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HODGE INTEGRALS AND HURWITZ NUMBERS VIA VIRTUAL LOCALIZATION
, 2000
"... Abstract. Ekedahl, Lando, Shapiro, and Vainshtein announced a remarkable formula ([ELSV]) expressing Hurwitz numbers (counting covers of P 1 with specified simple branch points, and specified branching over one other point) in terms of Hodge integrals. We give a proof of this formula using virtual l ..."
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Abstract. Ekedahl, Lando, Shapiro, and Vainshtein announced a remarkable formula ([ELSV]) expressing Hurwitz numbers (counting covers of P 1 with specified simple branch points, and specified branching over one other point) in terms of Hodge integrals. We give a proof of this formula using virtual localization on the moduli space of stable maps, and describe how the proof could be simplified by the proper algebrogeometric definition of a “relative
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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COUNTING CURVES ON RATIONAL SURFACES
"... Abstract. In [CH3], Caporaso and Harris derive recursive formulas counting nodal plane curves of degree d and geometric genus g in the plane (through the appropriate number of fixed general points). We rephrase their arguments in the language of maps, and extend them to other rational surfaces, and ..."
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Abstract. In [CH3], Caporaso and Harris derive recursive formulas counting nodal plane curves of degree d and geometric genus g in the plane (through the appropriate number of fixed general points). We rephrase their arguments in the language of maps, and extend them to other rational surfaces, and other specified intersections with a divisor. As applications, (i) we count irreducible curves on Hirzebruch surfaces in a fixed divisor class and of fixed geometric genus, (ii) we compute the higher genus GromovWitten invariants of (or equivalently, counting curves of any genus and divisor class on) del Pezzo surfaces of degree at least 3. In the case of the cubic surface in (ii), we first use a result of Graber to enumeratively interpret highergenus GromovWitten invariants of certain Knef surfaces, and then apply this to a degeneration of a cubic surface.
Genus 0 and 1 Hurwitz Numbers: Recursions, Formulas, AND GRAPHTHEORETIC INTERPRETATIONS
, 2000
"... We derive a closedform expression for all genus 1 Hurwitz numbers, and give a simple new graphtheoretic interpretation of Hurwitz numbers in genus 0 and 1. (Hurwitz numbers essentially count irreducible genus g covers of the sphere, with arbitrary specified branching over one point, simple branch ..."
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Cited by 15 (5 self)
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We derive a closedform expression for all genus 1 Hurwitz numbers, and give a simple new graphtheoretic interpretation of Hurwitz numbers in genus 0 and 1. (Hurwitz numbers essentially count irreducible genus g covers of the sphere, with arbitrary specified branching over one point, simple branching over other specified points, and no other branching. The problem is equivalent to counting transitive factorisations of permutations into transpositions.) These results prove a conjecture of Goulden, Jackson and Vainshtein, and extend results of Hurwitz and many others.
Recursions, formulas, and graphtheoretic interpretations of ramified coverings of the sphere by surfaces of genus 0
 and 1, Trans. Amer. Math. Soc
"... Abstract. We derive a closedform expression for all genus 1 Hurwitz numbers, and give a simple new graphtheoretic interpretation of Hurwitz numbers in genus 0 and 1. (Hurwitz numbers essentially count irreducible genus g covers of the sphere, with arbitrary speci ed branching over one point, simpl ..."
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Abstract. We derive a closedform expression for all genus 1 Hurwitz numbers, and give a simple new graphtheoretic interpretation of Hurwitz numbers in genus 0 and 1. (Hurwitz numbers essentially count irreducible genus g covers of the sphere, with arbitrary speci ed branching over one point, simple branching over other speci ed points, and no other branching. The problem is equivalent to counting transitive factorisations of permutations into transpositions.) These results prove a conjecture of Goulden and Jackson, and extend results of Hurwitz and many others. 1.
Hodge Integrals and Hurwitz . . .
, 2000
"... Ekedahl, Lando, Shapiro, and Vainshtein announced a remarkable formula ([ELSV]) expressing Hurwitz numbers (counting covers of P 1 with specified simple branch points, and specified branching over one other point) in terms of Hodge integrals. We give a proof of this formula using virtual localizat ..."
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Ekedahl, Lando, Shapiro, and Vainshtein announced a remarkable formula ([ELSV]) expressing Hurwitz numbers (counting covers of P 1 with specified simple branch points, and specified branching over one other point) in terms of Hodge integrals. We give a proof of this formula using virtual localization on the moduli space of stable maps, and describe how the proof could be simplified by the proper algebrogeometric definition of a “relative space".
TRANSITIVE FACTORIZATIONS IN THE HYPEROCTAHEDRAL GROUP
"... ABSTRACT. The classical Hurwitz Enumeration Problem has a presentation in terms of transitive factorisations in the symmetric group. This presentation suggests a generalization from type A to other £nite re¤ection groups and, in particular, to type B. We study this generalization both from a combina ..."
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ABSTRACT. The classical Hurwitz Enumeration Problem has a presentation in terms of transitive factorisations in the symmetric group. This presentation suggests a generalization from type A to other £nite re¤ection groups and, in particular, to type B. We study this generalization both from a combinatorial and a geometric point of view, with the prospect of providing a means of understanding more of the structure of the moduli spaces of maps with an S2symmetry. The type A case has been well studied and connects Hurwitz numbers to the moduli space of curves. We conjecture an analogous setting for the type B case that is studied here. 1.
A TOOL FOR STABLE REDUCTION OF CURVES ON SURFACES
"... Abstract. In the study of the geometry of curves on surfaces, the following question often arises: given a oneparameter family of divisors over a pointed curve, what does the central fiber look like after stable or nodal reduction? We present a lemma describing the dual graph of the limit. Contents ..."
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Abstract. In the study of the geometry of curves on surfaces, the following question often arises: given a oneparameter family of divisors over a pointed curve, what does the central fiber look like after stable or nodal reduction? We present a lemma describing the dual graph of the limit. Contents