Projectivity of the moduli of curves

Cheng, Raymond; Lian, Carl; Murayama, Takumi

doi:10.1017/9781009051897.003

Cited by 2 publications

(3 citation statements)

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“…(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.…”

Section: 2mentioning

confidence: 99%

Moduli spaces and geometric invariant theory: old and new perspectives

Hoskins¹

2023

Preprint

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Many moduli spaces are constructed as quotients of group actions; this paper surveys the classical theory, as well as recent progress and applications. We review geometric invariant theory for reductive groups and how it is used to construct moduli spaces, and explain two new developments extending this theory to non-reductive groups and to stacks, which enable the construction of new moduli spaces. VICTORIA HOSKINSWe also outline another significant development that extends ideas of (reductive) GIT to stacks, as pioneered by Alper, 48,7,52]. 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. I am indebted to both Peter Newstead and Frances Kirwan, as I learned the basics of reductive GIT from Peter Newstead's Tata lecture notes [79] and discussions with Frances Kirwan. I am very grateful to Greg Bérczi, Dominic Bunnett, Eloise Hamilton, Josh Jackson and Frances Kirwan for numerous conversations on non-reductive GIT. I would also like to thank the organisers of VBAC 2022 for soliciting this paper in honour of Peter Newstead.Conventions. Throughout we will assume k is an algebraically closed field and all schemes are assumed to be finite type k-schemes, unless otherwise stated. By a point, we will mean a k-point (or equivalently, a closed point).

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Section: 2mentioning

confidence: 99%

Moduli spaces and geometric invariant theory: old and new perspectives

Hoskins¹

2023

Preprint

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“…In [5] the projectivity of M g was established by giving a stack-theoretic treatment of Kollár's paper.…”

Section: Dedicated To the Memory Of Conjeevaram Srirangachari Seshadrimentioning

confidence: 99%

“…To show that this map is surjective, we must show that every semistable bundle E arises as an extension as in (5). By the first paragraph of the proof, E is globally generated.…”

Section: Boundedness Of This Family Now Follows From Example 214 Withmentioning

confidence: 99%

Projectivity of the moduli space of vector bundles on a curve

Alper¹,

Belmans

Bragg³

et al. 2022

Stacks Project Expository Collection

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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.

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Projectivity of the moduli of curves

Cited by 2 publications

References 18 publications

Moduli spaces and geometric invariant theory: old and new perspectives

Moduli spaces and geometric invariant theory: old and new perspectives

Projectivity of the moduli space of vector bundles on a curve

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