Projectivity of the moduli space of vector bundles on a curve

Abstract

We discuss the projectivity of the moduli space of semistable vector bundles on a curve of genus g ≥ 2. This is a classical result from the 1960s, obtained using geometric invariant theory. We outline a modern approach that combines the recent machinery of good moduli spaces with determinantal line bundle techniques. The crucial step producing an ample line bundle follows an argument by Faltings with improvements by Esteves-Popa. We hope to promote this approach as a blueprint for other projectivity arguments.

Moduli spaces and geometric invariant theory: old and new perspectives

“…Alper's notion of good and adequate moduli spaces of stacks enables GIT-free constructions of moduli spaces. This is even more tangible following the recent existence criteria of Alper-Halpern-Leistner-Heinloth [7], which equates the existence of moduli spaces to two simple valuative criteria and has been applied to various moduli problems [4,5,12,13].…”

“…More generally, given a scheme X with a G-linearisation L, Mumford defines a GIT quotient using invariant sections of positive powers of L whose nonvanishing locus is affine (so that one can take affine GIT quotients and glue them). This produces a good quotient of a 'semistable locus' ([74, Definition 1.7]), which in this situation is defined to be the set of points x ∈ X such that there exists σ ∈ H 0 (X, L ⊕r ) G for r > 0 with σ(x) = 0 and such that X σ is affine 4 . The semistable set and quotient obtained in this way are both quasi-projective (see [79,Theorem 3.21]).…”

“…3 By construction, the projective GIT quotient comes with a line bundle and this regrading does not change the GIT quotient but does change this line bundle. 4 If X is projective and L is ample, then this non-vanishing locus is always affine.…”

“…(3) Find an ample line bundle on M to show it is projective. This approach has been implemented for moduli of smooth projective curves in [24], where the second step uses the Keel-Mori Theorem and the third step follows Kollar's proof of projectivity using the determinant of a relative pluricanonical sheaf for the universal family; and for moduli of vector bundles on a smooth projective curve in characteristic zero in [4], where the second step uses Theorem 4.9 and the third step follows Faltings' proof of projectivity using a determinantal line bundle constructed from the universal family. For moduli of representations of an acyclic quiver (in arbitrary characteristic), a new moduli-theoretic proof of projectivity was given in [13], where again a determinantal line bundle is used.…”

“…Alper's notion of good and adequate moduli spaces of stacks enables GIT-free constructions of moduli spaces. This is even more tangible following the recent existence criteria of Alper-Halpern-Leistner-Heinloth [7], which equates the existence of moduli spaces to two simple valuative criteria and has been applied to various moduli problems [4,5,12,13].However, adequate and good moduli spaces are locally modelled on reductive GIT and require closed points to have reductive stabiliser groups. Thus, in some senses these two recent developments are orthogonal to each other and ideally there should eventually be an extension of non-reductive GIT to stacks.Acknowledgements.…”