Prym varieties of bi-elliptic curves

Abstract

Denote by $\mathscr{R}_g$ the set of pairs $\left(C^{\prime}, C\right)$ where $C$ is a (smooth) curve of genus $g$ and $C^{\prime}$ is an unramified double cover of $C$. One can then define a map from $\mathscr{R}_g$ to the moduli space $\mathscr{A}_{g-1}$ of abelian varieties by associating to a pair the corresponding Prym variety. This map is known to be generically injective for $g \geq 7$ but not injective for any value of $g$. In fact, the so-called tetragonal construction due to R. Donagi [cf. Bull. Amer. Math. Soc. (N.S.) 4 (1981), no. 2, 181-185; MR0598683] associates to a pair with $C$ tetragonal two new pairs in which the base curve is again tetragonal and with the same associated Prym variety. Donagi conjectured (without much evidence, as he himself admits) that these are the only counterexamples to the injectivity of the Prym map. More precisely, given two pairs $\left(C^{\prime}, C\right)$ and $\left(C_1^{\prime}, C_1\right)$ with the same associated Prym variety, there should be a chain of pairs of coverings with each pair in the chain tetragonally related to the next. Debarre showed that this result is true outside the trigonal and bielliptic locus if $g$ is at least 13 (and so the $g_4^1$ is unique). This paper shows that this is still true for a generic bielliptic curve, so long as one stays in the category of smooth curves. Consider now a pair $\left(C^{\prime}, C\right)$ with $C$ bielliptic, i.e. a double cover of an elliptic curve $E$. The Galois group of $C^{\prime}$ over $E$ is either $Z_2$ or $Z_2 \times Z_2$. In the latter case denote by Id, $i, i_1, i_2$ its elements with $C=C^{\prime} / i, C_1=C^{\prime} / i_1, C_2=C^{\prime} / i_2$. If the genus of $C_1$ (say $t$ ) is smaller than the genus of $C_2$, then the pair $\left(C^{\prime}, C\right)$ is said to be in $\mathscr{R}_{b, t}$. In the case $t=4$, the author gives a construction which associates to the pair $\left(C^{\prime}, C\right)$ an allowable double covering (which is not tetragonal) and with the same associated Prym variety. It is then shown that, in the larger class of allowable double coverings, these are the only exceptions to the tetragonal conjecture over any generic point of the set of bielliptic pairs.