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 Modular compactifications of the space of pointed elliptic curves II

2011 ◽  
Vol 147 (6) ◽  
pp. 1843-1884 ◽  
Author(s):  
David Ishii Smyth
Keyword(s):  
Elliptic Curves ◽  
Moduli Spaces ◽  
Minimal Model ◽  
Model Program ◽  
Minimal Model Program ◽  
Stable Curves ◽  
Log Minimal Model Program

AbstractWe prove that the moduli spaces of n-pointed m-stable curves introduced in our previous paper have projective coarse moduli. We use the resulting spaces to run an analogue of Hassett’s log minimal model program for $\overline {M}_{1,n}$.

scholarly journals On the fine Simpson moduli spaces of 1-dimensional sheaves supported on plane quartics

2018 ◽  
Vol 16 (1) ◽  
pp. 46-62
Author(s):  
Oleksandr Iena
Keyword(s):  
Parameter Space ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Blow Up ◽  
Projective Bundle ◽  
Blow Down ◽  
Plane Quartics ◽  
Common Parameter

AbstractA parametrization of the fine Simpson moduli spaces of 1-dimensional sheaves supported on plane quartics is given: we describe the gluing of the Brill-Noether loci described by Drézet and Maican, provide a common parameter space for these loci, and show that the Simpson moduli space M = M4m ± 1(ℙ2) is a blow-down of a blow-up of a projective bundle over a smooth moduli space of Kronecker modules. Two different proofs of this statement are given.

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 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.

scholarly journals Monads for framed sheaves on Hirzebruch surfaces

2015 ◽  
Vol 15 (1) ◽  
Author(s):  
Claudio Bartocci ◽  
Claudio L. S. Rava ◽  
Ugo Bruzzo
Keyword(s):  
Moduli Spaces ◽  
Algebraic Varieties ◽  
Hirzebruch Surfaces ◽  
Framed Sheaves ◽  
Torsion Free

AbstractWe define monads for framed torsion-free sheaves on Hirzebruch surfaces and use them to construct moduli spaces for these objects. These moduli spaces are smooth algebraic varieties, and we show that they are fine by constructing a universal monad.

scholarly journals Moduli of products of stable varieties

2013 ◽  
Vol 149 (12) ◽  
pp. 2036-2070 ◽  
Author(s):  
Bhargav Bhatt ◽  
Wei Ho ◽  
Zsolt Patakfalvi ◽  
Christian Schnell
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Minimal Model ◽  
Complex Numbers ◽  
Model Program ◽  
Minimal Model Program ◽  
Cotangent Complex ◽  
Geometric Methods ◽  
Local Results

AbstractWe study the moduli space of a product of stable varieties over the field of complex numbers, as defined via the minimal model program. Our main results are: (a) taking products gives a well-defined morphism from the product of moduli spaces of stable varieties to the moduli space of a product of stable varieties; (b) this map is always finite étale; and (c) this map very often is an isomorphism. Our results generalize and complete the work of Van Opstall in dimension$1$. The local results rely on a study of the cotangent complex using some derived algebro-geometric methods, while the global ones use some differential-geometric input.

scholarly journals Reachable sheaves on ribbons and deformations of moduli spaces of sheaves

2017 ◽  
Vol 28 (12) ◽  
pp. 1750086
Author(s):  
Jean-Marc Drézet
Keyword(s):  
Line Bundle ◽  
Moduli Spaces ◽  
Coherent Sheaf ◽  
Manuscripta Math ◽  
Projective Curve ◽  
Smooth Curves ◽  
Coherent Sheaves ◽  
Moduli Spaces Of Sheaves ◽  
Irreducible Components ◽  
Torsion Free

A primitive multiple curve is a Cohen–Macaulay irreducible projective curve [Formula: see text] that can be locally embedded in a smooth surface, and such that [Formula: see text] is smooth. In this case, [Formula: see text] is a line bundle on [Formula: see text]. If [Formula: see text] is of multiplicity 2, i.e. if [Formula: see text], [Formula: see text] is called a ribbon. If [Formula: see text] is a ribbon and [Formula: see text], then [Formula: see text] can be deformed to smooth curves, but in general a coherent sheaf on [Formula: see text] cannot be deformed in coherent sheaves on the smooth curves. It has been proved in [Reducible deformations and smoothing of primitive multiple curves, Manuscripta Math. 148 (2015) 447–469] that a ribbon with associated line bundle [Formula: see text] such that [Formula: see text] can be deformed to reduced curves having two irreducible components if [Formula: see text] can be written as [Formula: see text] where [Formula: see text] are distinct points of [Formula: see text]. In this case we prove that quasi-locally free sheaves on [Formula: see text] can be deformed to torsion-free sheaves on the reducible curves with two components. This has some consequences on the structure and deformations of the moduli spaces of semi-stable sheaves on [Formula: see text].

scholarly journals Framed sheaves on projective space and Quot schemes

2021 ◽  
Author(s):  
Alberto Cazzaniga ◽  
Andrea T. Ricolfi
Keyword(s):  
Projective Space ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Chern Character ◽  
Free Sheaf ◽  
Quot Scheme ◽  
Quot Schemes ◽  
Framed Sheaves ◽  
Torsion Free

AbstractWe prove that, given integers $$m\ge 3$$ m ≥ 3 , $$r\ge 1$$ r ≥ 1 and $$n\ge 0$$ n ≥ 0 , the moduli space of torsion free sheaves on $${\mathbb {P}}^m$$ P m with Chern character $$(r,0,\ldots ,0,-n)$$ ( r , 0 , … , 0 , - n ) that are trivial along a hyperplane $$D \subset {\mathbb {P}}^m$$ D ⊂ P m is isomorphic to the Quot scheme $$\mathrm{Quot}_{{\mathbb {A}}^m}({\mathscr {O}}^{\oplus r},n)$$ Quot A m ( O ⊕ r , n ) of 0-dimensional length n quotients of the free sheaf $${\mathscr {O}}^{\oplus r}$$ O ⊕ r on $${\mathbb {A}}^m$$ A m . The proof goes by comparing the two tangent-obstruction theories on these moduli spaces.

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