On a new compactification of moduli of vector bundles on a surface. V: Existence of a universal family (Q2841116)

This is the sixth in a series of papers by the author on compactifications of the moduli spaces of vector bundles on surfaces (the first was unnumbered).NEWLINENEWLINELet \(S\) be a smooth irreducible projective surface defined over an algebraically closed field of characteristic zero and let \(L\) be a fixed ample invertible sheaf on \(S\). In two previous papers in the series [Sb. Math. 202, No. 3, 413--465 (2011); translation from Mat. Sb. 2011, No. 3, 107--160 (2011; Zbl 1230.14060) and Math. Notes 90, No. 1, 142--148 (2011); translation from Mat. Zametki 90, No. 1, 143--150 (2011; Zbl 1237.14023)], the author defined, for fixed \(S\), \(L\), rank and Hilbert polynomial, a concept of admissible projective scheme \(\widetilde{S}\) with distinguished polarization \(\widetilde{L}\). She then formulated a definition of (semi)stability for pairs \(((\widetilde{S},\widetilde{L}),\widetilde{E})\), where \(\widetilde{E}\) is a locally free sheaf on \(\widetilde{S}\), and used this to construct a moduli functor. The usual definitions can be carried through to this situation and it was shown that the moduli functor possesses a coarse moduli scheme \(\widetilde{M}\). Moreover, there is a birational morphism \(\kappa:\overline{M}_{\mathrm{red}}\to\widetilde{M}_{\mathrm{red}}\), where \(\overline{M}\) is the union of the components of the Gieseker-Maruyama moduli scheme of semistable torsion-free sheaves with the given invariants which contain locally free sheaves.NEWLINENEWLINEIn the present paper, the author addresses the problem of the existence of a universal family over \(\widetilde{M}\). The results are essentially as expected although there are considerable technical problems in proving them. First assume that, for all semistable admissible pairs, there exists \(m\gg 0\) such that the induced immersions \(j:\widetilde{S}\to G(H^0(\widetilde{S},\widetilde{E}\otimes\widetilde{L}^m),r)\) (here \(r\) is the rank) have no non-trivial \(\text{PGL}(H^0(\widetilde{S},\widetilde{E}\otimes\widetilde{L}^m))\)-automorphisms. Then (Theorem 1) \(\widetilde{M}\) contains an open subscheme \(\widetilde{M}^s\) carrying a universal family. This subscheme coincides set-theoretically with the image \(\kappa(\overline{M}^s_{\mathrm{red}})\), where \(\overline{M}^s_{\mathrm{red}}\) is the open subscheme of GIT-stable points in \(\overline{M}_{\mathrm{red}}\). If \(\overline{M}\) carries a universal family, so does \(\widetilde{M}\). To cover the case where non-trivial automorphisms exist, the author introduces pseudofamilies; a pseudofamily is a collection of families parametrized by the sets of an étale open covering of \(\widetilde{M}\) with appropriate gluing conditions. Then (Theorem 2) \(\widetilde{M}\) contains an open subscheme \(\widetilde{M}^s\) carrying a universal pseudofamily. This subscheme coincides set-theoretically with the image \(\kappa(\overline{M}^s_{\mathrm{red}})\). If \(\overline{M}\) carries a universal family, then \(\widetilde{M}\) carries a universal pseudofamily. As a consequence (Corollary 1), if Maruyama's coprimality condition is satisfied, then \(\widetilde{M}\) carries a universal pseudofamily. Moreover, if the immersions \(j\) have no non-trivial automorphisms for all stable admissible pairs, then \(\widetilde{M}\) carries a universal family.