text: We introduce a space of stable meromorphic differentials with poles of prescribed orders and define its tautological cohomology ring. This space, just as the space of holomorphic differentials, is stratified according to the set of multiplicities of zeros of the differential. The main goal of this paper is to compute the Poincar 'e-dual cohomology classes of all strata. We prove that all these classes are tautological and give an algorithm to compute them. In a second part of the paper we study the Picard group of the strata. We use the tools introduced in the first part to deduce several relations in these Picard groups.
  @cite_11: The moduli space of canonical divisors (with prescribed zeros and poles) on nonsingular curves is not compact since the curve may degenerate. We define a proper moduli space of twisted canonical divisors in the moduli space of Deligne-Mumford stable pointed curves which includes the space of canonical divisors as an open subset. The theory leads to geometric combinatorial constraints on the closures of the moduli spaces of canonical divisors. In case the differentials have at least one pole (the strictly meromorphic case), the moduli spaces of twisted canonical divisors on genus g curves are of pure codimension g in the moduli spaces of stable pointed curves. In addition to the closure of the canonical divisors on nonsingular curves, the moduli spaces have virtual components. In the Appendix, a complete proposal relating the sum of the fundamental classes of all components (with intrinsic multiplicities) to a formula of Pixton is proposed. The result is a precise and explicit conjecture in the tautological ring for the weighted fundamental class of the moduli spaces of twisted canonical divisors. As a consequence of the conjecture, the classes of the closures of the moduli spaces of canonical divisors on nonsingular curves are determined (both in the holomorphic and meromorphic cases).
summary: * Moduli space of twisted canonical divisors. @cite_11 , Farkas and Pandharipande proposed another compactification of the strata. Let @math be non-integers such that @math . Let @math be a vector of positive integers of lenght @math and let @math be vector of non-negative integers of length @math that is complete for @math and @math . We recall that @math is the locus of smooth curves such that @math and that we denote by @math its closure in @math . @cite_11 , Farkas and Pandharipande defined the space of twisted canonical divisors denoted by @math . The space of twisted canonical divisors is a singular closed subspace of @math such that @math is one of the irreducible components of @math .