1,546 research outputs found
The universal K3 surface of genus 14 via cubic fourfolds
Using the isomorphism between the moduli space of polarized K3 surfaces of
genus 14 and the moduli space of special cubic fourfolds of discriminant 26, we
establish the rationality of the universal K3 surface of genus 14. Precisely,
we show that the universal K3 surface of genus 14 is a projective bundle over a
certain moduli space of nodal scrolls in P^5, whose rationality we prove using
a degenerate version of Mukai's structure theorem for curves of genus 8.Comment: 20 pages. Final version, to appear in the Journal de Mathematiques
Pures et Appliquee
Green's Conjecture for curves on arbitrary K3 surfaces
Green's Conjecture predicts than one can read off special linear series on an
algebraic curve, by looking at the syzygies of its canonical embedding. We
extend Voisin's results on syzygies of K3 sections, to the case of K3 surfaces
with arbitrary Picard lattice. This, coupled with results of Voisin and
Hirschowitz-Ramanan, provides a complete solution to Green's Conjecture for
smooth curves on arbitrary K3 surfaces.Comment: 13 pages. Minor revisions, to appear in Compositio Mathematic
Singularities of moduli of curves with a universal root
In a series of recent papers, Chiodo, Farkas and Ludwig carry out a deep
analysis of the singular locus of the moduli space of stable (twisted) curves
with an -torsion line bundle. They show that for and
pluricanonical forms extend over any desingularization. This
allows to compute the Kodaira dimension without desingularizing, as done by
Farkas and Ludwig for , and by Chiodo, Eisenbud, Farkas and Schreyer
for . Here we treat roots of line bundles on the universal curve
systematically: we consider the moduli space of curves with a line bundle
such that . New loci of canonical
and non-canonical singularities appear for any and
, we provide a set of combinatorial tools allowing us to completely
describe the singular locus in terms of dual graph. We characterize the locus
of non-canonical singularities, and for small values of we give an
explicit description.Comment: 30 pages, to appear in Documenta Mathematic
The generic Green-Lazarsfeld secant conjecture
Generalizing the well-known Green Conjecture on syzygies of canonical curves,
Green and Lazarsfeld formulated in 1986 the Secant Conjecture predicting that a
line bundle L of sufficiently high degree on a curve has a non-linear p-syzygy
if and only if L fails to be (p+1)-very ample. Via lattice theory for special
K3 surfaces, Voisin's solution of the classical Green Conjecture and
calculations on moduli stacks of pointed curves, we prove: (1) The
Green-Lazarsfeld Secant Conjecture in various degree of generality, including
its strongest possible form in the divisorial case in the universal Jacobian.
(2) The Prym-Green Conjecture on the naturality of the resolution of a general
Prym-canonical curve of odd genus.Comment: 24 pages. Final version, to appear in Inventiones Mat
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