ABSTRACT. We prove a localization formula for the moduli space of stable relative maps. As an application, we prove that all codimension i tautological classes on the moduli space of stable pointed curves vanish away from strata corresponding to curves with at least i − g + 1 genus 0 components. As consequences, we prove and generalize various conjectures and theorems about various moduli spaces of curves (due to Diaz, Faber, Getzler, Ionel, Looijenga, Pandharipande, and others). This theorem appears to be the geometric content behind these results; the rest is straightforward graph combinatorics. The theorem also suggests the importance of the stratification of the moduli space by number of rational components.

Double Hurwitz numbers count branched covers of CP 1 with fixed branch points, with simple branching required over all but two points 0 and ∞, and the branching over 0 and ∞ specified by partitions of the degree (with m and n parts, respectively). Single Hurwitz numbers (or more usually, Hurwitz numbers) have a rich structure, explored by many authors in fields as diverse as algebraic geometry, symplectic geometry, combinatorics, representation theory, and mathematical physics. The remarkable ELSV formula relates single Hurwitz numbers to intersection theory on the moduli space of curves. This connection has led to many consequences, including Okounkov and Pandharipande's proof of Witten's conjecture.In this paper, we determine the structure of double Hurwitz numbers using techniques from geometry, algebra, and representation theory. Our motivation is geometric: we give evidence that double Hurwitz numbers are top intersections on a moduli space of curves with a line bundle (a universal Picard variety). In particular, we prove a piecewise-polynomiality result analogous to that implied by the ELSV formula. In the case m = 1 (complete branching over one point) and n is arbitrary, we conjecture an ELSV-type formula, and show it to be true in ଁ I.P.G. and D.M.J. 44 I.P. Goulden et al. / Advances in Mathematics 198 (2005) 43 -92 genus 0 and 1. The corresponding Witten-type correlation function has a richer structure than that for single Hurwitz numbers, and we show that it satisfies many geometric properties, such as the string and dilaton equations, and an Itzykson-Zuber-style genus expansion ansatz. We give a symmetric function description of the double Hurwitz generating series, which leads to explicit formulae for double Hurwitz numbers with given m and n, as a function of genus. In the case where m is fixed but not necessarily 1, we prove a topological recursion on the corresponding generating series, which leads to closed-form expressions for double Hurwitz numbers and an analogue of the Goulden-Jackson polynomiality conjecture (an early conjectural variant of the ELSV formula). In a later paper (Faber's intersection number conjecture and genus 0 double Hurwitz numbers, 2005, in preparation), the formulae in genus 0 will be shown to be equivalent to the formulae for "top intersections" on the moduli space of smooth curves M g . For example, three formulae we give there will imply Faber's intersection number conjecture (in: Moduli of Curves and Abelian Varieties, Aspects of Mathematics, vol. E33, Vieweg, Braunschweig, 1999, pp. 109-129) in arbitrary genus with up to three points.

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...

ABSTRACT. We consider the question: "How bad can the deformation space of an object be?" The answer seems to be: "Unless there is some a priori reason otherwise, the deformation space may be as bad as possible." We show this for a number of important moduli spaces.More precisely, every singularity of finite type over Z (up to smooth parameters) appears on: the Hilbert scheme of curves in projective space; and the moduli spaces of smooth projective general-type surfaces (or higher-dimensional varieties), plane curves with nodes and cusps, stable sheaves, isolated threefold singularities, and more. The objects themselves are not pathological, and are in fact as nice as can be: the curves are smooth, the surfaces have very ample canonical bundle, the stable sheaves are torsion-free of rank 1, the singularities are normal and Cohen-Macaulay, etc. This justifies Mumford's philosophy that even moduli spaces of well-behaved objects should be arbitrarily bad unless there is an a priori reason otherwise.Thus one can construct a smooth curve in projective space whose deformation space has any given number of components, each with any given singularity type, with any given non-reduced behavior along various associated subschemes. Similarly one can give a surface over F p that lifts to Z/p 7 but not Z/p 8 . (Of course the results hold in the holomorphic category as well.)It is usually difficult to compute deformation spaces directly from obstruction theories. We circumvent this by relating them to more tractable deformation spaces via smooth morphisms. The essential starting point is Mnëv's Universality Theorem.

Hurwitz numbers, which count certain covers of the projective line (or, equivalently, factorizations of permutations into transpositions), have been extensively studied for over a century. The Gromov‐Witten potential F of a point, the generating series for descendent integrals on the moduli space of curves, is a central object of study in Gromov‐Witten theory. We define a slightly enriched Gromov‐Witten potential G (including integrals involving one ‘λ‐class’), and show that, after a non‐trivial change of variables, G = H in positive genus, where H is a generating series for Hurwitz numbers. We prove a conjecture of Goulden and Jackson on higher genus Hurwitz numbers, which turns out to be an analogue of a genus expansion ansatz of Itzykson and Zuber. As consequences, we have new combinatorial constraints on F, and a much more direct proof of the ansatz of Itzykson and Zuber. We can produce recursions and explicit formulas for Hurwitz numbers; the algorithm presented proves all such recursions. As examples we present surprisingly simple new recursions in genus 0 to 3. Similar recursions should exist for all genera. As we expect this paper also to be of interest to combinatorialists, we have tried to make it as self‐contained as possible, including reviewing some results and definitions well known in algebraic and symplectic geometry, and mathematical physics. 2000 Mathematical Subject Classification: primary 14H10, 81T40; secondary 05C30, 58D29.