Distinguishing Prymians from Jacobians

Abstract

1.1. Throughou t this paper we fix an algebraically closed field k of characteristic =4=2; all varieties considered are defined over k. We denote by C a connected curve with an involution i: C--+ C (i 2=Id). Throughou t this paper we also suppose that the pair ((~, i) satisfies the following conditions: (i) C has only ordinary double points; (ii) The fixed points of i are exactly the singular points, and at a singular point the two branches are not exchanged under i. In this situation, due to Mumford [M] in the non-singular case and to Beauville [B] in the general case, we have the principally polarized Prym variety, or Prymian for short, (P, ~), i.e. an Abelian variety P with a principal polarization ~. In this paper we are interested in establishing when (P, ~) is not a Jacobian of some smooth curve or a sum of them. Of course, we consider Jacobians with their principal polarizations. Note that (i), (ii) imply (iii) For any decomposi t ion (~ = C1 L) C a we have :~:(C1 ('~ C2) ~-~0 (mod 2). If (~ = (~1 w C2 with (~1 n t~ 2 = {p, q}, and C'i (i = 1, 2) is the curve obtained from Ci by identifying p, q, then by Lemma (4.11) [B] P~-P1 xP2, where Pi is the Prym variety associated to 01 with the involution induced by i. So in view of (iii) we may restrict our interest to curves which satisfy the following condit ion: (iv) For any non-trivial decomposi t ion C = (~1 u (~2, # C1 c~ C2>4. The main result is