Weierstrass Cycles in Moduli Spaces and the Krichever Map

We analyze cohomological properties of the Krichever map and use the results to study Weierstrass cycles in moduli spaces and the tautological ring. Let us consider a point p on a smooth projective connected curve C over C of genus g. We say that a natural number n is a non-gap if there exists a function that is holomorphic on C \p and has a pole of order n at the point p (in other words h0(O(np)) > h0(O((n−1)p))). It is obvious that the set of all non-gaps is a semigroup; it is easy to derive from RiemannRoch theorem that the number of gaps (the cardinality of the complement to the set of non-gaps in N) is equal to g. We denote by H the set consisting of 0 and of all integers n such that h0(O(np)) > h0(O((n− 1)p)) (in other words, we include 0 and all non-gaps into H). One says that H is the Weierstrass semigroup at p. One says that a subsemigroup H of N0 such that #(N0\H) = g and 0 ∈ H is a numerical semigroup of genus g; obviously any Weierstrass semigroup belongs to this class. (Here N0 stands for the semigroup of non-negative integers). The point p is a Weierstrass point if the first non-gap is ≤ g (i.e. H 6= {0, g + 1, g + 2, · · · }). There exist only a finite number of Weierstrass points on a curve. Instead of Weierstrass semigroup H, one can consider a decreasing sequence of integers such that si is the largest integer with h(KC(−sip)) = i. HereKC denotes the canonical line bundle on C. It follows from the Riemann-Roch theorem that this sequence (the Weierstrass sequence of the point p ) has the form si = ag−i+1 − 1 if 1 ≤ i ≤ g and si = g − 1 − i if i ≥ g + 1. Here 1 = a1 < · · · < ag denotes the increasing sequence of gaps. Notice that all these statements remain correct if p is a nonsingular point of an irreducible (not necessarily smooth) curve and the canonical line bundle is replaced by the dualizing sheaf ωC . (Every irreducible curve is a Cohen-Macaulay curve; hence it is not necessary to consider a complex of sheaves talking about the dualizing sheaf.) 1 Any numerical semigroup of genus g is a Weierstrass semigroup at a point on an irreducible curve of (arithmetic) genus g; see Section 3. Let us consider the moduli space Mg,1 of non-singular irreducible curves of genus g with one marked point (one can characterize this space as the universal curve). 2 If H is a numerical semigroup of genus g, we denote by MH the subset of Mg,1 consisting of curves with marked points having Weierstrass semigroup H. The closure WH = MH of the Weierstrass set MH in Mg,1 is called a Weierstrass cycle. Under some conditions, we 1All curves we consider are reduced irreducible projective curves. (in other words we work with projective integral curves) 2One can consider this space as an orbifold or as a moduli space of a stack. However, we are interested only in cohomology over C, therefore it is sufficient to consider it as a topological space.