Divisors on the Moduli Spaces of Stable Maps to Flag Varieties and Reconstruction

We determine generators for the codimension 1 Chow group of the moduli spaces of genus zero stable maps to flag varieties G/P . In the case of SL flags, we find all relations between our generators, showing that they essentially come from M0,n. In addition, we analyze the codimension 2 classes on the moduli spaces of stable maps to Grassmannians and prove a new codimension 2 relation. This will lead to a partial reconstruction theorem for the Grassmannian of 2 planes. In [12] and [13] we began the study of the tautological rings of the moduli spaces of stable maps to general flag varieties. We conjectured that the cohomology of these moduli spaces is entirely tautological and proved this conjecture for SL flags. In this note, we bring more evidence in favor of our conjecture, establishing it completely for general flag varieties in codimension one, using a different method. More precisely, let X be a projective homogeneous space. The coarse moduli spaces M0,n(X,β) parametrize marked stable maps to X in the cohomology class β ∈ H 2(X,Z). These moduli spaces are related by a complicated system of natural morphisms which we enumerate below: • forgetful morphisms: π : M0,S(X,β) → M0,T (X,β) defined whenever T ⊂ S; • evaluation morphisms to the target space, evi : M0,n(X,β) → X for all 1 ≤ i ≤ n; • gluing morphisms which produce maps with nodal domain curves, gl : M0,S1∪{⋆}(X,β1)×XM0,{•}∪S2(X,β2) → M 0,S1∪S2(X,β1 + β2). The system of tautological classes on M0,n(X,β) is defined as the smallest subring of H(M 0,n(X,β)) (or of the Chow ring A (M0,n(X,β))) with the following properties: • the system is closed under pushforwards and pullbacks by the natural morphisms; • all monomials in the evaluation classes ev i α for α ∈ H (X) are in the system. Typical examples of tautological classes are the following. Let α1, . . . , αp be cohomology classes on X. The class κ(α1, . . . , αp) is defined as the pushforward via the forgetful projection π : M0,n+p(X,β) → M 0,n(X,β): κ(α1, . . . , αp) = π⋆(ev ⋆ n+1α1 · . . . · ev ⋆ n+pαp). One of the main results of this note is the following: