scholarly journals Birational models of moduli spaces of coherent sheaves on the projective plane

2019 ◽  
Vol 23 (1) ◽  
pp. 347-426 ◽  
Author(s):  
Chunyi Li ◽  
Xiaolei Zhao
Keyword(s):  
Projective Plane ◽  
Moduli Spaces ◽  
Coherent Sheaves

scholarly journals On the Singular Sheaves in the Fine Simpson Moduli Spaces of 1-dimensional Sheaves

2017 ◽  
Vol 60 (3) ◽  
pp. 522-535 ◽  
Author(s):  
Oleksandr Iena ◽  
Alain Leytem
Keyword(s):  
Projective Plane ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Blow Up ◽  
Line Bundles ◽  
Hilbert Polynomial ◽  
Stable Sheaves ◽  
Closed Subvariety

AbstractIn the Simpson moduli space M of semi-stable sheaves with Hilbert polynomial dm − 1 on a projective plane we study the closed subvariety M' of sheaves that are not locally free on their support. We show that for d ≥4 , it is a singular subvariety of codimension 2 in M. The blow up of M along M' is interpreted as a (partial) modification of M \ M' by line bundles (on support).

RELATIVE GEOMETRIC INVARIANT THEORY AND UNIVERSAL MODULI SPACES

1996 ◽  
Vol 07 (02) ◽  
pp. 151-181 ◽  
Author(s):  
YI HU
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Invariant Theory ◽  
Hilbert Scheme ◽  
Geometric Invariant Theory ◽  
Geometric Invariant ◽  
Coherent Sheaves ◽  
Ample Cone

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.

scholarly journals Moduli spaces of framed flags of sheaves on the projective plane

2017 ◽  
Vol 118 ◽  
pp. 138-168
Author(s):  
Rodrigo A. von Flach ◽  
Marcos Jardim
Keyword(s):  
Projective Plane ◽  
Moduli Spaces

scholarly journals Variation of geometric invariant theory quotients and derived categories

2019 ◽  
Vol 2019 (746) ◽  
pp. 235-303 ◽  
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.

scholarly journals Birational geometry of moduli spaces of perverse coherent sheaves on blow-ups

2021 ◽  
Author(s):  
Naoki Koseki
Keyword(s):  
Moduli Spaces ◽  
Blow Up ◽  
Derived Categories ◽  
Model Program ◽  
Minimal Model Program ◽  
Blow Down ◽  
Coherent Sheaves ◽  
Framed Sheaves ◽  
Wall Crossing Formula ◽  
Torsion Free

AbstractIn order to study the wall-crossing formula of Donaldson type invariants on the blown-up plane, Nakajima–Yoshioka constructed a sequence of blow-up/blow-down diagrams connecting the moduli space of torsion free framed sheaves on projective plane, and that on its blow-up. In this paper, we prove that Nakajima–Yoshioka’s diagram realizes the minimal model program. Furthermore, we obtain a fully-faithful embedding between the derived categories of these moduli spaces.

scholarly journals Stability conditions and birational geometry of projective surfaces

2014 ◽  
Vol 150 (10) ◽  
pp. 1755-1788 ◽  
Author(s):  
Yukinobu Toda
Keyword(s):  
Moduli Spaces ◽  
Minimal Model ◽  
Derived Category ◽  
Stability Conditions ◽  
Birational Geometry ◽  
Projective Surface ◽  
Model Program ◽  
Minimal Model Program ◽  
Coherent Sheaves ◽  
Projective Surfaces

AbstractWe show that the minimal model program on any smooth projective surface is realized as a variation of the moduli spaces of Bridgeland stable objects in the derived category of coherent sheaves.

scholarly journals Stability conditions and related filtrations for (G,h)-constellations

2017 ◽  
Vol 28 (14) ◽  
pp. 1750098
Author(s):  
Ronan Terpereau ◽  
Alfonso Zamora
Keyword(s):  
Moduli Spaces ◽  
Derived Categories ◽  
Irreducible Representations ◽  
Quiver Representations ◽  
Geometric Invariant ◽  
Reductive Algebraic Group ◽  
Coherent Sheaves ◽  
Finite Dimensional ◽  
Git Quotient ◽  
Stability Notions

Given an infinite reductive algebraic group [Formula: see text], we consider [Formula: see text]-equivariant coherent sheaves with prescribed multiplicities, called [Formula: see text]-constellations, for which two stability notions arise. The first one is analogous to the [Formula: see text]-stability defined for quiver representations by King [Moduli of representations of finite-dimensional algebras, Quart. J. Math. Oxford Ser.[Formula: see text]2) 45(180) (1994) 515–530] and for [Formula: see text]-constellations by Craw and Ishii [Flops of [Formula: see text]-Hilb and equivalences of derived categories by variation of GIT quotient, Duke Math. J. 124(2) (2004) 259–307], but depending on infinitely many parameters. The second one comes from Geometric Invariant Theory in the construction of a moduli space for [Formula: see text]-constellations, and depends on some finite subset [Formula: see text] of the isomorphy classes of irreducible representations of [Formula: see text]. We show that these two stability notions do not coincide, answering negatively a question raised in [Becker and Terpereau, Moduli spaces of [Formula: see text]-constellations, Transform. Groups 20(2) (2015) 335–366]. Also, we construct Harder–Narasimhan filtrations for [Formula: see text]-constellations with respect to both stability notions (namely, the [Formula: see text]-HN and [Formula: see text]-HN filtrations). Even though these filtrations do not coincide in general, we prove that they are strongly related: the [Formula: see text]-HN filtration is a subfiltration of the [Formula: see text]-HN filtration, and the polygons of the [Formula: see text]-HN filtrations converge to the polygon of the [Formula: see text]-HN filtration when [Formula: see text] grows.

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