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44
STABLE MORPHISMS TO SINGULAR SCHEMES AND RELATIVE STABLE MORPHISMS
"... Let W/C be a degeneration of smooth varieties so that the special fiber has normal crossing singularity. In this paper, we first construct the stack of expanded degenerations of W. We then construct the moduli space of stable morphisms to this stack, which provides a degeneration of the moduli spa ..."
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Cited by 101 (5 self)
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Let W/C be a degeneration of smooth varieties so that the special fiber has normal crossing singularity. In this paper, we first construct the stack of expanded degenerations of W. We then construct the moduli space of stable morphisms to this stack, which provides a degeneration of the moduli spaces of stable morphisms associated to W/C. Using a similar technique, for a pair (Z, D) of smooth variety and a smooth divisor, we construct the stack of expanded relative pairs and then the moduli spaces of relative stable morphisms to (Z, D). This is the algebrogeometric analogue of DonaldsonFloer theory in gauge theory. The construction of relative GromovWitten invariants and the degeneration formula of GromovWitten invariants will be treated in the subsequent paper.
A DEGENERATION FORMULA OF GWINVARIANTS
, 2001
"... This is the sequel to the paper [Li]. In this paper, we construct the virtual moduli cycles of the degeneration of the moduli of stable morphisms constructed in [Li]. We also construct the virtual moduli cycles of the moduli of relative stable morphisms of a pair of a smooth divisor in a smooth var ..."
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Cited by 84 (4 self)
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This is the sequel to the paper [Li]. In this paper, we construct the virtual moduli cycles of the degeneration of the moduli of stable morphisms constructed in [Li]. We also construct the virtual moduli cycles of the moduli of relative stable morphisms of a pair of a smooth divisor in a smooth variety. Based on these, we prove a degeneration formula of the GromovWitten invariants.
Relative virtual localization and vanishing of tautological classes on moduli spaces of curves
 Duke Math. J
"... 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 ..."
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Cited by 68 (6 self)
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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 Getzler, Ionel, Faber, Looijenga, Pandharipande, Diaz, 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. CONTENTS
Using stacks to impose tangency conditions on curves
 math.AG/0210398. [Ch1] [Ch2] [Co87] [DM69] [FSZ] [Gr68] [Ha83
"... From a scheme Y, an effective Cartier divisor D ⊂ Y, and a positive integer r, we define a stack YD,r and work out some of its basic properties. The most important of these relates morphisms from a curve C into YD,r to morphisms from C into Y such that the order of contact of C with D is a multiple ..."
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Cited by 66 (5 self)
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From a scheme Y, an effective Cartier divisor D ⊂ Y, and a positive integer r, we define a stack YD,r and work out some of its basic properties. The most important of these relates morphisms from a curve C into YD,r to morphisms from C into Y such that the order of contact of C with D is a multiple of r at each point. This is a foundational paper whose results will be applied to the enumerative geometry of curves with tangency conditions in a future paper. 1
Another way to enumerate rational curves with torus actions
 Invent. Math
"... mirror symmetry to predict the numbers of rational curves of any degree on a quintic threefold [7]. This took geometers by surprise, as the best mathematical results at the time could only count the rational curves of degree three or less. Since then, the field of enumerative algebraic geometry has ..."
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Cited by 47 (5 self)
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mirror symmetry to predict the numbers of rational curves of any degree on a quintic threefold [7]. This took geometers by surprise, as the best mathematical results at the time could only count the rational curves of degree three or less. Since then, the field of enumerative algebraic geometry has
A desingularization of the main component of the moduli space of genusone stable maps into P^n
, 2007
"... We construct a natural smooth compactification of the space of smooth genusone curves with k distinct points in a projective space. It can be viewed as an analogue of a wellknown smooth compactification of the space of smooth genuszero curves, i.e. the space of stable genuszero maps M0,k(Pn, d). ..."
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Cited by 38 (12 self)
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We construct a natural smooth compactification of the space of smooth genusone curves with k distinct points in a projective space. It can be viewed as an analogue of a wellknown smooth compactification of the space of smooth genuszero curves, i.e. the space of stable genuszero maps M0,k(Pn, d). In fact, our compactification is obtained from the singular space of stable genusone maps M1,k(Pn, 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 M 0 1,k (Pn, d) of M1,k(Pn, d). A number of applications of these desingularizations in enumerative geometry and GromovWitten theory are described in the introduction, including the second author’s proof of physicists’ predictions for genus1 Gromov
The GromovWitten potential of a point, Hurwitz numbers, and Hodge integrals
 Proc. London Math. Soc
, 1999
"... 1.1. Recursions and GromovWitten theory 2 ..."
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Logarithmic GromovWitten invariants
 Journal of the American Mathematical Society
"... ar ..."
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Quantum Lefschetz Hyperplane Theorem
"... Abstract. The mirror theorem is generalized to any smooth projective variety X. That is, a fundamental relation between the Gromov–Witten invariants of X and Gromov–Witten invariants of complete intersections Y in X is established. Contents ..."
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Cited by 27 (4 self)
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Abstract. The mirror theorem is generalized to any smooth projective variety X. That is, a fundamental relation between the Gromov–Witten invariants of X and Gromov–Witten invariants of complete intersections Y in X is established. Contents