Number of equations defining the Kummer varieties of hyperelliptic curves

Abstract

The Jacobian of an hyperelliptic curve $C$ of genus $g$ over $\mathbb{C}$ is equipped with a symmetric theta divisor $Θ$ that defines a principal polarization. The line bundle $\mathcal{O}(2Θ)$ defines an embedding of the Kummer variety $K$ inside the projective space $\mathbb{P}^{2^g-1}$, which factorizes inside a certain orthogonal Grassmannian. We describe a minimal free resolution of $K$ inside this Grassmannian using geometric representation theory. As a consequence, we obtain a formula for the number of quartic equations defining $K$ in the projective space $\mathbb{P}^{2^g-1}$.