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Title: | The positive dimensional fibres of the Prym map |
Author: | Naranjo del Val, Juan Carlos |
Keywords: | Corbes algebraiques Geometria algebraica Varietats abelianes Algebraic curves Algebraic geometry Abelian varieties |
Issue Date: | 1-Jan-1996 |
Publisher: | Mathematical Sciences Publishers |
Abstract: | Let $C$ be an irreducible complex smooth curve of genus $g ; \pi: \bar{C} \rightarrow C$ a connected unramified double covering of $C$. The Prym variety associated to the covering is, by definition, the component of the origin of the kernel of the norm $\operatorname{map} P(\bar{C}, C)=\operatorname{Ker}\left(\mathrm{Nm}_\pi\right)^0 \subset J \bar{C}$, that is, a principally polarized abelian variety (p.p.a.v.) of dimension $g(\bar{C})-g=g-1$. One defines the Prym map $P_g: \mathscr{R}_g \rightarrow \mathscr{A}_{g-1},(\bar{C} \stackrel{\pi}{\rightarrow} C) \mapsto P(\bar{C}, C)$, where $\mathscr{R}_g$ is the coarse moduli space of the coverings $\pi$ as above and $\mathscr{A}_{g-1}$ stands for the coarse moduli space of p.p.a.v.'s of dimension $g-1$. It is well known that this map is generically injective for $g \geq 7$. On the other hand, this map is never injective. The coarse moduli space $\mathscr{R} \mathscr{H}_g$ of and the fibres of the restriction of $P_g$ to $\mathscr{R} \mathscr{H}_g$ have positive dimension. Let $\mathscr{R}_g$ be the coarse moduli space of the unramified double coverings $\pi: \bar{C} \rightarrow C$ such that $C$ is a smooth bi-elliptic curve of genus $g$. This variety has $[(g+1) / 2]+2$ irreducible components: $\mathscr{R} \mathscr{B}_g=\left(\bigcup_{t=0}^{[(g-1) / 2]} \mathscr{R}_{g, t}\right) \cup \mathscr{R}_{\mathscr{B}_g^{\prime}}$. In this note the author characterizes the fibres of positive dimension of the Prym map. Theorem. Assume $g \geq 13$. A fibre of $P_g$ is positivedimensional at $(\bar{C}, C)$ if and only if $C$ is either hyperelliptic or $(\bar{C}, C) \in \bigcup_{t \geq 1} \mathscr{R}_{g, t}$. |
Note: | Reproducció del document publicat a: https://doi.org/10.2140/pjm.1996.172.223 |
It is part of: | Pacific Journal of Mathematics, 1996, vol. 172, num. 1, p. 223-226 |
URI: | http://hdl.handle.net/2445/197420 |
Related resource: | https://doi.org/10.2140/pjm.1996.172.223 |
ISSN: | 0030-8730 |
Appears in Collections: | Articles publicats en revistes (Matemàtiques i Informàtica) |
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