trivial \(G\)-action on \(\mathfrak W\). Relative cotangents over \(BG\) give its equivariant cotangent morphism. Its cotangent morphism, restricted to truncations, gives \[Lk^*E_B\longrightarrow E_W\longrightarrow L_W.\] By functoriality of the truncation comparison, this composite equals \(Lk^*E_B\to Lk^*L_B\to L_W\).

To identify this map, use the tangent formula for arrows. On \(W\), \[R\!\mathop{\mathrm{Hom}}(S',Q)=\bigoplus_{i,j}R\!\mathop{\mathrm{Hom}}(S'_i,Q_j).\] Direct sum maps the two input tangents to the diagonal blocks. Trace vanishes on off-diagonal blocks, and the based Picard tangent has weight zero. To display the comparison, abbreviate these complexes by \(\mathcal A_0=\bigoplus_iR\!\mathop{\mathrm{Hom}}(S'_i,Q_i)\), \(\mathcal A=\bigoplus_{i,j}R\!\mathop{\mathrm{Hom}}(S'_i,Q_j)\), and \(\mathcal P=R\Gamma(\mathcal I_o)[1]\otimes\mathcal O_W\). The derivative gives the morphism of fiber triangles \[\begin{array}{ccccc} E_W^\vee&\longrightarrow&\mathcal A_0&\longrightarrow&\mathcal P\\ \downarrow&&\downarrow&&\Vert\\ Lk^*E_B^\vee&\longrightarrow&\mathcal A&\longrightarrow&\mathcal P, \end{array}\] whose middle map is diagonal inclusion. After taking weight zero, the middle and right maps are identities. The left map is therefore an isomorphism of perfect complexes in families. The remaining summands are exactly (12).

In particular (11) has Tor-amplitude \([0,1]\), as a direct summand of \(Lk^*E_B^\vee\). Lemma 7 makes \(E_W\to L_W\) a perfect obstruction theory. Dualizing the fiber-triangle comparison gives (10). The compatibility already established shows that this is the fixed obstruction morphism used in [18], and not just an identity of virtual tangent classes. ◻

Proof. Untwist by \(L_R\), and write \(I\subset\mathcal O_{X\times R}\) for the kernel. It is flat over \(R\), because the source and quotient are flat. Every geometric fiber \(I_u\) is torsion-free of rank one on a smooth surface and has projective dimension at most one. Locally choose a surjection from a vector bundle \(G\) onto \(I\). Its kernel \(E\) is flat over \(R\) and has locally free geometric fibers, so is locally free by the local criterion of flatness. Thus \[0\longrightarrow E\longrightarrow G\longrightarrow I\longrightarrow0\] is a resolution by vector bundles, and remains a resolution after any base change.

Set \(M=\det G\otimes(\det E)^{-1}=\det I\). If \(E\) has rank \(e\), wedging a section of \(G\) with the image of \(\bigwedge^eE\) gives the maximal-minor map \(G\to M\); it kills \(E\) and induces \(I\to M\). On the locus where \(I\) is invertible it is the canonical determinant identification. These local maps glue. To justify this last assertion over a nonreduced base, the fiber criterion for local freeness shows that the closed noninvertible locus has finite fibers over \(R\); properness makes this locus finite over \(R\). Any closed support with finite fibers in the smooth relative surface has grade at least two locally: choose two elements of its ideal which form a regular sequence on the surface fiber and lift them; successive applications of the local flatness criterion show that they remain a regular sequence in the total space and that the successive quotients are base-flat. Powers of these elements annihilate any coherent sheaf with that support; the regular-sequence calculation therefore makes its sheaf \(\mathcal{H}om\) and \(\mathcal Ext^1\) into a line bundle vanish. In particular a section of a line bundle supported there is zero. Hence two local minor maps which agree on the invertible locus agree everywhere.

On each geometric fiber the resulting map \(I_u\to M_u\) is the usual inclusion of a rank-one torsion-free sheaf into its determinant line: it is injective, and its cokernel has finite support. Fiberwise injectivity between base-flat modules now implies that \(I\to M\) is injective with base-flat cokernel. After twisting by \(M^{-1}\) it is an ideal \(J\subset\mathcal O_{X\times R}\), and \(\mathcal O/J\) is base-flat with finite fibers. Properness of \(X\) makes its support finite over \(R\). It therefore defines a finite flat subscheme \(Z\), with \(I=M\mathcal I_Z\).

The original inclusion \(I% % \mathrel{% \BeginAccSupp{method=hex,unicode,space=false,pdfliteral=direct,ActualText=21AA}% \lhook\joinrel\rightarrow% \EndAccSupp{pdfliteral=direct}% }% % \mathcal O_{X\times R}\) extends uniquely to a map \(M\to\mathcal O_{X\times R}\). Indeed the grade-two observation, applied to the support of \(M/I=M\mathcal O_Z\), gives \[\mathcal{H}om(M\mathcal O_Z,\mathcal O_{X\times R})= \mathcal Ext^1(M\mathcal O_Z,\mathcal O_{X\times R})=0.\] Applying \(\mathcal{H}om(-,\mathcal O_{X\times R})\) to \(0\to I\to M\to M\mathcal O_Z\to0\) proves the extension and uniqueness. Its restriction to every surface fiber is nonzero, hence injective. The same flatness criterion makes it an injection with base-flat cokernel. It is therefore a relative effective Cartier divisor inclusion, \(M=\mathcal O(-\mathcal C)\).

The determinant resolutions, minor maps, and short exact sequences all survive base change, the last because their quotients are base-flat. The regular sequences used for uniqueness also survive base change, because their successive quotients are base-flat. Thus uniqueness identifies the resulting factorization after every base change, including a nonnoetherian one. Conversely, multiplying a relative Cartier ideal by a finite flat point ideal gives a flat quotient: its quotient is an extension of \(\mathcal O_{\mathcal C}\) by \(\mathcal O_Z(-\mathcal C)\), both base-flat. This construction inverts the preceding one. More explicitly, any other line extending \(I\) with relatively finite cokernel is uniquely identified with \(M\): the same grade-two extension argument extends the identity on \(I\) in both directions, and uniqueness makes the composites identities. This proves the functorial factorization, and hence the asserted isomorphism of parameter schemes. ◻

Proof. The scalar map \(R\Gamma(\mathcal O_X)\to R\!\mathop{\mathrm{Hom}}(\mathcal I,\mathcal I)\) is split by trace, since \(\mathcal I\) has rank one. At a geometric point \(Z\), extending across its codimension-two support gives \[\mathop{\mathrm{Hom}}(I_Z,I_Z)=\mathbb C,\qquad \mathop{\mathrm{Hom}}(I_Z,I_Z(K))=H^0(K).\] For example, an endomorphism extends to the determinant line and is there multiplication by a global section; conversely multiplication preserves the ideal. By Serre duality the trace-free \(\mathop{\mathrm{Ext}}^2(I_Z,I_Z)\) vanishes, as does the trace-free \(\mathop{\mathrm{Ext}}^0\). Riemann–Roch gives \(\chi(I_Z,I_Z)=\chi(\mathcal O_X)-2r\); hence the trace-free \(\mathop{\mathrm{Ext}}^1\) has dimension \(2r\).

The boundary map of the point-ideal sequence embeds \[T_ZX^{[r]}=\mathop{\mathrm{Hom}}(I_Z,\mathcal O_Z) \longrightarrow \mathop{\mathrm{Ext}}^1(I_Z,I_Z).\] Indeed every map \(I_Z\to\mathcal O_X\) is a scalar multiple of inclusion, so its composite with \(\mathcal O_X\to\mathcal O_Z\) is zero. This boundary is the Kodaira–Spencer map of the universal ideal. Its trace vanishes, because the universal point ideal has constant determinant. Fogarty’s smoothness Theorem [16] gives \(\dim X^{[r]}=2r\), so this is an isomorphism onto the trace-free part. The family complexes are perfect and commute with derived base change. A map between them is a quasi-isomorphism if it is so on geometric fibers, proving (16).

For the other assertion, \(Z\) is finite flat of length \(r\), so \(\pi_*\mathcal O_Z(K-\mathcal C)\) is locally free of that rank, with arbitrary base change and no higher direct images. Relative Serre duality for the smooth proper surface projection identifies the dual of \(R\!\mathop{\mathrm{Hom}}(\mathcal O_Z,\mathcal O(\mathcal C))\) with \(\pi_*\mathcal O_Z(K-\mathcal C)[2]\). Dualizing gives precisely (18), including its shift and the indicated dual bundle. ◻

Proof. Here \(L_{Y/Z}=\Omega_{Y/Z}\) is a vector bundle in degree zero. Locally split the surjection from a two-term presentation of \(E_g\) onto this bundle. The remaining acyclic degree-zero terms can be cancelled; one obtains \[E_g\simeq \Omega_{Y/Z}\oplus\mathop{\mathrm{Ob}}_g^\vee[1],\] with comparison map the projection. This shows local freeness and the stated \(K\)-class, without requiring a global splitting. The relative intrinsic normal cone of a smooth map is \([0/T_{Y/Z}]\). The dual comparison morphism gives its global inclusion into \(h^1/h^0(E_g^\vee)\), with quotient bundle \(\mathop{\mathrm{Ob}}_g\). Locally this is the inclusion into \([\mathop{\mathrm{Ob}}_g/T_{Y/Z}]\) supplied by the splitting above; globally the subbundle excess formula gives its zero-section Gysin image as the Euler class of \(\mathop{\mathrm{Ob}}_g\). Applying this excess formula over each base cycle proves (20). ◻

Proof. The Picard scheme is discrete and reduced, and \(R\Gamma(\mathcal I_o)=0\). Thus the determinant condition leaves just the indicated products, with the sum of their two rank-one obstruction theories. For an effective line \(P\), multiplication by any nonzero section of \(P\) injects \(H^0(K-P)\) into \(H^0(K)=0\), so \(H^2(P)=0\). On its linear system, \(\mathcal O(\mathcal C)=P\boxtimes\mathcal O(1)\). The Euler sequence and (19) therefore give \[[T_{|P|\times X^{[r]}}]-[T_U^\mathrm{vir}] = [H^1(P)\otimes\mathcal O(1)]+[F].\] Apply Lemma 14. Equality of these vector bundle classes gives equality of their Chern classes, and hence (21); it does not assert a global splitting of the obstruction bundle.

Finally, \[\dim|P|-h^1(P)=h^0(P)-1-h^1(P)=m_P.\] If \(m_P<0\), the factor \(x^{h^1(P)}\) is already zero in \(\mathop{\mathrm{CH}}^*(|P|)\). Otherwise integration against that factor replaces extraction at degree \(\dim|P|\) by extraction at degree \(m_P\). The localized moving factors are expanded with nonzero equivariant constant terms, so introduce no negative hyperplane powers. Thus this argument applies also to the localized insertions used below. ◻

Proof. Use the actual projection of enhancements \(\mathfrak W\to\mathfrak U_b\) from Proposition 11. The tangent of its determinant fiber in the high direction is (22). The map there is the high arrow’s boundary followed by trace and based normalization. Lemma 8, with rank one and \(KD_a>K^2\), makes its degree-one map onto \(H^2(\mathcal O_X)\) surjective, so the fiber has amplitude \([0,1]\). Explicitly, an annihilating canonical form would annihilate \(Q_a\); at each generic curve point a quotient of one generator annihilated by that form has length at most its vanishing order. Thus \(KD_a\le K^2\), a contradiction.

Restrict the derived cotangent transitivity triangle to \(W\). It has a natural morphism to the classical transitivity triangle, giving \[\begin{array}{ccccc} Lf^*E_{U_b}&\longrightarrow&E_W&\longrightarrow&E_f\\ \downarrow&&\downarrow&&\downarrow\\ Lf^*L_{U_b}&\longrightarrow&L_W&\longrightarrow&L_{W/U_b}. \end{array}\] Each row continues by a shift of its first term. Here the absolute comparisons have cones in degrees at most \(-2\), by Lemma 7. If these cones are \(C_{U_b}\) and \(C_W\), and the relative comparison cone is \(C_f\), the diagram gives a triangle \[Lf^*C_{U_b}\longrightarrow C_W\longrightarrow C_f \longrightarrow Lf^*C_{U_b}[1].\] Derived pullback preserves the upper cohomological bound, and the shift \([1]\) lowers that bound. Hence \(C_f\) is also concentrated in degrees at most \(-2\). Combined with the amplitude calculation, this proves that \(E_f\) is a relative perfect obstruction theory.

We have thus produced the compatible triple, including its morphisms to the cotangent triangle, required in [25]. The morphisms are between quasiprojective schemes, so are of Deligne–Mumford type and have the required stratifications by global quotients. Virtual-pullback functoriality [25] gives (23). ◻

Proof. There is a morphism of derived moduli problems \[ g:\mathfrak D^{D_b}\times X^{[n_b]}\longrightarrow\mathfrak U_b, \qquad (M_b\to L_b,\mathcal I_b)\longmapsto (M_b\mathcal I_b\to M_b\to L_b). \tag{25}\] The ordinary smooth Hilbert scheme here carries its smooth enhancement, and its universal ideal is perfect. Tensoring and composing perfect families define the displayed map on derived bases as well. Lemma 12 identifies its truncation with the identity of \(U_b\).

To compute its differential, first forget the arrows and compare their source moduli. On the common truncation the map is \[ R\Gamma(\mathcal O_X)[1]\oplus T_{X^{[n_b]}} \longrightarrow R\!\mathop{\mathrm{Hom}}(\mathcal I_b,\mathcal I_b)[1]. \tag{26}\] The first summand acts by scalars and the second is the Kodaira–Spencer map of the ideal, so Lemma 13 makes this a quasi-isomorphism. Indeed, on each geometric fiber the scalar map identifies the trace summands, and the Hilbert map induces an isomorphism onto the trace-free \(\mathop{\mathrm{Ext}}^1\). The source derivative is therefore triangular with isomorphic diagonal blocks. Its perfect cone has zero geometric fibers, so vanishes; no choice of a global splitting is needed here. The scalar assertion follows directly by differentiating tensoring with a line. Before taking sections it is \(\mathcal O_X[1]\to R\mathcal{H}om(\mathcal I_b,\mathcal I_b)[1]\), and corresponds to the identity endomorphism.

The relative arrow tangent map over these source moduli is \[R\!\mathop{\mathrm{Hom}}(M_b,L_b)\longrightarrow R\!\mathop{\mathrm{Hom}}(M_b\mathcal I_b,L_b).\] The point-ideal sequence and (18) identify its cone with \[R\!\mathop{\mathrm{Hom}}(\mathcal O_{Z_b},\mathcal O(\mathcal C_b))[1] =F_b[-1].\] The comparison of the two source tangent triangles, using (26), therefore proves \[ T_{\mathrm{prod}}^\mathrm{vir}\longrightarrow T_{U_b}^\mathrm{vir} \longrightarrow F_b[-1] \longrightarrow T_{\mathrm{prod}}^\mathrm{vir}[1]. \tag{27}\]

Crucially, dualizing this is a morphism of perfect obstruction theories \(E_{U_b}\to E_{\mathrm{prod}}\), compatible with their maps to the same \(L_{U_b}\), because it came from (25). Thus the induced map of obstruction bundle stacks is an inclusion \[h^1/h^0(T_{\mathrm{prod}}^\mathrm{vir}) % % \mathrel{% \BeginAccSupp{method=hex,unicode,space=false,pdfliteral=direct,ActualText=21AA}% \lhook\joinrel\longrightarrow% \EndAccSupp{pdfliteral=direct}% }% % h^1/h^0(T_{U_b}^\mathrm{vir})\] with quotient vector bundle \(F_b\). For precision, on an affine open the extension in (27) splits: its class lies in \(\mathop{\mathrm{Hom}}(F_b,T_{\mathrm{prod}}^\mathrm{vir}[2])=0\), since \(F_b\) is locally free and the tangent has amplitude \([0,1]\). This describes the inclusion locally as a vector bundle substack; the quotient is globally the displayed \(F_b\).

Compatibility of the obstruction morphisms implies that the intrinsic normal cone embedded using the Quot theory is the image of its embedding using the product theory. The subbundle excess formula for zero-section Gysin maps therefore multiplies the latter virtual class by \(\mathop{\mathrm{e}}(F_b)\). One can see the formula on any base cycle by pulling to the larger bundle stack: the smaller stack is the zero locus of the quotient-bundle coordinate, so its inclusion multiplies the pullback class by the top Chern class of that quotient. This is the zero-section excess formula of [17]; its vector-bundle form is also [23]. We apply it to the bundle-stack presentations of the intrinsic cone in [5]. The product theory has class \([\mathop{\mathrm{Div}}^{D_b}(X)]^\mathrm{vir}\times[X^{[n_b]}]\). This proves (24) on the possibly singular divisor scheme. ◻

Proof. The first assertion is the vanishing of a Chow group in negative dimension. For the second, the universal divisor sequence yields a cosection \[ \sigma:\mathop{\mathrm{Ob}}_{\mathop{\mathrm{Div}}^P} =R^1\pi_*\mathcal O_{\mathcal C}(\mathcal C) \longrightarrow H^2(\mathcal O_X)\otimes\mathcal O \simeq\mathcal O. \tag{28}\] At a divisor \(C\) of line class \(P\) the long exact cohomology sequence ends in \[H^1(\mathcal O_C(C))\xrightarrow{\sigma_C}H^2(\mathcal O_X) \longrightarrow H^2(P)\longrightarrow0.\] The obstruction fibers agree with these groups because the perfect tangent has amplitude \([0,1]\). Thus \(\mathop{\mathrm{coker}}(\sigma_C)=H^2(P)=H^0(K-P)^\vee\). The cosection is not surjective exactly where \(K-P\) is effective. Kiem–Li localization [23] expresses the ordinary virtual class as the pushforward of a class on this degeneracy locus.

Suppose a degeneracy point exists and put \(r=KP\). Both \(P\) and \(K-P\) are effective. Since \(K\) is ample and \(m_P\ge0\), Hodge index gives \[0\le r\le K^2,\qquad r\le P^2\le\frac{r^2}{K^2}.\] For \(0<r<K^2\) these inequalities contradict one another. If \(r=0\), ampleness forces the effective divisor \(C\) to be empty. If \(r=K^2\), any effective divisor in \(K-P\) has zero intersection with \(K\), so is empty. In the latter case \(P\cong K\) as line bundles, and \(h^0(K)=1\) gives exactly one divisor \(z\).

For the empty divisor the tangent complex \(R\pi_*\mathcal O_\varnothing\) is zero. Its parameter scheme is therefore a reduced point with the ordinary point class. A nontrivial numerically trivial line has no effective divisor, again by ampleness. This proves all the support assertions. ◻

Proof. Let \(C_{\mathop{\mathrm{Pic}}}\) be the Picard component containing \(K\); it is a smooth projective curve because \(q=1\). Choose an open neighborhood \(V\) of the point \(K\). Upper semicontinuity, followed by shrinking, gives \(h^0(P)\le1\) for \(P\in V\). For \(P\ne K\) in this component, \(K-P\) is a nontrivial numerically trivial line, so \(h^2(P)=0\). Riemann–Roch gives \(\chi(P)=1\), and consequently \(h^0(P)=1+h^1(P)\ge1\). At \(K\), \(h^0(K)=1\) by assumption. Thus \(h^0(P)\) is constantly one on \(V\).

We verify the resulting scheme structure, including infinitesimal families. Represent the derived direct image of the based universal line over \(V\), locally, by vector bundles \[E^0\xrightarrow{d^0}E^1\xrightarrow{d^1}E^2.\] There are no negative terms because this is the direct image of a line bundle, and relative dimension two bounds the positive terms. The kernel of the fiber of \(d^0\) has constant dimension one. Since \(V\) is reduced, all minors above the constant rank vanish scheme-theoretically. After inverting a maximal nonzero minor, elementary row and column operations turn \(d^0\) into an identity block and a zero block. Its kernel is therefore a line subbundle and commutes with arbitrary, including nonreduced, base change. It is the bundle of sections of the universal line. The divisor functor over \(V\) is its projective bundle of nonzero sections up to scalar, and hence is \(V\) itself. Thus the divisor scheme is a smooth curve in a neighborhood of \(z\).

The divisor theory has virtual dimension \(m_K=0\). Lemma 14 makes its obstruction sheaf an obstruction line \(\mathcal E\) on this neighborhood. At \(z\) the map \(H^2(\mathcal O_X)\to H^2(K)\) in the divisor sequence is an isomorphism: its dual maps \(1\) to the nonzero canonical section. Hence the cosection \(\mathcal E\to\mathcal O_V\) vanishes at \(z\). By Lemma 18 it is nonzero at every other point nearby. On the smooth curve its germ is multiplication by a unit times a uniformizer to a finite positive power \(s\). It follows that the line with its cosection is isomorphic to \[\bigl(\mathcal O_V(-sz),\ \mathcal O_V(-sz)% % \mathrel{% \BeginAccSupp{method=hex,unicode,space=false,pdfliteral=direct,ActualText=21AA}% \lhook\joinrel\rightarrow% \EndAccSupp{pdfliteral=direct}% }% % \mathcal O_V\bigr).\]

Shrink to an affine neighborhood of \(z\). As in Lemma 14, the perfect obstruction theory there splits, with its comparison map, as \(\Omega_V\oplus\mathcal E^\vee[1]\to\Omega_V\). The localized intrinsic-cone calculation is thus the localized Euler class of the zero section in \(\mathcal E=\mathcal O_V(-sz)\). For clarity, in the regularizing formula [23], take the identity modification of \(V\), the quotient line \(\mathcal O_V(D)\) with \(D=-sz\), and its zero kernel. The lifted zero cone is \(V\), so the formula gives \[[D]\cap[V]=-s[z].\] This determines the coefficient of the global localized class: localized Gysin maps commute with restriction to an open neighborhood of the degeneracy locus, by restricting the regularizing modification, its cone cycles, and the ordinary Gysin maps. This excision is also stated in [12]. The support lemma shows that \(z\) is the only degeneracy point on the whole divisor component. Pushing its localized class forward, by [23], proves (29). The argument used smoothness of the Picard curve; it imposed no smoothness or reducedness condition on the canonical divisor itself. ◻

Proof. By Lemmas 16 and 17, the fixed class is obtained by applying \(f^!\) to \[\mathop{\mathrm{e}}(F_b)\cap \bigl([\mathop{\mathrm{Div}}^{D_b}(X)]^\mathrm{vir}\times[X^{[n_b]}]\bigr).\] The support and canonical-sign Lemmas make this input zero unless its divisor class is the point class of the empty divisor or \(-s[z]\).

Over each of the corresponding low Picard components, the high line determined by the determinant has \(D_a-K\) numerically \(2K\) or \(K\), respectively. It is therefore ample. Kodaira vanishing gives \(H^j(D_a)=0\) for \(j>0\), uniformly in that component. Use the based universal low line to obtain the based high line as \(D-D_b\). Its pushforward is a vector bundle of rank \(\chi(D_a)=1+m_{D_a}\), with arbitrary base change. Lemma 12 consequently expresses \(f\) here as its relative projective bundle of high divisors, followed by the product with the smooth Hilbert scheme \(X^{[n_a]}\). In particular \(f\) is smooth, even if the low divisor scheme away from the degeneracy point is singular.

Formula (22), together with (19), gives \[\begin{aligned} [T_f^\mathrm{vir}] ={}&[R\pi_*\mathcal O(\mathcal C_a)]-[F_a] +[T_{X^{[n_a]}}]\\ &-[R\Gamma(\mathcal O_X)] -[R\Gamma(\mathcal I_o)[1]]. \end{aligned}\] The evaluation sequence at \(o\) implies \[[R\Gamma(\mathcal O_X)] +[R\Gamma(\mathcal I_o)[1]]=1.\] Thus the two determinant and scalar terms together subtract exactly one scalar. Higher cohomology vanishes for the high line, and the relative projective Euler sequence now identifies \[[T_f^\mathrm{vir}]=[T_f]-[F_a].\] Lemma 14 proves, on all base cycles, \[ f^!\gamma=\mathop{\mathrm{e}}(F_a)\cap f^*\gamma. \tag{30}\] Here the use of the \(K\)-class is legitimate: the compatible relative perfect obstruction theory has already been constructed, and smoothness identifies its virtual operation with the Euler class of a vector bundle. Its Chern classes are determined by the displayed \(K\)-class.

Apply (30) to the point-supported low input class. Compatibility of smooth pullback with closed pushforward restricts it to the fiber over that low divisor point, while retaining \(X^{[n_b]}\). The projection formula similarly restricts every operational insertion to the fiber. The universal divisors there are \(D_i\boxtimes\mathcal O(1_i)\), with the low factor a point; in particular \(x_b=0\). The high projective space has dimension \(h^0(D_a)-1=m_{D_a}\), proving the extraction rule. The universal point ideals, quotient complexes, and their derived Hom complexes restrict by derived base change for the smooth proper surface projection. All of these operations commute with adjoining the marked surface factors. This proves the asserted fiber formula with the multipliers already calculated. ◻

Proof. The geometric description and universal sequence are the consecutive nested-scheme construction of [15]. A quotient \(\mathcal I_{Z'}\twoheadrightarrow\mathbb C_x\) determines the extra length-one subscheme and conversely. In families its quotient line is \(\mathcal L\), giving the projectivization and (38). The smoothness and dimension of the consecutive nested scheme are the surface Hilbert-scheme facts recorded there. Over the open locus of \(r\) distinct points, forgetting which point is residual has degree \(r\), proving the pushforward formula for \(\psi\). The universal ideals are flat over their Hilbert bases, so their ordinary pullbacks in this sequence agree with their derived pullbacks.

Here is also the pushdown calculation, to fix its convention and sign. The universal ideal has projective dimension at most one: it is flat over the smooth Hilbert base and its surface fibers are torsion-free ideals. A sufficiently negative sum of ample line bundles surjects onto it; the kernel is locally free, by the fiberwise projective dimension bound and the local criterion of flatness. This gives the displayed global resolution. Write \(a=\mathop{\mathrm{rk}}U_1\), so \(\mathop{\mathrm{rk}}U_0=a+1\), and let \(p:\mathbb P(U_0)\to X^{[r-1]}\times X\). The inclusion \(N_r\subset\mathbb P(U_0)\) is the zero locus of the section of \(p^*U_1^\vee\otimes\mathcal O(1)\) induced by \(U_1\to U_0\). Its codimension is \(a\), because both \(N_r\) and the base of \(\sigma\) have dimension \(2r\). The ambient space is smooth, so the section is regular and its top Chern class represents \([N_r]\).

For \(\xi=c_1(\mathcal O_{\mathbb P(U_0)}(1))\) the quotient convention gives \(p_*(\xi^{a+k})=[c(U_0^\vee)^{-1}]_k\) for \(k\geq0\), and zero for smaller exponents. Hence the left side of (39) equals \[\begin{split} p_*\bigl(\xi^k c_a(p^*U_1^\vee\otimes\mathcal O(1))\bigr) &=\sum_{b=0}^{\min(a,k)}c_b(U_1^\vee) [c(U_0^\vee)^{-1}]_{k-b}\\ &=\left[\frac{c(U_1^\vee)}{c(U_0^\vee)}\right]_k. \end{split}\] This is also [15]. In particular, for \(r=2\) the map \(\sigma\) is the blowup of \(X\times X\) along its diagonal, \(\zeta=-E\) for its exceptional divisor, and the formula gives \(\sigma_*(\zeta^2)=-[\Delta_X]\). Flat base change and the projection formula give all the assertions with extra factors. ◻

Proof. We induct on \(n_1+n_2\). When this sum is positive, choose a positive length \(r\) in a fixed order of the Hilbert factors and pull back along \(\psi\) of Lemma 21, with all other factors unchanged. All spaces used here are smooth, so Chow pullbacks are defined. Projection formula and \(\psi_*[N_r]=r[X^{[r]}]\) express the original pushforward as \(1/r\) times the pushforward of this pullback.

In every surface slot, replace the pulled-back ideal by (38) in \(K\)-theory. Its extra term is the graph of the new residual point, with the line twist \(\mathcal L\). For that graph, Grothendieck–Riemann–Roch gives \[ \mathop{\mathrm{ch}}(j_*\mathcal L) =j_*\bigl(e^{\zeta}\rho^*\mathop{\mathrm{td}}(T_X)^{-1}\bigr). \tag{40}\] Indeed the graph is a regular embedding of codimension two with normal bundle \(\rho^*T_X\). It is the pullback of the diagonal under \((\rho,\mathop{\mathrm{id}}_X):N_r\times X\to X\times X\). This square is Tor independent: the two local equations of the diagonal cut out the graph regularly, in the same codimension. Thus the graph term in (40) is a polynomial in \(\zeta\), the equality diagonal between the new point slot and the slot in use, and tangent classes. For a perfect complex, the degree-\(k\) Chern character of its derived dual is \((-1)^k\mathop{\mathrm{ch}}_k\); derived duals therefore introduce no new kind of class.

After this substitution the integrand is a polynomial in \(\zeta\) whose coefficients pull back from \(X^{[r-1]}\times X\) and the unchanged factors. Apply (39) to each power of \(\zeta\). Its right side is a universal polynomial in the Chern classes of \(\mathcal I_{r-1}\), equivalently in its Chern characters with rational coefficients. The Hilbert length has decreased by one; the added surface factor records the residual point. The two output factors have not been pushed forward at any step. Repeating the construction ends with Hilbert schemes of length zero, whose ideals are the structure sheaf. Only surface factors, their diagonals, and the specified decorations remain. The choices of Hilbert-factor order, slot order, and polynomial expansion order can be fixed once for all inputs. This proves the assertion. We have adapted the geometric recursion underlying the integral formula of [15]: here the two output factors remain unintegrated throughout a sequence of Chow identities. ◻

Proof. We first put the integrand into the form required by Lemma 22. For the surface projection \(\pi_X:M\times X\to M\), Grothendieck–Riemann–Roch gives \[\begin{align*} \mathop{\mathrm{ch}}\bigl(R\!\mathop{\mathrm{Hom}}(\mathcal I_i,\mathcal I_j(P))\bigr) &=\pi_{X*}\bigl(\mathop{\mathrm{ch}}(\mathcal I_i^\vee) \mathop{\mathrm{ch}}(\mathcal I_j)e^P\mathop{\mathrm{td}}(T_X)\bigr), \tag{43}\\ \mathop{\mathrm{ch}}(P^{[n_i]}) &=\pi_{X*}\bigl((1-\mathop{\mathrm{ch}}(\mathcal I_i))e^P\mathop{\mathrm{td}}(T_X)\bigr). \tag{44}\end{align*}\] The dual in (43) is the derived dual. These identities hold in Chow, since the ideals are perfect and projection is smooth and proper. The constant complex \(R\Gamma(P)\) has \(K\)-class \(\chi(P)[\mathcal O_M]\), with \(\chi(P)=1+P(P-K)/2\). The Chern classes in the Euler factors are universal polynomials in the resulting Chern characters. The incidence insertion is \([Z_i]=-\mathop{\mathrm{ch}}_2(\mathcal I_i)\). Thus, after the finite truncation required by (36), every term is a polynomial in the classes of the preceding Lemma and in pushforwards of such classes.

Products of surface pushforwards introduce only additional surface slots. More precisely, if \(a,b\) are classes on \(M\times X\), the projection formula and flat base change identify \(\pi_{X*}(a)\pi_{X*}(b)\) with the pushforward of \(a_{(1)}b_{(2)}\) from \(M\times X\times X\). The same statement holds over the base \(M\) together with the two marked factors and can be iterated for any finite product. Apply Lemma 22 to the resulting expressions.

It remains to perform the surface pushforwards. This reduction will both prove the Chow decomposition and specify the coefficient to be compared across surfaces. Represent a monomial in equality diagonals by a graph \(\mathcal G\): its vertices are the surface slots and each factor \([\Delta_{uv}]\) is an edge between slots \(u,v\), with repeated factors recorded as repeated edges. For each connected component \(C\), let \(v_C,e_C\) count its vertices and edges. Let \[\iota:\prod_{C\text{ component of }\mathcal G}X \longrightarrow X^{\{\text{vertices of }\mathcal G\}}\] assign the same coordinate to all vertices in a component, and let \(\operatorname{pr}_C\) project its domain to the factor indexed by \(C\). Then \[ \prod_{uv\text{ edge of }\mathcal G}[\Delta_{uv}] =\iota_*\left( \prod_{C\text{ component of }\mathcal G} \bigl(\operatorname{pr}_C^*c_2(T_X)\bigr)^{e_C-v_C+1} \right). \tag{45}\] Indeed, a spanning tree in \(C\) imposes \(v_C-1\) transverse equalities, giving its small diagonal. Each of the remaining \(e_C-v_C+1\) edges imposes an equality already satisfied. The diagonal self-intersection formula contributes \(c_2(T_X)\) for each such edge. The exponent is zero for an isolated vertex as well as for a tree. By projection formula, every line or tangent decoration is then restricted to the copy of \(X\) belonging to its component.

There are three possibilities for the subsequent pushforward:

1. A component containing neither output is integrated over \(X\). It contributes zero unless its decoration has codimension two; otherwise it contributes the intersection number of that decoration.

2. A component containing exactly one output maps isomorphically to that output and leaves its decoration there. When the outputs belong to different components, their decorations therefore give an external product of total codimension two, multiplied by the intersection numbers from components of type (i).

3. A component containing both outputs maps to their diagonal. It contributes \(\Delta_{X*}(\eta)\), multiplied by the same kind of intersection numbers. Since \(\Delta_X\) already has codimension two, the codimension-two part of the answer uses only \(\eta\in\mathop{\mathrm{CH}}^0(X)=\mathbb Q[X]\). Thus this case gives a scalar multiple of \([\Delta_X]\).

Fix the recursion and edge-processing orders once for all inputs. Carry out this reduction on the formal decorated graphs, replacing components of type (i) by formal intersection numbers. Define \(c_*\) as the sum of the scalar coefficients in case (iii), and assign the terms with separated outputs to \(\Gamma_*\). Only after this assignment do we evaluate the intersection numbers and Chow classes on \(X\). Equation (45) and the three cases prove (41) in Chow and specify the promised choice of its coefficient.

Every codimension-two decoration integrated over a surface is a linear combination of \(c_2(T_X)\) and products of the lines in (42), since \(c_1(T_X)=-K\). The scalar Euler exponents are determined by the same data through Riemann–Roch. Hence the rule gives the same \(c_*\) on any two surfaces with those numerical data. We do not require the resulting diagonal decomposition to be unique on an individual surface: relations between its diagonal and external products are never used to select \(c_*\).

Finally, when both point lengths are zero, \(M\) is a point, \(\mathcal E=1\), and \(E_{ij}=0\). There is no incidence term in \(J\). The two copies of \(J\) involve only classes on their separate output factors and the scalar variables \(x_i,w_i\). The graph procedure has no edge joining the outputs, so it selects \(c_*=0\). None of this formal calculation requires sections of \(T\) or \(H\), proving the last assertion as well. ◻

Proof. The finite-type fixed locus has only finitely many discrete decompositions and Hilbert polynomials, as in Section 4. Proposition 15 gives the first list and discards negative \(m_i\). Proposition 20 gives the second list, its fiber restrictions, and its multiplicities. The moving factor and insertions were computed in (34) and (35), so these are exactly the contributions to the ordinary codimension-two part of localization of (31). That part vanishes by (32). Substitute (41) for each contribution to obtain (46). ◻

Proof. For \(Z\in\mathop{\mathrm{CH}}^2(X\times X)\), use its correspondence action \[Z_*(z)=\operatorname{pr}_{2*} \bigl(\operatorname{pr}_1^*z\cdot Z\bigr).\] The diagonal acts as the identity. If \(Z=a\boxtimes b\) has total codimension two and \(a\) has positive codimension, then \(z\cdot a=0\) on the surface for every zero-cycle \(z\), by dimension. If \(a\) has codimension zero, connectedness of \(X\) gives \(a=\lambda[X]\) and \(Z_*(z)=\lambda\deg(z)b\). Every summand of \(\Gamma\) therefore annihilates degree-zero zero-cycles. Acting by (46) gives \(cz=0\) in \(\mathop{\mathrm{CH}}_0(X)\otimes\mathbb Q\). Since \(c\ne0\), this implies \(z=0\). ◻

Proof. In the odd case choose diagonal coordinates and write \(K_0=(u;v_1,\ldots,v_b)\). The characteristic property says that all these integers are odd. Coordinate sign changes and permutations allow us to assume \[u>0,\qquad v_1\geq\cdots\geq v_b>0, \qquad u^2=9-b+\sum_{i=1}^b v_i^2.\] If \(v_i=1\), the class \(-e_i\), for the corresponding negative basis vector \(e_i\), has square \(-1\) and pairing \(1\) with \(K_0\). We show by descent on \(u\) that this case can always be reached by integral lattice isometries.

For \(b=1\), the equation \((u-v_1)(u+v_1)=8\) forces \((u,v_1)=(3,1)\): both factors are positive even integers and the second is strictly larger than the first. Suppose next that \(b\geq3\) and that every \(v_i\geq3\). Put \(\ell=v_1+v_2+v_3\). The ordering of the \(v_i\) gives \[\begin{align*} \ell^2-u^2 &=2(v_1v_2+v_1v_3+v_2v_3) -\sum_{i=4}^b v_i^2-(9-b)\\ &\geq (9-b)(v_3^2-1)>0. \end{align*}\] On the other hand, \(\ell^2\leq3(v_1^2+v_2^2+v_3^2)<3u^2<4u^2\). The vector \(z=(1;1,1,1,0,\ldots,0)\) has square \(-2\). Its integral reflection \(x\mapsto x+(xz)z\) changes the positive coordinate of \(K_0\) to \[u'=2u-\ell,\qquad 0<u'<u.\]

For \(b=2\) and \(v_1,v_2\geq3\), put \(\ell=v_1+v_2\). Here \[\ell^2-u^2=2v_1v_2-7>0, \qquad \ell^2\leq2(u^2-7)<2u^2<\frac94u^2.\] The vector \(z=(1;1,1)\) has square \(-1\), so its reflection is the integral isometry \(x\mapsto x+2(xz)z\). It changes the positive coordinate to \[u'=3u-2\ell,\qquad 0<u'<u.\] In either case the reflected vector remains characteristic and of square \(9-b\). Rechoose the signs and order of its negative coordinates and repeat. The positive integer \(u\) strictly decreases, so the process must reach a coordinate \(v_i=1\). Pulling the corresponding vector back through the isometries gives the asserted class, which lifts to a line bundle.

In the even case choose an isotropic basis \(f,g\) with \(fg=1\). Writing \(K_0=af+bg\), the characteristic property makes \(a,b\) even, and \(2ab=K_0^2=8\). After changing both basis signs if necessary, \(a=b=2\). Either \(f\) or \(g\) gives \(h_0\). ◻

Proof. Write \[P=\pi^*P_0+\sum_{j=1}^e a_jE_j, \qquad r_0=K_0P_0,\] where the \(a_j\) are integers. Pushing forward an effective divisor representing \(P\) shows that \(P_0\) is effective; consequently \(r_0\geq0\) by nefness of \(K_0\). The inequality \(2m_P=P^2-KP\geq0\) gives \[ 0\leq\sum_{j=1}^e a_j(a_j-1) \leq P_0^2-r_0 <\frac{r_0^2}{k_0}-r_0. \tag{53}\] The first inequality holds for all integral \(a_j\), including negative ones. The last is strict Hodge index: \(\alpha P\ne0\) implies that \(P_0\) is not numerically proportional to \(K_0\). Together with \(r_0\geq0\), this forces \(r_0>k_0\).

We next prove \(KP>0\). If \(e=0\), this is immediate. Otherwise put \(A_0=\sum_j a_j\) and suppose \(KP=r_0-A_0\leq0\). By Cauchy–Schwarz, \[\sum_j a_j(a_j-1) \geq\frac{A_0^2}{e}-A_0 \geq\frac{r_0^2}{e}-r_0 >\frac{r_0^2}{k_0}-r_0.\] Indeed \(A_0\geq r_0>k_0>e\), and \(u\mapsto u^2/e-u\) is increasing on \([r_0,\infty)\). This contradicts (53). Thus \(KP>0\).

Since \(K^2=k>0\), the intersection form is negative definite on \(K^\perp\). Also \(\alpha P\ne0\) and \(\alpha K=0\) show that \(P\) is not numerically proportional to \(K\). Therefore \[0<KP\leq P^2<\frac{(KP)^2}{k},\] where the middle inequality is \(m_P\geq0\). Dividing by \(KP>0\) proves \(KP>k\). This application of Hodge index requires the positive square of \(K\); all effectivity bounds above used the nef class \(\pi^*K_0\). ◻

Proof. Since \(\alpha T=-\alpha H\ne0\), Lemma 28 applies to both \(H\) and \(T\). It gives \[r>k,\qquad dk-r>k,\qquad r\leq s<\frac{r^2}{k}.\] Furthermore \(s-r\) is even by Riemann–Roch.

If \(k=1\), then \(d=4\) and the first two bounds force \(r=2\). The inequalities \(2\leq s<4\) and \(s\equiv2\pmod2\) force \(s=2\). Equation (55) gives \(N=0\).

For \(2\leq k\leq5\), write \(r=k+j\). Since \(d=3\), the two strict canonical bounds give \(1\leq j\leq k-1\). Use the parity condition by writing \(s=r+2m\), where \(m=m_H\) is a nonnegative integer. The strict Hodge bound gives \[ 2m<\frac{r^2}{k}-r=j+\frac{j^2}{k}<2j. \tag{57}\] Consequently \(m\leq j-1\). Substituting into (55) yields \[ 0\leq N=3(1-j)+2m\leq1-j. \tag{58}\] It follows that \(j=1\), then \(m=0\) and \(N=0\), so \(r=s=k+1\). This integral bound excludes every \(j\geq2\), including the boundary values for \(k=3\) and \(k=4\), without a separate numerical enumeration.

In every case \(r=s=k+1\) and \(N=0\). Since \(n_1,n_2\geq0\), they both vanish. Finally \[2m_T=(dK-H)^2-K(dK-H) =d(d-1)k-2(d-1)(k+1),\] which equals \(0\) for \((k,d)=(1,4)\) and \(2(k-2)\) for \(d=3\). This proves all assertions. ◻

Proof. Equation (50) gives \(KH=H^2=k+1\). The calculations in the preceding proof then give \(m_H=0\), \(m_T=p\), and \(N=0\). Consequently \[\chi(H)=1,\qquad \chi(T)=p+1>0.\] Let \(B_0=\pi^*K_0\), a nef divisor on \(X\). Its pairings with the Serre-dual line bundles are \[B_0(K-H)=-a<0,\qquad B_0(K-T)=(2-d)k_0+a<0.\] For the second inequality, in the odd case its value is \(-1\) when \(k_0=1\) and \(1-k_0<0\) when \(k_0\geq2\); in the even case it is \(-8+2=-6\). Thus neither \(K-H\) nor \(K-T\) has a section, since an effective divisor has nonnegative intersection with a nef divisor. Serre duality gives \(H^2(H)=H^2(T)=0\), and Riemann–Roch now yields \[h^0(H)=\chi(H)+h^1(H)>0,\qquad h^0(T)=\chi(T)+h^1(T)>0.\] Both line bundles are therefore effective. Finally \(\alpha H=\alpha h'=\alpha^2<0\) by (52). Thus this pair occurs in the fixed locus with both point subschemes empty, by Lemma 12. ◻

Proof of Proposition 26. Use the data (54) and pair the Chow relation of Theorem 24 with \(\alpha\boxtimes\alpha\) in cohomology. We compute the contributions using (36) and (48).

For any contribution with \(\alpha H=0\), the class \(\alpha\) is orthogonal to \(K,L_1,L_2,T,H\). Thus its external-product part has zero pairing, and it contributes \(c_*\alpha^2\) by Proposition 23.

For every remaining contribution, Lemma 29 gives \(n_1=n_2=0\). Its chosen diagonal coefficient \(c_*\) is therefore zero by Proposition 23. Its point Hilbert factors and fixed Euler factors are trivial. Formula (35) gives \[\mu_\alpha=(\alpha H)(-w_1+w_2+x_1-x_2) =(\alpha H)(-1+x_1-x_2).\] Since \(m_T=p\) and \(m_H=0\), its full pairing is \[\begin{align*} &[x_1^p x_2^0]\, \mathcal R(x_1,x_2)(x_1+x_2)^p (\alpha H)^2(-1+x_1-x_2)^2\\ &\hspace{35mm}=(\alpha H)^2\mathcal R(0,0). \end{align*}\] Here all denominators in \(\mathcal R\) have nonzero constant term. Setting \(x_2=0\) and extracting \(x_1^p\) therefore uses only the constant term of the factors multiplying \(x_1^p\).

We compute this constant exactly. Since there are no point factors, (34) reduces at \((x_1,x_2)=(0,0)\) to the scalar weight powers. The \(i=2,j=1\) weight is \(1\), and the \(i=1,j=2\) weight is \(-1\), with \[P_{12}=T+K,\qquad B_{12}=T-H+K.\] Thus \[\begin{align*} \mathcal R(0,0) &=(-1)^{\chi(T-H+K)-\chi(T+K)}\\ &=(-1)^{-TH+(H^2-KH)/2} =(-1)^{-TH}. \end{align*}\] The second equality follows by expanding Riemann–Roch, and the last uses \(H^2=KH\). Moreover \[TH=(dK-H)H=(d-1)(k+1) =\begin{cases}6,&k=1,\\2(k+1),&2\leq k\leq5,\end{cases}\] which is even. Hence every such term has \[ \mathcal R(0,0)=1. \tag{60}\]

Let the sum below run over the effective ordered line decompositions occurring in the fixed locus with \(\alpha H\ne0\), retaining distinct torsion twists as distinct terms. This is a finite sum, as the fixed locus has only finitely many component data. It is nonempty by Lemma 30. Summing the two kinds of contributions therefore gives \[ 0=c\alpha^2+\sum_{\alpha H\ne0}(\alpha H)^2, \qquad c=-\frac{\displaystyle\sum_{\alpha H\ne0}(\alpha H)^2}{\alpha^2}>0. \tag{61}\] In this equality \(c\) is the coefficient already chosen in the Chow relation: the second group of terms has \(c_*=0\), and the first group accounts for its entire sum. The strict positivity follows from \(\alpha^2<0\) and the nonempty sum of positive real squares.

Corollary 25 now gives the rational degree isomorphism on \(X\). Lemma 4 passes it to \(X_0\), proving the Proposition. ◻

Proof. Apply the construction on \(Y\) with the data (62). Proposition 20 first replaces the low divisor virtual class by the empty-divisor point with coefficient \(1\), or by the canonical-divisor point with coefficient \(-s\). The compatible virtual pullback along the high morphism is its smooth pullback multiplied by the high point-obstruction Euler class. Thus the surviving terms are computed on \[|3K_Y|\times Y^{[n_1]}\times Y^{[n_2]} \quad\text{or}\quad |2K_Y|\times Y^{[n_1]}\times Y^{[n_2]},\] with total point lengths \(20\) and \(2\), respectively, and with both fixed point-obstruction Euler factors retained. The high linear systems have dimensions \(27\) and \(9\). The low divisor is fixed, so its hyperplane variable is zero. Both orders occur, and the canonical multiplier is \(-s\) in each order.

The projection formula restricts all operational insertions to these actual fibers. Derived base change likewise restricts the moving complexes, giving exactly Equation (34). Consequently the remaining formulas are those defining \(c_{\mathrm{out}}\) and \(c_{\mathrm{mid}}\), with their indicated multipliers. This restriction precedes the use of universality: no integration over a varying Picard component is being compared by numerical invariants.

After that restriction, the equality of the numerical data in (66) with those for \(X\) permits Proposition 23 to identify the chosen coefficients. The resulting Chow relation on \(Y^2\) has diagonal coefficient \(c_{\mathrm{out}}-s c_{\mathrm{mid}}\). Pair it with \(\alpha\boxtimes\alpha\). Every external term vanishes: bidegree allows only a divisor on each factor to pair nontrivially, and each such divisor is orthogonal to \(\alpha\). The diagonal pairs to \(\alpha^2\). Equation (67) now gives (68). ◻

Proof. Compute on \(Y\) with the class \(\alpha\) in (67), omitting the overall canonical multiplier \(-s\). We compute the order \(T=2K_Y\), \(H=K_Y\) first, and verify equality with the reverse order at the end. For this order, \[n_1+n_2=2,\qquad m_T=9,\qquad m_H=0.\] The three possible point-length distributions are \((2,0)\), \((1,1)\), and \((0,2)\). Taking the coefficient of \(x_2^0\) means setting \(x_2=0\): every moving weight in Equation (34) has nonzero constant term \(1\) or \(-1\) after \(t=1/2\), so all its inverse Euler factors expand with nonnegative powers of \(x_1,x_2\). This remains true for virtual bundles, by expanding their Chern classes at the nonzero weight. If \(n_2>0\), the low fixed factor at \(x_2=0\) is \[e\bigl((\mathcal O_Y^{[n_2]})^\vee\bigr)=0.\] Indeed the constant function \(1\) is a nowhere-zero section of the rank-\(n_2\) tautological bundle, so its top Chern class, and hence that of its dual, vanishes. Equivalently its Euler polynomial is divisible by \(x_2\); there is no negative power of \(x_2\) in the other factors to cancel it. Therefore only \((n_1,n_2)=(2,0)\) contributes.

Put \[U=(-K_Y)^{[2]},\qquad N=\mathcal O_Y^{[2]},\qquad x=x_1, \qquad \mu=\int_Y[Z_1]\alpha\in H^2(Y^{[2]},\mathbb R).\] All the other terms of Equation (35) integrate to zero against \(\alpha\), by its orthogonality to divisors and by degree. Consequently \(\mu\) is the class used in the pairing formula (48).

For this order, the line data in the moving ratio are \[P_{12}=2K_Y,\qquad P_{21}=K_Y,\qquad B_{12}=K_Y,\qquad B_{21}=-K_Y.\] Riemann–Roch gives \(\chi(K_Y)=1\) and \(\chi(2K_Y)=\chi(-K_Y)=10\). The scalar part of the moving ratio is therefore \[ \frac{(x-1)^1}{(x-1)^{10}} \frac{(1-x)^{10}}{1^1}=-(1-x). \tag{69}\] Since the second ideal is \(\mathcal O_Y\), its two point-complex terms are \[ E_{12}(K_Y)=N^\vee,\qquad E_{21}(-K_Y)=U \quad\text{in }K(Y^{[2]}). \tag{70}\] For the first identity, apply \(R\!\mathop{\mathrm{Hom}}(-,K_Y)\) to the point-ideal sequence: the difference \([R\Gamma(K_Y)]-[R\!\mathop{\mathrm{Hom}}(\mathcal I_1,K_Y)]\) is \([R\!\mathop{\mathrm{Hom}}(\mathcal O_{Z_1},K_Y)]=[N^\vee[-2]]=[N^\vee]\). For the second, apply derived sections to that sequence tensored by \(-K_Y\). The even shift in the first identity contributes no minus sign in \(K\)-theory.

For any rank-two bundle \(F\) one has \(e_{-u}(F^\vee)=e_u(F)\). Hence the moving point numerator \(e_{x-1}(U^\vee)\) cancels the denominator \(e_{1-x}(U)\), and the remaining denominator \(e_{x-1}(N^\vee)\) equals \(e_{1-x}(N)\). Including the high fixed Euler factor \(e_x(U^\vee)\) and the insertion \(x^7\), the pairing for this order is \[ I=-[x^2]\int_{Y^{[2]}} \mu^2(1-x)\frac{e_x(U^\vee)}{e_{1-x}(N)}. \tag{71}\]

Here is the required homogeneous expansion. Set \(u_i=c_i(U)\) and \(\nu_i=c_i(N)\). Then \[e_x(U^\vee)=x^2-u_1x+u_2,\qquad e_{1-x}(N)=(1-x)^2+\nu_1(1-x)+\nu_2,\] and, through degree two in the Chern classes on \(Y^{[2]}\), \[\frac{1-x}{e_{1-x}(N)} =\frac1{1-x}-\frac{\nu_1}{(1-x)^2} +\frac{\nu_1^2-\nu_2}{(1-x)^3} +\text{terms of Chern degree at least three}.\] The class \(\mu^2\) has real cohomological degree four and \(Y^{[2]}\) has complex dimension four. Thus only Chern degree two in the remaining factor can contribute to the integral. Multiplication by \(x^2-u_1x+u_2\) and extraction of \(x^2\) in that degree gives \(u_2+2u_1\nu_1+\nu_1^2-\nu_2\). Equation (71) therefore becomes \[ I=-\int_{Y^{[2]}}\mu^2 \bigl(c_2(U)+2c_1(U)c_1(N)+c_1(N)^2-c_2(N)\bigr). \tag{72}\]

We have reduced the middle coefficient to four tautological Chern terms in (72). To evaluate them, consider the consecutive nested Hilbert scheme \[\widetilde{Y^2}=Y^{[1,2]}=\operatorname{Bl}_{\Delta_Y}(Y^2), \qquad \pi:\widetilde{Y^2}\longrightarrow Y^2, \qquad \psi:\widetilde{Y^2}\longrightarrow Y^{[2]}.\] The first surface coordinate records the length-one subscheme and the second the residual point. The morphism \(\psi\) is proper and generically of degree two, since either point of a reduced length-two subscheme can be selected as its length-one subscheme. The projectivization description and the tautological sequence are those of [15], specialized to length one. Write \(E\) for the exceptional divisor and \(p_i=\operatorname{pr}_i\circ\pi\). The tautological quotient line of the projectivized diagonal ideal is \(\mathcal O(-E)\). Consequently, for any line bundle \(P\) on \(Y\), the tautological sequence is \[ 0\longrightarrow p_2^*P\otimes\mathcal O(-E) \longrightarrow\psi^*(P^{[2]}) \longrightarrow p_1^*P\longrightarrow0. \tag{73}\] It follows either from the universal point-ideal sequence, by tensoring with \(P\) and pushing along the finite universal subscheme, or directly from the quotient recording the residual point.

Use subscripts for pullbacks by \(p_i\), and set \(P=-K_Y\). The length-two incidence cycle pulls back to the sum of the two point graphs, so \[ \begin{gathered} \psi^*\mu=\alpha_1+\alpha_2,\qquad \psi^*c_1(N)=-E,\qquad \psi^*c_2(N)=0,\\ \psi^*c_1(U)=P_1+P_2-E,\qquad \psi^*c_2(U)=P_1P_2-P_1E. \end{gathered} \tag{74}\] The incidence assertion also follows by taking the codimension-two Chern character of the universal point-ideal sequence: its \(\mathcal O(-E)\) twist does not alter the graph’s fundamental codimension-two term. The bundle identities in (74) follow at once from (73), including the sign of \(E\).

For completeness, the exceptional-divisor pushforwards needed here are \[ \pi_*E=0,\qquad \pi_*(E^2)=-[\Delta_Y]. \tag{75}\] The first vanishes because \(E\) maps to a smaller-dimensional cycle. For the second, \(E\) is a \(\mathbb P^1\)-bundle over \(\Delta_Y\) and \(\mathcal O(E)|_E=\mathcal O_E(-1)\), whose degree on each fiber is \(-1\). This proves the sign directly.

The bracket in Equation (72) now pulls back to \[P_1P_2-(3P_1+2P_2)E+3E^2.\] All terms linear in \(E\) integrate to zero by (75) and the projection formula. The remaining term involving \(P\) satisfies \[\int_{Y^2}(\alpha_1+\alpha_2)^2P_1P_2 =2(\alpha\cdot P)^2=0;\] the two other terms in the square exceed the dimension of one surface factor. Since \(\psi_*[\widetilde{Y^2}]=2[Y^{[2]}]\), the factor of two from this cover must also be included. Thus \[\begin{align*} I &=-\frac12\int_{\widetilde{Y^2}} (\alpha_1+\alpha_2)^2\,3E^2\\ &=\frac32\int_{\Delta_Y}(\alpha_1+\alpha_2)^2 =\frac32\int_Y(2\alpha)^2 =6\alpha^2. \end{align*}\] The external terms pair to zero, so Equation (48) and \(\alpha^2\neq0\) show that this order has universal diagonal coefficient \(6\).

Finally exchange the two orders. Since \(L_1=L_2=0\), exchanging the two summands is the same as replacing the equivariant parameter \(t\) by \(-t\) before evaluation. The ordinary codimension-two output component has parameter power \[p+2-v=7+2-13=-4.\] This exponent is even, so the exchange preserves its pairing at \(t=1/2\). The second order therefore also has coefficient \(6\). Both orders and all three point-length distributions have now been accounted for, and \(c_{\mathrm{mid}}=6+6=12\). ◻

Proof of Proposition 31. Combine Equations (65) and (68) with Lemma 33. On the original surface \(X\) they give \[c=(s+\tau_X)c_{\mathrm{mid}}=12(s+\tau_X)>0,\] because \(s>0\) and \(\tau_X\geq0\). ◻

Proof of Theorem 1. For a surface of Kodaira dimension below two, the assertion follows from Bloch–Kas–Lieberman as in Proposition 5. For a surface of general type, pass to a smooth minimal model \(X_0\). If \(K_{X_0}^2\le8\), Proposition 26 gives a nonzero diagonal coefficient in Theorem 24 on a point blowup of \(X_0\). If \(K_{X_0}^2=9\), Proposition 31 gives such a coefficient on \(X_0\) itself. In either case Corollary 25 and Lemma 4 yield \[\mathop{\mathrm{CH}}_0(S)^0\otimes_{\mathbb Z}\mathbb Q=0.\]

Let \(z\in\mathop{\mathrm{CH}}_0(S)^0\). Vanishing after tensoring with \(\mathbb Q\) means that some nonzero integer annihilates \(z\). Roitman’s torsion Theorem makes the Albanese map injective on torsion zero-cycles over \(\mathbb C\) [6]; see also [30]. But \(\mathop{\mathrm{Alb}}(S)=0\), since \(q(S)=0\). Therefore \(z=0\). Thus the integral degree map is injective. Any complex point of \(S\) has degree one, so it is also surjective. ◻

Proof of Corollary 2. If \(S\) is not of general type, Bloch, Kas, and Lieberman prove that the kernel of \(\operatorname{alb}_S\) vanishes with integral coefficients [8]. Surjectivity gives the claim.

Suppose \(S\) is of general type, and let \(S_0\) be a smooth minimal model. The inequalities \(K_{S_0}^2>0\) and \(\deg c_2(S_0)>0\), together with Noether’s formula, give \(1-q(S_0)+p_g(S_0)>0\) [29]. By Lemma 4, \(p_g(S_0)=p_g(S)=0\) and \(q(S_0)=q(S)\), so the inequality forces \(q(S)=0\). Theorem 1 now gives \(\mathop{\mathrm{CH}}_0(S)^0=0\), while \(\mathop{\mathrm{Alb}}(S)(\mathbb C)=0\). Thus \(\operatorname{alb}_S\) is an isomorphism. ◻

Proof of Corollary 3. Let \(T(S)\) be the kernel of the rational Albanese map \[\mathop{\mathrm{CH}}_0(S)^0\otimes_{\mathbb Z}\mathbb Q \longrightarrow \mathop{\mathrm{Alb}}(S)(\mathbb C)\otimes_{\mathbb Z}\mathbb Q.\] Corollary 2 gives \(T(S)=0\). Kahn–Murre–Pedrini’s equivalence [22] then gives \(t_2(S)=0\) and Kimura finite-dimensionality of \(h(S)_{\mathbb Q}\). Their hypotheses hold over \(\mathbb C\), an algebraically closed characteristic-zero field of infinite transcendence degree over \(\mathbb Q\). This step does not require \(q(S)=0\).

Now assume \(q(S)=0\). Kahn–Murre–Pedrini’s surface Chow–Künneth decomposition [22] is \[h(S)_{\mathbb Q}\simeq \mathbf 1\oplus h_1(S)\oplus\mathbb L^{\oplus\rho(S)} \oplus t_2(S)\oplus h_3(S)\oplus\mathbb L^2,\] where \(\rho(S)\) is the Picard number. The endomorphism identifications for \(h_1(S)\) and \(h_3(S)\) with those of the Albanese and Picard varieties show that these two motives vanish when \(q(S)=0\). The Betti realization of the middle decomposition, with \(t_2(S)=0\), gives \(\rho(S)=b_2(S)\). This gives the asserted decomposition. Tensoring it proves the assertion for all powers of \(S\).

For a partition \(\nu=1^{a_1}\cdots n^{a_n}\) of \(n\ge1\), put \(\ell(\nu)=\sum_j a_j\), \(\Sigma_\nu=\prod_j\mathfrak S_{a_j}\), and \(S^{(\nu)}=S^{\ell(\nu)}/\Sigma_\nu\). The formula of de Cataldo–Migliorini [14] gives \[h(S^{[n]})_{\mathbb Q}\simeq \bigoplus_{\nu\vdash n}h(S^{(\nu)})_{\mathbb Q} \otimes\mathbb L^{\,n-\ell(\nu)}.\] Here \(h(S^{(\nu)})_{\mathbb Q}\) is the rational quotient motive of the possibly singular symmetric-product quotient. In the quotient formalism of [14], it is the image on \(h(S^{\ell(\nu)})_{\mathbb Q}\) of the averaging projector \[e_\nu=\frac{1}{|\Sigma_\nu|} \sum_{\sigma\in\Sigma_\nu}\Gamma_\sigma.\] Since \(\mathop{\mathrm{Hom}}(\mathbb L^a,\mathbb L^b)\) is zero for \(a\ne b\) and is \(\mathbb Q\) for \(a=b\), every idempotent summand of a finite Tate sum is again a finite Tate sum. Each term in the formula is therefore pure Tate, as is \(h(S^{[n]})_{\mathbb Q}\). The case \(n=0\) is immediate.

Finally, on a summand \(\mathbb L^a\), the rational Chow group and even Betti cohomology each have one generator in codimension \(a\) and vanish in other codimensions; cycle class identifies these generators, and odd Betti cohomology vanishes. Compatibility of cycle class with correspondences transfers these assertions to the stated pure Tate motives. Thus the displayed cycle-class maps are isomorphisms of rational vector spaces. This uses the decomposition in rational Chow motives, not finite-dimensionality alone. ◻