THE MODULI SPACE OF CURVES OF GENUS 2 COVERING ELLIPTIC-CURVES

We consider the moduli space S(n) of curves C of genus 2 with the property: C has a ''maximal'' map f of degree n to an elliptic curve E. Here, the term ''maximal'' means that the map f : C --> E doesn't factor over an unramified cover of E. By Torelli map S(n) is viewed as a subset of the moduli space A2 of principally polarized abelian surfaces. On the other hand the Humbert surface H(DELTA) of invariant DELTA is defined as a subvariety of A2(C), the set of C-valued points of A2. The purpose of this paper is to relate S(n) with H(DELTA).