Singularities of moduli spaces of sheaves on K3 surfaces and Nakajima quiver varieties

The aim of this paper is to study the singularities of certain moduli spaces of sheaves on K3 surfaces by means of Nakajima quiver varieties. The singularities in question arise from the choice of a non--generic polarization, with respect to which we consider stability, and admit natural symplectic resolutions corresponding to choices of general polarizations. For sheaves that are pure of dimension one, we show that these moduli spaces are, locally around a singular point, isomorphic to a quiver variety and that, via this isomorphism, the natural symplectic resolutions correspond to variations of GIT quotients of the quiver variety.Comment: 40 pages; final version; As pointed out to us by Z. Zhang, we prove quadraticity and not formality of the Kuranishi family. Quadraticity is all we need for our main theorem. The current version reflects this correction. A few other improvements in exposition and correction of typo

Teichmueller space via Kuranishi families

We construct Teichmueller space by patching together Kuranishi families. We also discuss the basic properties of Teichmueller space, and in particular show that our construction leads to simplifications in the proof of Teichmueller's theorem asserting that the genus g Teichmueller space is homeomorphic to a (6g-6)-dimensional ball.Comment: 26 pages; minor mistakes corrected, references added and correcte

Mukai's program for curves on a K3 surface

Let C be a general element in the locus of curves in M_g lying on some K3 surface, where g is congruent to 3 mod 4 and greater than or equal to 15. Following Mukai's ideas, we show how to reconstruct the K3 surface as a Fourier-Mukai transform of a Brill-Noether locus of rank two vector bundles on C.Comment: Final version. To appear in "Algebraic Geometry

Rank two vector bundles on polarised Halphen surfaces and the Gauss-Wahl map for du Val curves

A genus-g du Val curve is a degree-3g plane curve having 8 points of multiplicity g, one point of multiplicity g-1, and no other singularity. We prove that the corank of the Gauss-Wahl map of a general du Val curve of odd genus (>11) is equal to one. This, together with the results of [1], shows that the characterisation of Brill-Noether-Petri curves with non-surjective Gauss-Wahl map as hyperplane sections of K3 surfaces and limits thereof, obtained in [3], is optimal

Relative Prym varieties associated to the double cover of an Enriques surface

Given an Enriques surface T , its universal K3 cover f : S → T , and a genus g linear system |C| on T, we construct the relative Prym variety PH = Prymv,H(D/C), where C → |C| and D → |f∗C| are the universal families, v is the Mukai vector (0, [D], 2−2g) and H is a polarization on S. The relative Prym variety is a (2g−2)-dimensional possibly singular variety, whose smooth locus is endowed with a hyperk ̈ahler structure. This variety is constructed as the closure of the fixed locus of a symplectic birational involution defined on the moduli space Mv,H (S). There is a natural Lagrangian fibration η : PH → |C|, that makes the regular locus of PH into an integrable system whose general fiber is a (g − 1)-dimensional (principally polarized) Prym variety, which in most cases is not the Jacobian of a curve. We prove that if |C| is a hyperelliptic linear system, then PH admits a symplectic resolution which is birational to a hyperk ̈ahler manifold of K3[g−1]-type, while if |C| is not hyperelliptic, then PH admits no symplectic resolution. We also prove that any resolution of PH is simply connected and, when g is odd, any resolution of PH has h2,0-Hodge number equal to one

Generalized boundary strata classes

We describe a generalization of the usual boundary strata classes in the Chow ring of $\overline{\mathcal{M}}_{g,n}$. The generalized boundary strata classes additively span a subring of the tautological ring. We describe a multiplication law satisfied by these classes and check that every double ramification cycle lies in this subring.Comment: For the Proceedings of the 2017 Abel Symposium, 10 page

On the base locus of the generalized theta divisor

We present a new class of examples of base points for the generalized theta divisor on the moduli space of semistable vector bundles of trivial determinant on a compact Riemann surface and we prove that for sufficiently large rank the base locus is positive dimensional

Determinantal Characterization of Canonical Curves and Combinatorial Theta Identities

We characterize genus g canonical curves by the vanishing of combinatorial products of g+1 determinants of Brill-Noether matrices. This also implies the characterization of canonical curves in terms of (g-2)(g-3)/2 theta identities. A remarkable mechanism, based on a basis of H^0(K_C) expressed in terms of Szego kernels, reduces such identities to a simple rank condition for matrices whose entries are logarithmic derivatives of theta functions. Such a basis, together with the Fay trisecant identity, also leads to the solution of the question of expressing the determinant of Brill-Noether matrices in terms of theta functions, without using the problematic Klein-Fay section sigma.Comment: 35 pages. New results, presentation improved, clarifications added. Accepted for publication in Math. An