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The irreducibility of the space of curves of given genus
 Publ. Math. IHES
, 1969
"... Fix an algebraically closed field k. Let Mg be the moduli space of curves of genus g over k. The main result of this note is that Mg is irreducible for every k. Of course, whether or not M s is irreducible depends only on the characteristic of k. When the characteristic s o, we can assume that k ~ ..."
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Cited by 512 (2 self)
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Fix an algebraically closed field k. Let Mg be the moduli space of curves of genus g over k. The main result of this note is that Mg is irreducible for every k. Of course, whether or not M s is irreducible depends only on the characteristic of k. When the characteristic s o, we can assume that k
Counting Higher Genus Curves with Crosscaps
, 2004
"... We compute all loop topological string amplitudes on orientifolds of local CalabiYau manifolds, by using geometric transitions involving SO/Sp ChernSimons theory, localization on the moduli space of holomorphic maps with involution, and the topological vertex. In particular we count Klein bottles ..."
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Cited by 3 (0 self)
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We compute all loop topological string amplitudes on orientifolds of local CalabiYau manifolds, by using geometric transitions involving SO/Sp ChernSimons theory, localization on the moduli space of holomorphic maps with involution, and the topological vertex. In particular we count Klein bottles and projective planes with any number of handles in some CalabiYau orientifolds.
ON THE FREYMAZUR CONJECTURE OVER LOW GENUS CURVES
"... Abstract. The FreyMazur conjecture states that an elliptic curve over Q is determined up to isogeny by its ptorsion Galois representation for p ≥ 17. We study a geometric analog of this conjecture, and show that the map from isogeny classes of “fake elliptic curves”—abelian surfaces with quaternio ..."
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Cited by 1 (1 self)
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with quaternionic multiplication—to their ptorsion Galois representations is at most twotoone over function fields of small genus curves for sufficiently large p relative to the genus. Moreover, if the Shimura curve parameterizing such abelian surfaces has genus> 1, then the map is in fact onetoone. 1.
KodairaSpencer theory of gravity and exact results for quantum string amplitudes
 Commun. Math. Phys
, 1994
"... We develop techniques to compute higher loop string amplitudes for twisted N = 2 theories with ĉ = 3 (i.e. the critical case). An important ingredient is the discovery of an anomaly at every genus in decoupling of BRST trivial states, captured to all orders by a master anomaly equation. In a particu ..."
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Cited by 545 (60 self)
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’ of holomorphic curves of higher genus curves in Calabi–Yau manifolds. It is shown that topological amplitudes can also be reinterpreted as computing corrections to superpotential terms appearing in the effective 4d theory resulting from compactification of standard 10d superstrings on the corresponding N = 2
The geometric Bogomolov conjecture for small genus curves
"... Abstract. The Bogomolov Conjecture is a finiteness statement about algebraic points of small height on a smooth complete curve defined over a global field. We verify an effective form of the Bogomolov Conjecture for all curves of genus at most 4 over a function field of characteristic zero. We recov ..."
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Cited by 4 (1 self)
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Abstract. The Bogomolov Conjecture is a finiteness statement about algebraic points of small height on a smooth complete curve defined over a global field. We verify an effective form of the Bogomolov Conjecture for all curves of genus at most 4 over a function field of characteristic zero. We
UNRAMIFIED COVERS OF GALOIS COVERS OF LOW GENUS CURVES
"... Abstract. Let X → Y be a Galois covering of curves, where the genus of X is ≥ 2 and the genus of Y is ≤ 2. We prove that under certain hypotheses, X has an unramified cover that dominates a hyperelliptic curve; our results apply, for instance, to all tamely superelliptic curves. Combining this with ..."
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Cited by 2 (0 self)
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Abstract. Let X → Y be a Galois covering of curves, where the genus of X is ≥ 2 and the genus of Y is ≤ 2. We prove that under certain hypotheses, X has an unramified cover that dominates a hyperelliptic curve; our results apply, for instance, to all tamely superelliptic curves. Combining
GromovWitten classes, quantum cohomology, and enumerative geometry
 Commun. Math. Phys
, 1994
"... The paper is devoted to the mathematical aspects of topological quantum field theory and its applications to enumerative problems of algebraic geometry. In particular, it contains an axiomatic treatment of Gromov–Witten classes, and a discussion of their properties for Fano varieties. Cohomological ..."
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Cited by 484 (3 self)
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prediction of some numerical characteristics of the space of algebraic curves in V, especially of genus zero, eventually endowed with a parametrization and marked points. It turned out that
Results 1  10
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81,815