Abstract:
Let $X$ denote a fixed smooth projective curve of genus $1$ defined over an algebraically closed field $\mathbb{K}$ of arbitrary characteristic $\boldsymbol{p}\neq2$. For any positive integer $n$, we consider
the moduli space $H(X,n)$ of degree-$n$ finite separable covers of $X$ by a hyperelliptic curve with three marked
Weierstrass points. We parameterize $H(X,n)$ by a suitable space of rational fractions and apply it to studying the (finite) subset of degree-$n$ hyperelliptic tangential covers of $X$. We find a polynomial characterization for the corresponding rational fractions and deduce a square system of polynomial equations whose solutions parameterize these covers. Furthermore, we also obtain nonsquare systems parameterizing hyperelliptic $d$-osculating covers for any $d>1$.
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