Tautological rings of stable map spaces

Abstract

We find a set of generators and relations for the system of extended tautological rings associated to the moduli spaces of stable maps in genus zero, admitting a simple geometrical interpretation. In particular, when the target is P n , these give a complete presentation for the cohomology and Chow rings in the cases with/without marked points.Let d ∈ H 2 (X) be a curve class on a smooth projective variety X. The space M 0,0 (X, d) parametrizes maps from rational smooth or nodal connected curves into X with image class d, such that any contracted component contains at least 3 nodes. Over M 0,0 (X, d) there exists a tower of moduli spaces of stable maps with marked points and morphisms M 0,m+1 (X, d) → M 0,m (X, d) forgetting one marked point. Moreover, M 0,m+1 (X, d), together with an evaluation map ev m+1 : M 0,m+1 (X, d) → X and m natural sections σ i : M 0,m (X, d) → M 0,m+1 (X, d), form the universal family over M 0,m (X, d).In this paper we investigate the relation between the structure of the cohomology ring of the variety X and that of M 0,m (X, d), which is less than obvious in particular when m = 0. When marked points exist, pullback by the natural evaluation maps ev i : M 0,m (X, d) → X generates a set of classes on the moduli space. A system of tautological rings for {M 0,m (X, d)} m is con-* Corresponding author. E-mail addresses: A.Mustata@ucc.ie (A.M. Mustaţǎ), Andrei.mustata@ucc.ie (A. Mustaţǎ).

Universal relations on stable map spaces in genus zero

“…At the end of this paper we illustrate this procedure in the case d = 2. The case of general degree d is the subject of [MM3], based on the results of the present paper.…”

“…We view this step as prototypical for studies of Chow quotients by SL 2 -action from an intersection-theoretical point of view. The description of H * (M 0,0 (P n , d)) is completed in [MM3].…”

Towards the cohomology of moduli spaces of higher genus stable maps

“…For g = 0 it turns out that M 0,n (P r , d) carries a natural structure of smooth orbifold, in particular its rational cohomology is well-defined and well-behaved. Indeed, in the last few years there has been a flurry of research about the cohomological properties of M 0,n (P r , d) after the first steps moved in [2]: it is worth mentioning at least the contributions [1], [3], [20], [21], [22], [16], [17], [18], [19], [9], [4], [5], [6].…”

“…Indeed, in the last few years there has been a flurry of research about the cohomological properties of M 0,n (P r , d) after the first steps moved in [2]: it is worth mentioning at least the contributions [1], [3], [20], [21], [22], [16], [17], [18], [19], [9], [4], [5], [6].…”