Counter-examples of High Clifford Index to Prym-torelli

from the moduli space Rg of (non-trivial) étale double coverings of curves of genus g to the moduli space Ag−1 of principally polarized abelian varieties of dimension g − 1, called the Prym map. It was shown independently by Friedman-Smith [10], Kanev [13], Welters [20] and Debarre [6], that prg is generically injective for g ≥ 7. On the other hand, Mumford showed in [16, p. 346], that the Prym variety of an étale double cover of a hyperelliptic curve is the product of the Jacobians of two other hyperelliptic curves (one of which can have genus 0). Via a dimension count, this implies that the Prym map has positive dimensional fibers on the locus of hyperelliptic Jacobians. Then Beauville remarked in [2, Contre-exemple 7.13 p. 388] that prg is not injective for 3 ≤ g ≤ 10 at some non-hyperelliptic curves. In [7] Donagi gave a construction showing that prg is not injective at any étale double cover of a curve X admitting a map X → P of degree 4 under some generality assumptions. Moreover, he conjectured (see [7, Conjecture 4.1] or [18, p. 253]) that prg is injective at any κ : X̃ → X, whenever X does not admit a g 4. Verra showed in [19] that pr10 is not injective at any étale double cover of a general plane sextic. In [17], Naranjo found examples of admissible double covers of nodal curves where the Prym map fails to be injective which were not obtained from Donagi’s construction. However, the curves which either admit a g 4 or are plane sextics (more precisely, have a g 2 6) are exactly the curves of Clifford index ≤ 2. Furthermore, Naranjo’s examples are double covers of specializations of bielliptic curves, hence specializations of curves of Clifford index ≤ 2. So, in [14], Lange and Sernesi ask whether prg is injective at κ : X̃ → X whenever X is of Clifford index ≥ 3. It is the aim of this paper to show that this is not the case. Our main result is the following theorem.