Geodesic
Systems of Polynomial Equations Defining Hyperelliptic $d$-Osculating Covers
Funkcionalʹnyj analiz i ego priloženiâ, Tome 49 (2015) no. 1, pp. 49-61

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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 $d>1$.
Keywords: finite separable covers, hyperelliptic curves, Weierstrass points. 
@article{FAA_2015_49_1_a3,
     author = {A. Treibich},
     title = {Systems of {Polynomial} {Equations} {Defining} {Hyperelliptic} $d${-Osculating} {Covers}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {49--61},
     publisher = {mathdoc},
     volume = {49},
     number = {1},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2015_49_1_a3/}
}
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A. Treibich. Systems of Polynomial Equations Defining Hyperelliptic $d$-Osculating Covers. Funkcionalʹnyj analiz i ego priloženiâ, Tome 49 (2015) no. 1, pp. 49-61. http://geodesic.mathdoc.fr/item/FAA_2015_49_1_a3/