A generic Torelli-type theorem for singular algebraic curves with an involution

Abstract

We prove a generic Torelli-type theorem for a special class of singular algebraic curves with an involution. In order to obtain this result we introduce an appropriate mixed Hodge structure on the anti-invariant part of the first homology group, and study its properties. Let X be an irreducible projective algebraic curve with an involution a. Suppose X has only ordinary singularities and let X be its singular locus. Let 7r: N -, X be the normalization of X and let T: N -, N be the involution induced by a. We suppose that the following condition is satisfied: The set of fixed points of a coincides with X and the involution T is without fixed points. Following J. Carlson [2] one can introduce a polarized mixed Hodge structure (PMHS) on the anti-invariant part of H1 (X, Z) with respect to a, denoted by H7 (X, Z): 0 -O HJ (N, Z) H1 (X, Z) -A -, 0, where H7 (N, Z) is the anti-invariant part of H1 (N, Z) with respect to z and has a polarized Hodge structure (PHS) of weight -1; A has PHS of weight 0. It turns out that the latter is isomorphic to the lattice generated by a root-system of the type Dn when #{z7K (X)} > 2. Using the generic Torelli theorem for the Prym map as proven by FriedmanSmith [4] and Kanev [5], the pair (N, T) is uniquely determined by its Prym variety (equivalently by the PHS of H7 (N, Z)) if the following condition is satisfied. N/T is a sufficiently general curve of genus g > 7. We prove the following result: Theorem 7. Let X be the curve which satisfies (*). Suppose that N/T is a general curve of genus g > 15. Then N, T, and the set 7 1 (X) are uniquely determined by the PMHS of HJ (X, Z). Received by the editors October 1, 1989 and, in revised form, March 29, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 1 4C30; Secondary 1 4H99.