The abelian fibration on the Hilbert cube of a K3 surface of genus 9

In this paper we construct an abelian fibration over P-3 on the Hilbert cube of the primitive K3 surface of genus 9. After the abelian fibration constructed by Hassett and Tschinkel on the Hilbert square on the primitive K3 surface of genus 5, this is the second example where the abelian fibration is constructed directly on Hilb(n)S. The recent more general result of Sawon proves the existence of an abelian fibration on the Hilbert scheme Hilb(n)S of a primitive K3 surface S of degree 2g - 2 = m(2)(2n - 2). Our example provides an alternative proof in the case m = 2, n = 3. Furthermore we identify the general fiber with the Hilbert scheme of twisted cubic curves in a Fano 3-fold of genus 9, and interpret the addition law on this variety.