All-n surface Hilbert symmetric-group kernel equivalence and norm-line compatibility

Assume AC and DC. Let S be any smooth connected projective complex surface and $n\ge 1$. Put $Y=S^{[n]}$, $M=S^n$, $Q=Sy{m}^n$ S and $X=(Y{\times }_{Q}M{)}_{\text{red}}$. With its natural $S_n$ linearization, the actual isospectral kernel $O_X$ defines an equivalence Φ:Dᵇ(Coh $Y)\to D_{S_n}^b(Coh$ M), $\mathrm{\Phi }(F)=R{\pi }_{M,\ast }(O_X{\otimes }^L{\pi }_Y\ast F)$. Its universal $p_{\ast }{O}_X$ is a rank-n! vector bundle with regular-representation fibers, and Y is the reduced free-orbit $S_n$-Hilbert component. No symplectic or trivial-canonical assumption is imposed. For any line A on S, let $A_{(n)}$ be the descent of $A^{⊠n}$ to Q and $A_n$ its Hilbert–Chow pullback; then $\mathrm{\Phi }(F\otimes A_n)=\mathrm{\Phi }(F)\otimes A^{⊠n}$, coherently with powers and specified torsion trivializations. The transform naturally intertwines surface automorphisms; exterior products of equivariantly linearized Fourier–Mukai kernels have the canonical $S_n$ and wreath compatibilities. No global tilting-bundle assertion on a projective Hilbert scheme is made.

All-n isospectral Hilbert Cohen–Macaulayness and the n!-flat family

Cohen–Macaulay isospectral schemes give all-n surface Hilbert kernel equivalence

The Axiom of Choice

The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain