No abstract

Abstract. We give a description of the relative Hilbert scheme of lines in the Dwork pencil of quintic threefolds. We describe the corresponding relative Hilbert scheme associated to the mirror family of quintic threefolds.

Abstract. We introduce a factorization for the map between moduli spaces of stable maps which forgets one marked point. This leads to a study of universal relations in the cohomology of stable map spaces in genus zero.

Abstract. For a proper local embedding between two Deligne-Mumford stacks Y and X, we find, under certain mild conditions, a new (possibly non-separated) DeligneMumford stack X , with an etale, surjective and universally closed map to the target X, and whose fiber product with the image of the local embedding is a finite union of stacks with corresponding etale, surjective and universally closed maps to Y . Moreover, a natural set of weights on the substacks of X allows the construction of a universally closed push-forward, and thus a comparison between the Chow groups of X and X. We apply the construction above to the computation of the Chern classes of a weighted blow-up along a regular local embedding via deformation to a weighted normal cone and localization. We describe the stack X in the case when X is the moduli space of stable maps with local embeddings at the boundary. We apply the methods above to find the Chern classes of the stable map spaces.

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).