9 D ec 1 99 8 STABLE MAPS OF GENUS ZERO TO FLAG SPACES

Topological quantum field theory led recently to a spectacular progress in numerical algebraic geometry. It was shown that generating functions of certain char-actertistic numbers of modular spaces of stable algebraic curves with labelled points satisfy remarkable differential equations of KP–type (E. Witten, M. Kontsevich). In a later series of developments, this was generalized, partly conjecturally, to the spaces of maps of curves into algebraic varieties leading to the Mirror Conjecture and the construction of quantum cohomology. The key technical notion in the context of algebraic geometry is that of a stable map introduced by M. Kontsevich (cf. [K] and [BM]) following the earlier work by M. Gromov in symplectic geometry. It provides a natural compactification of spaces of maps, in the same way as stable curves compactify moduli spaces. We will be working over a ground field. Let W be an algebraic variety. 0.1. Definition. A stable map (to W) is a structure (C; x 1 ,. .. , x n ; f) consisting of the following data.) is a connected complete reduced curve with n ≥ 0 labelled pairwise distinct non–singular points x i and at most ordinary double singular points. b). f : C → W is a morphism having no non–trivial first order infinitesimal au-tomorphisms identical on W and x i 's (stability). This means that every irreducible component of C of genus zero (resp. 1) has at least three (resp. one) special points (inverse images of singular and labelled points) on its normalization. A family of stable maps parametrized by a noetherian scheme S is a structure consisting of a flat proper morphism π : C → S, n sections x i : S → C, and a morphism f : C → W whose restriction to each geometric fiber of π is a stable map in the sense of the previous Definition. Families of stable maps form an algebraic stack in the sense of [DM]. For fixed n ≥ 0, g ≥ 0 and an algebraic homology class of dimension two β, denote by M g,n (W, β) the substack of maps for which g is the arithmetic genus of C and β = f * ([C]). For the proof of the following theorem see [K], [BM] and [FP]. 1 2 0.2. Theorem. a). If W is projective, then M g,n (W, β) is a proper separated algebraic stack of finite type. b). Assume that …