Poincaré Polynomials of Hyperquot Schemes

with dim Vi = si. The space Mord(P 1,F(n; s)) of morphisms from P1 to F(n; s) of multidegree d = (d1, ..., dl) can be viewed as the space of successive quotients of VP1 of vector bundles of rank ri and degree di, where ri := n − si and VP1 := V ⊗ OP1 is a trivial rank n vector bundle over P1. Its compactification, the hyperquot scheme which we denote HQd = HQd(F(n; s)), parametrizes flat families of successive quotient sheaves of VP1 of rank ri and degree di. It is a generalization of Grothendieck’s Quot scheme [G]. There has been much interest in compactifications of moduli spaces of maps, for example the stable maps of Kontsevich. Hyperquot schemes are another natural such compactification. Indeed, most of what is known so far about the quantum cohomology of Grassmanians and flag varieties has been obtained by using Quot scheme compactifications. They have been used by Bertram to study Gromov-Witten invariants and a quantum Schubert calculus [B1] [B2] for Grassmannians, and by Ciocan-Fontanine and Kim to study Gromov-Witten invariants and the quantum cohomology ring of flag varieties and partial flag varieties [K] [C-F1] [C-F2] [FGP], see also [C1] [C2]. The paper is organized in the following way: In section 2, we give some properties of hyperquot schemes, including a description of the Zariski tangent space to HQd at a point. In section 3, we consider a torus action on the hyperquot scheme. By the theorems of Bialynicki-Birula, the fixed points of this action give a cell decomposition of HQd [BB1][BB2]. The fixed point data is organized to give a generating function for the topological Euler characteristic of HQd. In section 4, we use the Zariski tangent space to HQd at a point as described in section 2 to compute tangent weights at the fixed points. This