Geometric Invariant Theory of Syzygies, with Applications to Moduli Spaces

We expose in detail the principle that the relative geometric invariant theory of equivariant morphisms is related to the GIT for linearizations near the boundary of the G-effective ample cone. We then apply this principle to construct and reconstruct various universal moduli spaces. In particular, we constructed the universal moduli space over [Formula: see text] of Simpson’s p-semistable coherent sheaves and a canonical rational morphism from the universal Hilbert scheme over [Formula: see text] to a compactified universal Picard.

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Birational geometry of the moduli space of quartic surfaces

Compositio Mathematica ◽

10.1112/s0010437x19007516 ◽

2019 ◽

Vol 155 (9) ◽

pp. 1655-1710

Author(s):

Radu Laza ◽

Kieran O’Grady

Keyword(s):

Moduli Space ◽

Moduli Spaces ◽

Invariant Theory ◽

Geometric Invariant Theory ◽

Dimensional Case ◽

Geometric Invariant ◽

Type Iv ◽

Log Canonical ◽

Symmetric Varieties ◽

The Relationship

By work of Looijenga and others, one understands the relationship between Geometric Invariant Theory (GIT) and Baily–Borel compactifications for the moduli spaces of degree-$2$ $K3$ surfaces, cubic fourfolds, and a few other related examples. The similar-looking cases of degree-$4$ $K3$ surfaces and double Eisenbud–Popescu–Walter (EPW) sextics turn out to be much more complicated for arithmetic reasons. In this paper, we refine work of Looijenga in order to handle these cases. Specifically, in analogy with the so-called Hassett–Keel program for the moduli space of curves, we study the variation of log canonical models for locally symmetric varieties of Type IV associated to $D$-lattices. In particular, for the $19$-dimensional case, we conjecturally obtain a continuous one-parameter interpolation between the GIT and Baily–Borel compactifications for the moduli of degree-$4$ $K3$ surfaces. The analogous $18$-dimensional case, which corresponds to hyperelliptic degree-$4$ $K3$ surfaces, can be verified by means of Variation of Geometric Invariant Theory (VGIT) quotients.

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Variation of geometric invariant theory quotients and derived categories

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2015-0096 ◽

2019 ◽

Vol 2019 (746) ◽

pp. 235-303 ◽

Cited By ~ 8

Author(s):

Matthew Ballard ◽

David Favero ◽

Ludmil Katzarkov

Keyword(s):

Moduli Spaces ◽

Invariant Theory ◽

General Framework ◽

Derived Categories ◽

Rational Curves ◽

Geometric Invariant Theory ◽

Geometric Invariant ◽

Coherent Sheaves ◽

Orthogonal Decompositions ◽

The Relationship

Abstract We study the relationship between derived categories of factorizations on gauged Landau–Ginzburg models related by variations of the linearization in Geometric Invariant Theory. Under assumptions on the variation, we show the derived categories are comparable by semi-orthogonal decompositions and we completely describe all components appearing in these semi-orthogonal decompositions. We show how this general framework encompasses many well-known semi-orthogonal decompositions. We then proceed to give applications of this complete description. In this setting, we verify a question posed by Kawamata: we show that D-equivalence and K-equivalence coincide for such variations. The results are applied to obtain a simple inductive description of derived categories of coherent sheaves on projective toric Deligne–Mumford stacks. This recovers Kawamata’s theorem that all projective toric Deligne–Mumford stacks have full exceptional collections. Using similar methods, we prove that the Hassett moduli spaces of stable symmetrically-weighted rational curves also possess full exceptional collections. As a final application, we show how our results recover and extend Orlov’s σ-model/Landau–Ginzburg model correspondence.

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Semi-continuity of stability for sheaves and variation of Gieseker moduli spaces

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2016-0022 ◽

2019 ◽

Vol 2019 (749) ◽

pp. 227-265 ◽

Cited By ~ 3

Author(s):

Daniel Greb ◽

Julius Ross ◽

Matei Toma

Keyword(s):

Finite Number ◽

Moduli Spaces ◽

Stability Condition ◽

Invariant Theory ◽

Geometric Invariant Theory ◽

Stability Conditions ◽

Geometric Invariant ◽

Moduli Spaces Of Sheaves ◽

Stable Sheaves ◽

The Stability

Abstract We investigate a semi-continuity property for stability conditions for sheaves that is important for the problem of variation of the moduli spaces as the stability condition changes. We place this in the context of a notion of stability previously considered by the authors, called multi-Gieseker-stability, that generalises the classical notion of Gieseker-stability to allow for several polarisations. As such we are able to prove that on smooth threefolds certain moduli spaces of Gieseker-stable sheaves are related by a finite number of Thaddeus-flips (that is flips arising for Variation of Geometric Invariant Theory) whose intermediate spaces are themselves moduli spaces of sheaves.

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NOTES ON COHERENT SYSTEMS

Revista de Matemática Teoría y Aplicaciones ◽

10.15517/rmta.v28i1.42154 ◽

2020 ◽

Vol 28 (1) ◽

pp. 1-38

Author(s):

ALEXANDER H.W. SCHMITT

Keyword(s):

Moduli Spaces ◽

Invariant Theory ◽

Vector Bundles ◽

Geometric Invariant Theory ◽

Geometric Invariant ◽

Coherent Systems ◽

Alternative Approach

We present an alternative approach to semistability and moduli spaces for coherent systems associated with decorated vector bundles. In this approach, it seems possible to construct a Hitchin map. We relate some examples to classical problems from geometric invariant theory.

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