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Rita Pardini : Linear systems on irregular varieties

        I will report on joint work M.A. Barja (UPC, Barcelona) and L. Stoppino (Universita' dell'Insubria, Como - Italy).
        Given a generically finite map a:X--> A, where X is a smooth projective variety and A is an abelian variety, and given a line bundle L on X, we study the linear system |L+P|, where P is a general element of Pic^0(A). We prove that up to taking base change with a suitable multiplication map A-->A, the map given by |L+P| is independent of P and induces a factorization of the map a. When L is the canonical bundle of X, this factorization is a new geometrical object intrinsically attached to the variety X.
        The factorization theorem also allows us to improve the known Clifford-Severi and Castelnuovo type numerical inequalities for line bundles on X, under certain assumptions on the map a:X-->A. A key tool in these proofs is the introduction of a real function, the continuous rank function, that also allows us to simplify considerably the proof of the general Clifford-Severi inequality.

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