scholarly journals THE EFFECTIVE CONE OF MODULI SPACES OF SHEAVES ON A SMOOTH QUADRIC SURFACE

2017 ◽  
Vol 232 ◽  
pp. 151-215 ◽  
Author(s):  
TIM RYAN
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Chern Character ◽  
Hilbert Schemes ◽  
Quadric Surface ◽  
Moduli Spaces Of Sheaves ◽  
Smooth Quadric Surface ◽  
Effective Cone

Let $\unicode[STIX]{x1D709}$ be a stable Chern character on $\mathbb{P}^{1}\times \mathbb{P}^{1}$, and let $M(\unicode[STIX]{x1D709})$ be the moduli space of Gieseker semistable sheaves on $\mathbb{P}^{1}\times \mathbb{P}^{1}$ with Chern character $\unicode[STIX]{x1D709}$. In this paper, we provide an approach to computing the effective cone of $M(\unicode[STIX]{x1D709})$. We find Brill–Noether divisors spanning extremal rays of the effective cone using resolutions of the general elements of $M(\unicode[STIX]{x1D709})$ which are found using the machinery of exceptional bundles. We use this approach to provide many examples of extremal rays in these effective cones. In particular, we completely compute the effective cone of the first fifteen Hilbert schemes of points on $\mathbb{P}^{1}\times \mathbb{P}^{1}$.

scholarly journals Moduli of stable sheaves supported on curves of genus three contained in a quadric surface

2020 ◽  
Vol 20 (4) ◽  
pp. 507-522
Author(s):  
Mario Maican
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Betti Numbers ◽  
Free Resolutions ◽  
Quadric Surface ◽  
Arithmetic Genus ◽  
Blow Down ◽  
Stable Sheaves ◽  
Stable Pairs ◽  
Smooth Quadric Surface

AbstractWe study the moduli space of stable sheaves of Euler characteristic 1 supported on curves of arithmetic genus 3 contained in a smooth quadric surface. We show that this moduli space is rational. We compute its Betti numbers by studying the variation of the moduli spaces of α-semi-stable pairs. We classify the stable sheaves using locally free resolutions or extensions. We give a global description: the moduli space is obtained from a certain flag Hilbert scheme by performing two flips followed by a blow-down.

scholarly journals K-MODULI OF CURVES ON A QUADRIC SURFACE AND K3 SURFACES

2021 ◽  
pp. 1-41
Author(s):  
KENNETH ASCHER ◽  
KRISTIN DEVLEMING ◽  
YUCHEN LIU
Keyword(s):  
Complete Intersection ◽  
Moduli Space ◽  
Moduli Spaces ◽  
K3 Surfaces ◽  
Moduli Of Curves ◽  
Quadric Surface ◽  
Intersection Curves

Abstract We show that the K-moduli spaces of log Fano pairs $\left(\mathbb {P}^1\times \mathbb {P}^1, cC\right)$ , where C is a $(4,4)$ curve and their wall crossings coincide with the VGIT quotients of $(2,4)$ , complete intersection curves in $\mathbb {P}^3$ . This, together with recent results by Laza and O’Grady, implies that these K-moduli spaces form a natural interpolation between the GIT moduli space of $(4,4)$ curves on $\mathbb {P}^1\times \mathbb {P}^1$ and the Baily–Borel compactification of moduli of quartic hyperelliptic K3 surfaces.

The Effective Cone of the Kontsevich Moduli Space

2008 ◽  
Vol 51 (4) ◽  
pp. 519-534 ◽  
Author(s):  
Izzet Coskun ◽  
Joe Harris ◽  
Jason Starr
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Complete Characterization ◽  
Stable Maps ◽  
Linear Combinations ◽  
Effective Divisors ◽  
Effective Cone

AbstractIn this paper we prove that the cone of effective divisors on the Kontsevich moduli spaces of stable maps, , stabilize when r ≥ d. We give a complete characterization of the effective divisors on . They are non-negative linear combinations of boundary divisors and the divisor of maps with degenerate image.

scholarly journals Stable Sheaves on a Smooth Quadric Surface with Linear Hilbert Bipolynomials

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Edoardo Ballico ◽  
Sukmoon Huh
Keyword(s):  
Moduli Spaces ◽  
Free Resolution ◽  
Quadric Surface ◽  
Special Cases ◽  
Stable Sheaves ◽  
Smooth Quadric Surface

We investigate the moduli spaces of stable sheaves on a smooth quadric surface with linear Hilbert bipolynomial in some special cases and describe their geometry in terms of the locally free resolution of the sheaves.

scholarly journals Moduli of vector bundles on higher-dimensional base manifolds — Construction and variation

2016 ◽  
Vol 27 (07) ◽  
pp. 1650054 ◽  
Author(s):  
Daniel Greb ◽  
Julius Ross ◽  
Matei Toma
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Vector Bundles ◽  
Recent Progress ◽  
Geometric Construction ◽  
Quiver Representations ◽  
Natural Morphism ◽  
Moduli Spaces Of Sheaves ◽  
Higher Dimensional ◽  
Moduli Of Vector Bundles

We survey recent progress in the study of moduli of vector bundles on higher-dimensional base manifolds. In particular, we discuss an algebro-geometric construction of an analogue for the Donaldson–Uhlenbeck compactification and explain how to use moduli spaces of quiver representations to show that Gieseker–Maruyama moduli spaces with respect to two different chosen polarizations are related via Thaddeus-flips through other “multi-Gieseker”-moduli spaces of sheaves. Moreover, as a new result, we show the existence of a natural morphism from a multi-Gieseker moduli space to the corresponding Donaldson–Uhlenbeck moduli space.

scholarly journals Refined Verlinde formulas for Hilbert schemes of points and moduli spaces of sheaves on K3 surfaces

2020 ◽  
Vol Volume 4 ◽  
Author(s):  
Lothar Göttsche
Keyword(s):  
Moduli Spaces ◽  
Generating Functions ◽  
K3 Surfaces ◽  
Line Bundles ◽  
Hilbert Schemes ◽  
Moduli Spaces Of Sheaves ◽  
Elliptic Genera ◽  
Determinant Line Bundles

We compute generating functions for elliptic genera with values in line bundles on Hilbert schemes of points on surfaces. As an application we also compute generating functions for elliptic genera with values in determinant line bundles on moduli spaces of sheaves on K3 surfaces.

SEMISTABLE SHEAVES WITH SYMMETRIC ON A QUADRIC SURFACE

2016 ◽  
Vol 227 ◽  
pp. 86-159 ◽  
Author(s):  
TAKESHI ABE
Keyword(s):  
Projective Plane ◽  
Moduli Spaces ◽  
Rational Maps ◽  
Quadric Surface ◽  
Plane Case ◽  
Moduli Spaces Of Sheaves ◽  
Strange Duality

For moduli spaces of sheaves with symmetric $c_{1}$ on a quadric surface, we pursue analogy to some results known for moduli spaces of sheaves on a projective plane. We define an invariant height, introduced by Drezet in the projective plane case, for moduli spaces of sheaves with symmetric $c_{1}$ on a quadric surface and describe the structure of moduli spaces of height zero. Then we study rational maps of moduli spaces of positive height to moduli spaces of representation of quivers, effective cones of moduli spaces, and strange duality for height-zero moduli spaces.

scholarly journals Non-commutative thickening of moduli spaces of stable sheaves

2017 ◽  
Vol 153 (6) ◽  
pp. 1153-1195 ◽  
Author(s):  
Yukinobu Toda
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Moduli Spaces Of Sheaves ◽  
Closed Point ◽  
Stable Sheaves

We show that the moduli spaces of stable sheaves on projective schemes admit certain non-commutative structures, which we call quasi-NC structures, generalizing Kapranov’s NC structures. The completion of our quasi-NC structure at a closed point of the moduli space gives a pro-representable hull of the non-commutative deformation functor of the corresponding sheaf developed by Laudal, Eriksen, Segal and Efimov–Lunts–Orlov. We also show that the framed stable moduli spaces of sheaves have canonical NC structures.

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.

scholarly journals LOCALIZATION BY -PERIODIC COMPLEXES AND VIRTUAL STRUCTURE SHEAVES

2020 ◽  
pp. 1-30
Author(s):  
Jeongseok Oh ◽  
Bhamidi Sreedhar
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Chern Character ◽  
Chern Characters ◽  
Periodic Complexes

Abstract In [12], Kim and the first author proved a result comparing the virtual fundamental classes of the moduli spaces of $\varepsilon $ -stable quasimaps and $\varepsilon $ -stable $LG$ -quasimaps by studying localized Chern characters for $2$ -periodic complexes. In this paper, we study a K-theoretic analogue of the localized Chern character map and show that for a Koszul $2$ -periodic complex it coincides with the cosection-localized Gysin map of Kiem and Li [11]. As an application, we compare the virtual structure sheaves of the moduli space of $\varepsilon $ -stable quasimaps and $\varepsilon $ -stable $LG$ -quasimaps.

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