Mathematics – Algebraic Geometry
Scientific paper
2006-01-11
Mathematics
Algebraic Geometry
35 pages, to appear in Advances in Geometry, part of the author's Phd Thesis
Scientific paper
The Weddle surface is classically known to be a birational (partially desingularized) model of the Kummer surface. In this note we go through its relations with moduli spaces of abelian varieties and of rank two vector bundles on a genus 2 curve. First we construct a moduli space A\_2(3)^- parametrizing abelian surfaces with a symmetric theta structure and an odd theta characteristic. Such objects can in fact be seen as Weddle surfaces. We prove that A\_2(3)^- is rational. Then, given a genus 2 curve C, we give an interpretation of the Weddle surface as a moduli space of extensions classes (invariant with respect to the hyperelliptic involution) of the canonical sheaf \omega of C with \omega^{-1}. This in turn allows to see the Weddle surface as a hyperplane section of the secant variety Sec(C) of the curve C tricanonically embedded in P^4.
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