A desingularization of the main component of the moduli space of genus-one stable maps into ℙⁿ

Abstract

A desingularization of the main component of the moduli space of genus-one stable maps into P n RAVI VAKIL ALEKSEY ZINGERWe construct a natural smooth compactification of the space of smooth genus-one curves with k distinct points in a projective space. It can be viewed as an analogue of a well-known smooth compactification of the space of smooth genus-zero curves, that is, the space of stable genus-zero maps S M 0;k .P n ; d /. In fact, our compactification is obtained from the singular space of stable genus-one maps S M 1;k .P n ; d / through a natural sequence of blowups along "bad" subvarieties. While this construction is simple to describe, it requires more work to show that the end result is a smooth space.As a bonus, we obtain desingularizations of certain natural sheaves over the "main" irreducible component S M 0 1;k .P n ; d/ of S M 1;k .P n ; d/. A number of applications of these desingularizations in enumerative geometry and Gromov-Witten theory are described in the introduction, including the second author's proof of physicists' predictions for genus-one Gromov-Witten invariants of a quintic threefold. 14D20; 53D991 Introduction Background and applicationsThe space of degree-d genus-g curves with k distinct marked points in P n is generally not compact, but admits a number of natural compactifications 1 . Among the most prominent compactifications is the moduli space of stable genus-g maps, S M g;k .P n ; d/, constructed by Gromov [9] and Fulton-Pandharipande [6]. It has found numerous applications in classical enumerative geometry and is a central object in Gromov-Witten theory. However, most applications in enumerative geometry and some results in GWtheory have been restricted to the genus-zero case. The reason for this is essentially that the genus-zero moduli space has a particularly simple structure: it is smooth and contains the space of smooth genus-zero curves as a dense open subset. hand, the moduli spaces of positive-genus stable maps fail to satisfy either of these two properties. In fact, S M g;k .P n ; d / can be arbitrarily singular according to Vakil [19]. It is thus natural to ask whether these failings can be remedied by modifying S M g;k .P n ; d/, preferably in a way that leads to a range of applications. As announced in [20] and shown in this paper, the answer is yes if gD1.We denote by M 1;k .P n ; d/ the subset of S M 1;k .P n ; d/ consisting of the stable maps that have smooth domains. This space is smooth and contains the space of genusone curves with k distinct marked points in P n as a dense open subset, provided d 3. However, M 1;k .P n ; d / is not compact. Let S M 0 1;k .P n ; d/ be the closure of M 1;k .P n ; d/ in the compact space S M 1;k .P n ; d/. While S M 0 1;k .P n ; d/ is not smooth, it turns out that a natural sequence of blowups along loci disjoint from M 1;k .P n ; d/ leads to a desingularization of S M 0 1;k .P n ; d/, which will be denoted byThe situation is as good as one could possibly hope. A general strategy when attempting to desingularize some space i...

Towards the cohomology of moduli spaces of higher genus stable maps

“…, n} and H k ( i∈I X i , Q) = 0 for every odd k < N. Let . Here the first equality holds by definition, the second one by [25], Lemma 2.3 (2), and the third one by [25], Lemma 2.3 (1). Now, it is well-know how to compare cohomology groups after a blow-up along a smooth subvariety of an orbifold (see [11], p. 605 and Proposition on p. 606, and [14], footnote on p. 514, for the orbifold case).…”

“…However, at least in the case g = 1, the beautiful construction performed in [25] has recently opened a new frontier to the research in the field. Namely, in [25] it is shown that the closure M 0 1,n (P r , d) of the stratum M 0 1,n (P r , d) corresponding to stable maps with smooth domain allows an orbifold desingularizationwhich can be explicitely described as a sequence of blow-ups along smooth centers. Moreover, the natural (C * ) r+1 -action on M 0 1,n (P r , d)…”

“…As a consequence, a cohomological investigation of the underlying topological spaces has never been addressed and it seems to be completely out of reach. However, at least in the case g = 1, the beautiful construction performed in [25] has recently opened a new frontier to the research in the field. Namely, in [25] it is shown that the closure M 0 1,n (P r , d) of the stratum M 0 1,n (P r , d) corresponding to stable maps with smooth domain allows an orbifold desingularization…”

“…We also need the following technical result: please refer to [25], Section 2.1, for all non standard definitions and notation. Lemma 1.…”

“…with i, j ∈ N (see [25] [25], p. 25). We just point out a crucial fact: from the definition of M 1,ρ (see [25], p. 23) it follows that…”