Singularities of moduli spaces of sheaves on K3 surfaces and Nakajima quiver varieties (original) (raw)
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On Some Moduli Spaces of Bundles on K3 Surfaces
Monatshefte Fur Mathematik, 2005
We give infinitely many examples in which the moduli space of rank 2 H-stable sheaves on a K3 surface S endowed by a polarization H of degree 2g – 2, with Chern classes c 1 = H and c 2 = g – 1, is birationally equivalent to the Hilbert scheme S[g – 4] of zero dimensional subschemes of S of length g – 4. We get in this way a partial generalization of results from [5] and [1].
Archiv der Mathematik, 2003
Fix a smooth very ample curve C on a K3 or abelian surface X. Let M denote the moduli space of pairs of the form (F, s), where F is a stable sheaf over X whose Hilbert polynomial coincides with that of the direct image, by the inclusion map of C in X, of a line bundle of degree d over C, and s is a nonzero section of F. Assume d to be sufficiently large such that F has a nonzero section. The pullback of the Mukai symplectic form on moduli spaces of stable sheaves over X is a holomorphic 2-form on M. On the other hand, M has a map to a Hilbert scheme parametrizing 0-dimensional subschemes of X that sends (F, s) to the divisor, defined by s, on the curve defined by the support of F. We prove that the above 2-form on M coincides with the pullback of the symplectic form on Hilbert scheme.
A Note on Formality and Singularities of Moduli Spaces
2010
This paper studies formality of the differential graded algebra RHom(E,E)RHom(E,E)RHom(E,E), where EEE is a semistable sheaf on a K3 surface. The main tool is Kaledin's theorem on formality in families. For a large class of sheaves EEE, this DG algebra is formal, therefore we have an explicit description of the singularity type of the moduli space of semistable sheaves at the point represented by EEE. This paper also explains why Kaledin's theorem fails to apply in the remaining case.
Quivers and moduli spaces of pointed curves of genus zero
arXiv: Algebraic Geometry, 2015
We construct moduli spaces of representations of quivers over arbitrary schemes and show how moduli spaces of pointed curves of genus zero like the Grothendieck-Knudsen moduli spaces overlineM0,n\overline{M}_{0,n}overlineM0,n and the Losev-Manin moduli spaces overlineLn\overline{L}_noverlineLn can be interpreted as inverse limits of moduli spaces of representations of certain bipartite quivers. We also investigate the case of more general Hassett moduli spaces overlineM0,a\overline{M}_{0,a}overlineM0,a of weighted pointed stable curves of genus zero.
Local structure of the moduli space of vector bundles over curves
Commentarii Mathematici Helvetici, 1996
Local structure of the moduli space of vector bundles over curves Yves LASZLO (*) 0. Introduction.-Let X be a smooth, projective and connected curve (over an algebraically closed field of characteristic zero) of genus g(X) ≥ 2. Let x be a (closed) point of X and SU(r, d) the moduli space of semi-stable vector bundles on X of rank r ≥ 2 and determinant O(dx). As usual, the geometric points of SU(r, d) correspond to S-equivalence classes [E] where E is a semi-stable rank r bundle of determinant O(dx) (another semi-stable bundle F is said to be S-equivalent to F if the graded objects gr(E) and gr(F) are isomorphic). The singular locus of SU(r, d) consists exactly of the non stable points (except if r = g(X) = 2 and d = 0. In this case, SU(r, d) = P 3 [N-R1]). In particular, except in the exceptional case above, SU(r, d) is smooth if and only if r and d are relatively prime. General facts about the action of reductive groups ensure that SU(r, d) is Cohen-Macaulay [E-H], normal and that the singularities are rational [B]. The principal aim of this paper is to give additional information about the singularities, essentially the description of the completion of the local ring at a non smooth point of SU(r, d) and to compute the multiplicity and the tangent cones at those singular points [E] which are not too bad, i.e. the corresponding graded object gr(E) of [E] has only two non isomorphic stable summands (or equivalently Aut gr(E) = G m × G m). Further, we give a complete description in the rank 2 case (proposition II.2, corollary II.3 and theorem III.4). As a corollary, we get the local form of the so called Coble quartic and prove that the Kummer variety of the Jacobian of a genus 3 non hyperelliptic curve is schematically defined by 8 cubics, the partials derivatives of the Coble quartic (theorem III.6). One could also give partial information at least if Aut gr(E) is a torus, or by using results of [P], if Aut gr(E) = Gl r (k) (the latter case essentialy means that gr(E) is the trivial bundle). But it seems to be difficult and somewhat messy to calculate for instance the multiplicity. In the remaining part of the paper, we compute the multiplicity of a generalized theta divisor of SU X (2, O) at a point [L ⊕ L ∨ ] , (*) The author was partially supported by the European Science Project "Geometry of Algebraic Varieties", Contract no. SCI-0398-C(A).
$DG$-models of projective modules and Nakajima quiver varieties
Homology, Homotopy and Applications, 2007
Associated to each finite subgroup Γ of SL 2 (C) there is a family of noncommutative algebras O τ (Γ), which is a deformation of the coordinate ring of the Kleinian singularity C 2 /Γ. We study finitely generated projective modules over these algebras. Our main result is a bijective correspondence between the set of isomorphism classes of rank one projective modules over O τ and a certain class of quiver varieties associated to Γ. We show that this bijection is naturally equivariant under the action of a "large" Dixmier-type automorphism group G. Our construction leads to a completely explicit description of ideals of the algebras O τ .
On the new compactification of moduli of vector bundles on a surface. II
Sbornik: Mathematics, 2009
A new compactification for the scheme of moduli for Gieseker-stable vector bundles with prescribed Hilbert polynomial, on the smooth projective polarized surface (S, L), is constructed. We work over the field k =k of characteristic zero. Families of locally free sheaves on the surface S are completed with locally free sheaves on schemes which are modifications of S. Gieseker-Maruyama moduli space has a birational morphism onto the new moduli space. We propose the functor for families of pairs "polarized scheme-vector bundle" with moduli space of such type.