Systems of polynomial equations for hyperelliptic \(d\)-osculating covers (Q1942079)

From the abstract: Let \(X\) denote a fixed smooth projective curve of genus \(1\), defined over an algebraically closed field of characteristic different from \(2\). 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 parametrizing \(H(X,n)\) by a suitable space of rational fractions, obtaining a polynomial characterization of those having order of osculation \(d \geq 1\). We deduce systems of polynomial equations, whose set of solutions codifies all degree \(n\) hyperelliptic \(d\) osculating covers of \(X\).