scholarly journals Moduli spaces of twisted sheaves on a projective variety

2019 ◽  
Author(s):  
Kōta Yoshioka
Keyword(s):  
Moduli Spaces ◽  
Projective Variety ◽  
Twisted Sheaves

scholarly journals NON-SYMPLECTIC INVOLUTIONS ON MANIFOLDS OF -TYPE

2020 ◽  
pp. 1-25
Author(s):  
CHIARA CAMERE ◽  
ALBERTO CATTANEO ◽  
ANDREA CATTANEO
Keyword(s):  
Moduli Spaces ◽  
Quadratic Forms ◽  
Symplectic Manifolds ◽  
Hilbert Schemes ◽  
Small Rank ◽  
Twisted Sheaves ◽  
Formula Type

We study irreducible holomorphic symplectic manifolds deformation equivalent to Hilbert schemes of points on a $K3$ surface and admitting a non-symplectic involution. We classify the possible discriminant quadratic forms of the invariant and coinvariant lattice for the action of the involution on cohomology and explicitly describe the lattices in the cases where the invariant lattice has small rank. We also give a modular description of all $d$ -dimensional families of manifolds of $K3^{[n]}$ -type with a non-symplectic involution for $d\geqslant 19$ and $n\leqslant 5$ and provide examples arising as moduli spaces of twisted sheaves on a $K3$ surface.

scholarly journals MODULI SPACE OF STABLE MAPS TO PROJECTIVE SPACE VIA GIT

2010 ◽  
Vol 21 (05) ◽  
pp. 639-664 ◽  
Author(s):  
YOUNG-HOON KIEM ◽  
HAN-BOM MOON
Keyword(s):  
Projective Space ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Projective Variety ◽  
Blow Up ◽  
Stable Maps ◽  
Homogeneous Polynomials ◽  
Birational Map

We compare the Kontsevich moduli space [Formula: see text] of stable maps to projective space with the quasi-map space ℙ( Sym d(ℂ2) ⊗ ℂn)//SL(2). Consider the birational map [Formula: see text] which assigns to an n tuple of degree d homogeneous polynomials f1, …, fn in two variables, the map f = (f1 : ⋯ : fn) : ℙ1 → ℙn-1. In this paper, for d = 3, we prove that [Formula: see text] is the composition of three blow-ups followed by two blow-downs. Furthermore, we identify the blow-up/down centers explicitly in terms of the moduli spaces [Formula: see text] with d = 1, 2. In particular, [Formula: see text] is the SL(2)-quotient of a smooth rational projective variety. The degree two case [Formula: see text], which is the blow-up of ℙ( Sym 2ℂ2 ⊗ ℂn)//SL(2) along ℙn-1, is worked out as a preliminary example.

scholarly journals Chow instability of certain projective varieties

1983 ◽  
Vol 92 ◽  
pp. 39-50 ◽  
Author(s):  
Shihoko Ishii
Keyword(s):  
Moduli Spaces ◽  
Projective Variety ◽  
Ample Divisor ◽  
Projective Varieties

A pair (X, D) of a projective variety X and a very ample divisor D on X is called stable (resp. semi-stable, resp. unstable) if the Chow point corresponding to the embedding is SL(N + 1)-stable (resp. semi-stable, resp. unstable). The criterion for stability is one of the most important steps in proving the existence of moduli spaces.

scholarly journals On the Euler characteristics of certain moduli spaces of 1-dimensional subschemes

2017 ◽  
Author(s):  
◽  
Mazen M. Alhwaimel
Keyword(s):  
Generating Function ◽  
Moduli Spaces ◽  
Projective Variety ◽  
Smooth Projective Variety ◽  
Euler Characteristics ◽  
Torus Actions ◽  
3 Dimensional ◽  
Hodge Polynomials

Generalizing the ideas in [LQ] and using virtual Hodge polynomials as well as torus actions, we compute the Euler characteristics of some moduli spaces of 1-dimensional closed subschemes when the ambient smooth projective variety admits a Zariskilocally trivial fibration to a codimension-1 base. As a consequence, we partially verify a conjecture of W.-P. Li and Qin [LQ]. We also calculate the generating function for the number of certain punctual 3-dimensional partitions, which is used to compute the above Euler characteristics.

Unramified Brauer group of the moduli spaces of PGLr(ℂ)-bundles over curves

2014 ◽  
Vol 12 (8) ◽  
Author(s):  
Indranil Biswas ◽  
Amit Hogadi ◽  
Yogish Holla
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Projective Variety ◽  
Brauer Group ◽  
Projective Curve ◽  
Connected Component ◽  
Smooth Complex ◽  
Complex Projective Variety ◽  
Complex Projective ◽  
Complex Projective Curve

AbstractLet X be an irreducible smooth complex projective curve of genus g, with g ≥ 2. Let N be a connected component of the moduli space of semistable principal PGLr (ℂ)-bundles over X; it is a normal unirational complex projective variety. We prove that the Brauer group of a desingularization of N is trivial.

scholarly journals Verra Four-Folds, Twisted Sheaves, and the Last Involution

2018 ◽  
Vol 2019 (21) ◽  
pp. 6661-6710 ◽  
Author(s):  
Chiara Camere ◽  
Grzegorz Kapustka ◽  
Michał Kapustka ◽  
Giovanni Mongardi
Keyword(s):  
Moduli Spaces ◽  
Hilbert Scheme ◽  
K3 Surfaces ◽  
Symplectic Manifolds ◽  
K3 Surface ◽  
Enriques Surfaces ◽  
Twisted Sheaves

Abstract We study the geometry of some moduli spaces of twisted sheaves on K3 surfaces. In particular we introduce induced automorphisms from a K3 surface on moduli spaces of twisted sheaves on this K3 surface. As an application we prove the unirationality of moduli spaces of irreducible holomorphic symplectic manifolds of K3[2]-type admitting non-symplectic involutions with invariant lattices U(2) ⊕ D4(−1) or U(2) ⊕ E8(−2). This complements the results obtained in [43], [13], and the results from [29] about the geometry of irreducible holomorphic symplectic (IHS) four-folds constructed using the Hilbert scheme of (1, 1) conics on Verra four-folds. As a byproduct we find that IHS four-folds of K3[2]-type with Picard lattice U(2) ⊕ E8(−2) naturally contain non-nodal Enriques surfaces.

scholarly journals On the equations and classification of toric quiver varieties

2016 ◽  
Vol 146 (2) ◽  
pp. 265-295 ◽  
Author(s):  
Mátyás Domokos ◽  
Dániel Joó
Keyword(s):  
Projective Space ◽  
Moduli Spaces ◽  
Toric Variety ◽  
Projective Variety ◽  
Homogeneous Ideal ◽  
Quiver Representations ◽  
Canonical Embedding ◽  
Quiver Varieties ◽  
Fixed Dimension

Toric quiver varieties (moduli spaces of quiver representations) are studied. Given a quiver and a weight, there is an associated quasi-projective toric variety together with a canonical embedding into projective space. It is shown that for a quiver with no oriented cycles the homogeneous ideal of this embedded projective variety is generated by elements of degree at most 3. In each fixed dimension d up to isomorphism there are only finitely many d-dimensional toric quiver varieties. A procedure for their classification is outlined.

scholarly journals Some Quot schemes in tilted hearts and moduli spaces of stable pairs

2021 ◽  
Author(s):  
Franco Rota
Keyword(s):  
Moduli Spaces ◽  
Projective Variety ◽  
Smooth Projective Variety ◽  
Stability Conditions ◽  
Verlinde Formula ◽  
Quot Schemes ◽  
Text Structures ◽  
Stable Pairs ◽  
Technical Framework ◽  
Invent Math

For a smooth projective variety [Formula: see text], we study analogs of Quot schemes using hearts of non-standard [Formula: see text]-structures of [Formula: see text]. The technical framework uses families of [Formula: see text]-structures as studied in A. Bayer, M. Lahoz, E. Macrì, H. Nuer, A. Perry and P. Stellari, Stability conditions in families, preprint (2019), arXiv:1902.08184. We provide several examples and suggest possible directions of further investigation, as we reinterpret moduli spaces of stable pairs, in the sense of M. Thaddeus, Stable pairs, linear systems and the Verlinde formula, Invent. Math. 117(2) (1994) 317–353; D. Huybrechts and M. Lehn, Stable pairs on curves and surfaces, J. Algebraic Geom. 4(1) (1995) 67–104, as instances of Quot schemes.

scholarly journals Demailly’s notion of algebraic hyperbolicity: geometricity, boundedness, moduli of maps

2020 ◽  
Vol 296 (3-4) ◽  
pp. 1645-1672 ◽  
Author(s):  
Ariyan Javanpeykar ◽  
Ljudmila Kamenova
Keyword(s):  
Moduli Spaces ◽  
Projective Variety ◽  
Closed Field ◽  
Projective Varieties ◽  
Complex Projective Variety ◽  
Hyperbolic Complex ◽  
Closed Subvariety ◽  
Algebraically Closed Fields ◽  
Complex Projective ◽  
Smooth Projective Varieties

Abstract Demailly’s conjecture, which is a consequence of the Green–Griffiths–Lang conjecture on varieties of general type, states that an algebraically hyperbolic complex projective variety is Kobayashi hyperbolic. Our aim is to provide evidence for Demailly’s conjecture by verifying several predictions it makes. We first define what an algebraically hyperbolic projective variety is, extending Demailly’s definition to (not necessarily smooth) projective varieties over an arbitrary algebraically closed field of characteristic zero, and we prove that this property is stable under extensions of algebraically closed fields. Furthermore, we show that the set of (not necessarily surjective) morphisms from a projective variety Y to a projective algebraically hyperbolic variety X that map a fixed closed subvariety of Y onto a fixed closed subvariety of X is finite. As an application, we obtain that $${{\,\mathrm{Aut}\,}}(X)$$ Aut ( X ) is finite and that every surjective endomorphism of X is an automorphism. Finally, we explore “weaker” notions of hyperbolicity related to boundedness of moduli spaces of maps, and verify similar predictions made by the Green–Griffiths–Lang conjecture on hyperbolic projective varieties.

scholarly journals Change of Polarization for Moduli of Sheaves on Surfaces as Bridgeland Wall-crossing

2018 ◽  
Vol 2020 (7) ◽  
pp. 2007-2033
Author(s):  
Aaron Bertram ◽  
Cristian Martinez
Keyword(s):  
Moduli Spaces ◽  
Stability Conditions ◽  
One Dimensional ◽  
Wall Crossing ◽  
Moduli Of Sheaves ◽  
Twisted Sheaves

Abstract We prove that the “Thaddeus flips” of L-twisted sheaves constructed by Matsuki and Wentworth explaining the change of polarization for Gieseker semistable sheaves on a surface can be obtained via Bridgeland wall-crossing. Similarly, we realize the change of polarization for moduli spaces of one-dimensional Gieseker semistable sheaves on a surface by varying a family of stability conditions.

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