Moduli Spaces Of Semistable Sheaves Research Articles

Compactification ofMℙ3(0,2) and Poncelet pairs of conics

Let M(0, 2) denote the quasi-projective variety of isomorphism classes of stable rank 2 vector bundles on P3(C) with C1=0 and C2=2 . In this paper we study a natural (irreducible) compactification of M(0, 2) and describe explicitly the sheaves on P3 which occur in the closure of M(0, 2) in the moduli space of semi-stable sheaves on P3 with c1= 0, c2=2 and c3=0.

Sur l'espace de modules des faisceaux semi stables de rang 2, de classes de Chern (0,3) sur ${\Bbb P}_2$

The moduli space of semi-stable sheaves of rank-2 and Chern classes (0,3) on the projective plane ℙ 2 , denoted by M=M(0,3), is a smooth irreducible projective variety of dimension 9. In M, the points which come from sheaves which are not locally free give a hypersurface ∂M. In this article we show that all complete surfaces in M must meet the boundary ∂M; in other words, there does not exist a family of vector bundles which is parametrized by a complete surface in M. The essential point of the proof is the construction of a birational morphism Φ:M→, Gr from M to the grassmannian Gr of nets of conics in ℙ 2 ; this allows us to identify M with a blow-up of Gr along a surface. This gives us a precise description of cohomology algebra of M and we use this to determine the fundamental class of a complete surface in M which does not meet ∂M. We then show that this value of the fundamental class can not arise.

The Euclid-Fourier-Mukai algorithm for elliptic surfaces

We describe the birational correspondences, induced by the Fourier-Mukai functor, between moduli spaces of semistable sheaves on elliptic surfaces with sections, using the notion of $P$-stability in the derived category. We give explicit conditions to determine whether these correspondences are isomorphisms. This is indeed not true in general and we describe the cases where the birational maps are Mukai flops. Moreover, this construction provides examples of new compactifications of the moduli spaces of vector bundles via sheaves with torsion and via complexes. We finally get for any fixed dimension an isomorphism between the Picard groups of the moduli spaces.

Moduli spaces of sheaves on K3 surfaces and symplectic stacks

We view the moduli space of semistable sheaves on a K3 surface as a global quotient stack, and compute its cotangent complex in terms of the universal sheaf on the Quot scheme. Relevant facts on the classical and reduced Atiyah classes are reviewed. We also define the notion of a symplectic stack, and show that it includes all moduli stacks of semistable sheaves on K3 surfaces.

Moduli Spaces of Semistable Sheaves of Dimension $1$ on $\mathbb{P}^2$

Let $M(d,\chi)$ be the moduli space of semistable sheaves of rank 0, Euler characteristic $\chi$ and first Chern class $dH (d>0)$, with $H$ the hyperplane class in $\mathbb{P}^2$. We give a description of $M(d,\chi)$, viewing each sheaf as a class of matrices with entries in $\bigoplus_{i\geq0}H^0(\mathcal{O}_{\mathbb{P}^2}(i))$. We show that there is a big open subset of $M(d,1)$ isomorphic to a projective bundle over an open subset of a Hilbert scheme of points on $\mathbb{P}^2.$ Finally we compute the classes of M(4,1), M(5,1) and M(5,2) in the Grothendieck group of varieties, especially we conclude that M(5,1) and M(5,2) are of the same class.