  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.
