[Paper Review] Counter-examples of high Clifford index to Prym-Torelli

Elham Izadi, Herbert Lange|arXiv (Cornell University)|Jan 20, 2010

Algebraic Geometry and Number Theory7 references4 citations

TL;DR

This paper constructs counter-examples to the Prym-Torelli theorem for curves of arbitrarily high Clifford index by leveraging the tetragonal construction on étale double covers of curves with a degree-$n$ map to a base curve. It proves that the Prym map fails to be injective even when the underlying curve has Clifford index exceeding any given bound, using explicit isogenies and cohomological computations to show that certain curves in Prym varieties represent multiples of the minimal class.

ABSTRACT

We give examples of curves of arbitrarily high Clifford index such that the Prym map is not injective at any of their étale double covers.

Motivation & Objective

• To disprove the conjecture that the Prym map is injective at étale double covers of curves with Clifford index ≥ 3.
• To construct explicit examples of curves with arbitrarily high Clifford index where the Prym map fails to be injective.
• To generalize Donagi’s tetragonal construction to arbitrary base curves $Y$, not just $\mathbb{P}^1$, and prove isomorphism of associated Prym varieties.
• To study the cohomology classes of curves in Prym varieties and show they represent multiples of the minimal class $[\Theta]^{g-2}/(g-2)!$.
• To provide a new, explicit proof of Donagi’s theorem on isomorphism of Prym varieties under the tetragonal construction, valid for any base curve $Y$.

Proposed method

• Constructs a tower of curves $\widetilde{X} \xrightarrow{\kappa} X \xrightarrow{\rho_n} Y$ where $\kappa$ is an unramified double cover and $\rho_n$ is a simply ramified $n$-sheeted cover.
• Defines the curve $\widetilde{C} \subset \widetilde{X}^{(n)}$ as the fiber product of $\widetilde{X}^{(n)} \to X^{(n)}$ and $Y \to X^{(n)}$, parametrizing lifts of points in $Y$ to $\widetilde{X}$.
• Uses the involution $\sigma$ on $\widetilde{C}$ to decompose it into two components $\widetilde{C}_1$ and $\widetilde{C}_2$, each giving rise to a Prym variety via the tetragonal construction.
• Establishes an isogeny $\alpha: P \to P_\nu$ between the Prym variety $P$ of $\kappa: \widetilde{X} \to X$ and the generalized Prym $P_\nu$ of $\widetilde{X}_\nu \to X_\nu$, with kernel of order $2^{ng_Y - 1}$.
• Proves that the isogeny $\alpha$ satisfies $\alpha^*\Theta_{P_\nu} = 2\Theta_P$, leading to the dual isogeny $\beta: P_\nu \to P$ with $\beta^*\Theta_P = 2\Theta_{P_\nu}$.
• Applies Poincaré duality and degree computations to relate pushforwards of cohomology classes, ultimately showing $[\widetilde{C}_i] = 2^{n-1} \frac{[\Theta_P]^{g_X - 2}}{(g_X - 2)!}$.

Experimental results

Research questions

• RQ1 Is the Prym map generically injective for étale double covers of curves with Clifford index ≥ 3?
• RQ2 Can the tetragonal construction be generalized to base curves $Y$ other than $\mathbb{P}^1$ to produce isomorphic Prym varieties?
• RQ3 Do curves in Prym varieties represent multiples of the minimal cohomology class $[\Theta]^{g-2}/(g-2)!$?
• RQ4 Can explicit isogenies between Prym varieties be constructed and shown to be isomorphisms using correspondence theory?
• RQ5 What is the precise cohomological class of the image of $\widetilde{C}_i$ in the Prym variety $P$?

Key findings

• For any integer $N$, there exists a curve $X$ of Clifford index at least $N$ such that the Prym map is not injective at any étale double cover of $X$, disproving the conjecture that high Clifford index implies injectivity.
• The Prym varieties associated to the two components $\widetilde{C}_1$ and $\widetilde{C}_2$ of the curve $\widetilde{C} \subset \widetilde{X}^{(n)}$ are isomorphic under small generality assumptions, even when $Y$ is not $\mathbb{P}^1$.
• The isogeny $\alpha: P \to P_\nu$ has degree $2^{ng_Y - 1}$, and its dual $\beta$ satisfies $\beta^*\Theta_P = 2\Theta_{P_\nu}$, leading to a precise cohomological computation of pushforwards.
• The class of the image of $\widetilde{C}_i$ in the Prym variety $P$ is $[\widetilde{C}_i] = 2^{n-1} \frac{[\Theta_P]^{g_X - 2}}{(g_X - 2)!}$, showing it represents a multiple of the minimal class.
• The proof of isomorphism between Prym varieties provides a third proof of Donagi’s theorem in the case $Y = \mathbb{P}^1$, independent of degeneration and cohomological methods.
• The construction works for any $n \geq 3$, generalizing the tetragonal construction to higher-degree covers and yielding curves in Prym varieties with controlled cohomology classes.