669 research outputs found
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
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
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
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
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
A remark on du Val linear systems
Let , be the genus du Val linear system on a Halphen surface
of index . We prove that the Clifford index is constant on smooth
curves . Let be the gonality of .
When (the relevant case), we show
that , and that the gonality is realized by the
Weierstrass linear series , which is totally ramified at one
point. The proof of the first statement follows closely the path indicated by
Green and Lazarsfeld for a similar statement regarding K3 surfaces.Comment: 12 page
Increasing trees and Kontsevich cycles
It is known that the combinatorial classes in the cohomology of the mapping
class group of punctures surfaces defined by Witten and Kontsevich are
polynomials in the adjusted Miller-Morita-Mumford classes. The leading
coefficient was computed in [Kiyoshi Igusa: Algebr. Geom. Topol. 4 (2004)
473-520]. The next coefficient was computed in [Kiyoshi Igusa: math.AT/0303157,
to appear in Topology]. The present paper gives a recursive formula for all of
the coefficients. The main combinatorial tool is a generating function for a
new statistic on the set of increasing trees on 2n+1 vertices. As we already
explained in the last paper cited this verifies all of the formulas conjectured
by Arbarello and Cornalba [J. Alg. Geom. 5 (1996) 705--749]. Mondello
[math.AT/0303207, to appear in IMRN] has obtained similar results using
different methods.Comment: Published by Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol8/paper26.abs.htm
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