Dimension estimates for Hilbert schemes and effective base point freeness on moduli spaces of vector bundles on curves

AbstractBy using a Fourier-Mukai transform for sheaves on K3 surfaces we show that for a wide class of K3 surfaces X the Hilbert schemes Hilbn(X) can be identified for all n ≥ 1 with moduli spaces of Gieseker stable vector bundles on X. We also introduce a new Fourier-Mukai type transform for such surfaces.

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Perverse bundles and Calogero–Moser spaces

Compositio Mathematica ◽

10.1112/s0010437x0800359x ◽

2008 ◽

Vol 144 (6) ◽

pp. 1403-1428 ◽

Cited By ~ 8

Author(s):

David Ben-Zvi ◽

Thomas Nevins

Keyword(s):

Moduli Spaces ◽

Vector Bundles ◽

Hilbert Schemes ◽

Simple Description ◽

Quiver Varieties ◽

Complex Curves ◽

Noncommutative Version ◽

Cotangent Bundles ◽

Rank One ◽

Torsion Free

AbstractWe present a simple description of moduli spaces of torsion-free 𝒟-modules (𝒟-bundles) on general smooth complex curves, generalizing the identification of the space of ideals in the Weyl algebra with Calogero–Moser quiver varieties. Namely, we show that the moduli of 𝒟-bundles form twisted cotangent bundles to moduli of torsion sheaves on X, answering a question of Ginzburg. The corresponding (untwisted) cotangent bundles are identified with moduli of perverse vector bundles on T*X, which contain as open subsets the moduli of framed torsion-free sheaves (the Hilbert schemes T*X[n] in the rank-one case). The proof is based on the description of the derived category of 𝒟-modules on X by a noncommutative version of the Beilinson transform on P1.

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THE HYPERELLIPTIC THETA MAP AND OSCULATING PROJECTIONS

Nagoya Mathematical Journal ◽

10.1017/nmj.2020.37 ◽

2020 ◽

pp. 1-23

Author(s):

MICHELE BOLOGNESI ◽

NÉSTOR FERNÁNDEZ VARGAS

Keyword(s):

Moduli Space ◽

Moduli Spaces ◽

Vector Bundles ◽

Hyperelliptic Curve ◽

Geometric Description ◽

Birational Model ◽

Rank 2 ◽

Semistable Vector Bundles ◽

Ramification Locus

Abstract Let C be a hyperelliptic curve of genus $g \geq 3$ . In this paper, we give a new geometric description of the theta map for moduli spaces of rank 2 semistable vector bundles on C with trivial determinant. In order to do this, we describe a fibration of (a birational model of) the moduli space, whose fibers are GIT quotients $(\mathbb {P}^1)^{2g}//\text {PGL(2)}$ . Then, we identify the restriction of the theta map to these GIT quotients with some explicit degree 2 osculating projection. As a corollary of this construction, we obtain a birational inclusion of a fibration in Kummer $(g-1)$ -varieties over $\mathbb {P}^g$ inside the ramification locus of the theta map.

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NON-SYMPLECTIC INVOLUTIONS ON MANIFOLDS OF -TYPE

Nagoya Mathematical Journal ◽

10.1017/nmj.2019.43 ◽

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.

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ON THE STABILITY OF SYZYGY BUNDLES

International Journal of Mathematics ◽

10.1142/s0129167x1100688x ◽

2011 ◽

Vol 22 (04) ◽

pp. 515-534 ◽

Cited By ~ 1

Author(s):

IUSTIN COANDĂ

Keyword(s):

Projective Space ◽

Complete Intersection ◽

Vector Bundles ◽

Elementary Proof ◽

Base Point ◽

Vector Spaces ◽

Similar Argument ◽

Vanishing Theorem ◽

The Stability ◽

Koszul Cohomology

We are concerned with the problem of the stability of the syzygy bundles associated to base-point-free vector spaces of forms of the same degree d on the projective space of dimension n. We deduce directly, from M. Green's vanishing theorem for Koszul cohomology, that any such bundle is stable if its rank is sufficiently high. With a similar argument, we prove the semistability of a certain syzygy bundle on a general complete intersection of hypersurfaces of degree d in the projective space. This answers a question of H. Flenner [Comment. Math. Helv.59 (1984) 635–650]. We then give an elementary proof of H. Brenner's criterion of stability for monomial syzygy bundles, avoiding the use of Klyachko's results on toric vector bundles. We finally prove the existence of stable syzygy bundles defined by monomials of the same degree d, of any possible rank, for n at least 3. This extends the similar result proved, for n = 2, by L. Costa, P. Macias Marques and R. M. Miro-Roig [J. Pure Appl. Algebra214 (2010) 1241–1262]. The extension to the case n at least 3 has been also, independently, obtained by P. Macias Marques in his thesis [arXiv:0909.4646/math.AG (2009)].

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COHERENT SYSTEMS AND MODULAR SUBAVRIETIES OF $\mathcal{SU}_C(r)$

International Journal of Mathematics ◽

10.1142/s0129167x12500371 ◽

2012 ◽

Vol 23 (04) ◽

pp. 1250037 ◽

Cited By ~ 4

Author(s):

MICHELE BOLOGNESI ◽

SONIA BRIVIO

Keyword(s):

Moduli Spaces ◽

Vector Bundles ◽

Low Rank ◽

Coherent Systems ◽

Projective Varieties ◽

Complex Curve ◽

Smooth Complex ◽

Moduli Theory ◽

Modular Varieties ◽

Small Genus

Let C be an algebraic smooth complex curve of genus g > 1. The object of this paper is the study of the birational structure of certain moduli spaces of vector bundles and of coherent systems on C and the comparison of different type of notions of stability arising in moduli theory. Notably we show that in certain cases these moduli spaces are birationally equivalent to fibrations over simple projective varieties, whose fibers are GIT quotients (ℙr-1)rg// PGL (r), where r is the rank of the considered vector bundles. This allows us to compare different definitions of (semi-)stability (slope stability, α-stability, GIT stability) for vector bundles, coherent systems and point sets, and derive relations between them. In certain cases of vector bundles of low rank when C has small genus, our construction produces families of classical modular varieties contained in the Coble hypersurfaces.

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