Algebraic Geometry and Combinatorics Seminar: Szego kernels and the Scorza map on moduli spaces of spin curves

Abstract: The Scorza correspondence was first studied by Scorza. Starting with a spin curve of genus 3 (i.e., a curve of genus 3 with an even theta-chracteristic with no global sections), Scorza used his correspondence to construct a second plane quartic which gave a birational map from the moduli space of curves of genus 3 to the moduli space of spin curves of genus 3. Scorza’s results were further used by Mukai to construct the family of Fano threefolds of genus 12 and degree 22. Scroza’s correspondence is in fact well-defined in all genera. We determine the limits of the Scorza correspondence at generic points of the vanishing theta-null divisor and at generic points of boundary divisors. We further show that the Scorza quartic can be defined using Wirtinger duality which shows that it can, in a certain form, be defined for principally polarized abelian varieties with a theta-characteristic. We further show that limit of the Scorza quartic at abelian varieties with vanishing theta-nulls is twice the quadric tangent cone to the theta divisor at the vanishing theta-null.

Host: Matt Kerr