HYPERELLIPTIC d-OSCULATING COVERS AND RATIONAL SURFACES

Let d be a positive integer, K an algebraically closed field of characteristic p not equal 2 and X an elliptic curve defined over K. We consider the hyperelliptic curves equipped with a projection over X, such that the natural image of X in the Jacobian of the curve osculates to order d to the embedding of the curve, at a Weier-strass point. We first study the relations between the degree n, the arithmetic genus g and the osculating degree d of such covers. We prove that they are in a one-to-one correspondence with rational curves of linear systems in a rational surface and deduce (d - 1)-dimensional families of hyperelliptic d-osculating covers, of arbitrary big genus g if p = 0 or such that 2g < p(2d+ 1) if p>2. It follows at last, (g + d - 1)-dimensional families of solutions of the KdV hierarchy, doubly periodic with respect to the d-th variable.