Systèmes dʼéquations polynomiales pour les revêtements hyperelliptiques d-osculants

Abstract

Let X denote a fixed smooth projective curve of genus 1, defined over an algebraically closed field K of arbitrary characteristic p≠2. For any positive integer n, we will consider the moduli space H(X,n) of degree-n finite separable covers of X by a hyperelliptic curve marked at a triplet of Weierstrass points. We start parameterizing H(X,n) by a suitable space of rational fractions, obtaining a polynomial characterization of those having order of osculation d (d⩾1). We then deduce systems of polynomial equations, whose set of solutions codifies all degree-n hyperelliptic d-osculating covers of X.