Results 1  10
of
16
Reduction of the Hurwitz space of metacyclic covers
, 2004
"... We compute the stable reduction of some Galois covers of the projective line branched at three points. These covers are constructed using Hurwitz spaces parameterizing metacyclic covers. The reduction is determined by a certain hypergeometric differential equation. This generalizes the result of Del ..."
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We compute the stable reduction of some Galois covers of the projective line branched at three points. These covers are constructed using Hurwitz spaces parameterizing metacyclic covers. The reduction is determined by a certain hypergeometric differential equation. This generalizes the result of Deligne and Rapoport on the reduction of the modular curve X(p).
ON THE TANGENT SPACE OF THE DEFORMATION FUNCTOR OF CURVES WITH AUTOMORPHISMS
, 2007
"... Abstract. We provide a method to compute the dimension of the tangent space to the global infinitesimal deformation functor of a curve together with a subgroup of the group of automorphisms. The computational techniques we developed are applied to several examples including Fermat curves, pcyclic c ..."
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Abstract. We provide a method to compute the dimension of the tangent space to the global infinitesimal deformation functor of a curve together with a subgroup of the group of automorphisms. The computational techniques we developed are applied to several examples including Fermat curves, pcyclic covers of the affine line and to LehrMatignon curves. The aim of this paper is the study of equivariant equicharacteristic infinitesimal deformations of a curve X of genus g, admitting a group of automorphisms. This paper is the result of my attempt to understand the works of J.Bertin A.Mézard [1] and of G. Cornelissen F. Kato [3]. Let X be a smooth algebraic curve, defined over an algebraically closed field of characteristic p ≥ 0. The infinitesimal deformations of the curve X, without considering compatibility with the group action, correspond to directions on the vector space H 1 (X, TX) which constitutes the tangent space to the deformation functor of the curve X [8]. All elements in H 1 (X, TX) give rise to unobstructed deformations, since X is onedimensional and the second cohomology vanishes.
The prank stratification of ArtinSchreier curves
"... ABSTRACT. We study a moduli spaceASg for ArtinSchreier curves of genus g over an algebraically closed field k of characteristic p. We study the stratification ofASg by prank into strataASg.s of ArtinSchreier curves of genus g with prank exactly s. We enumerate the irreducible components ofASg,s ..."
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ABSTRACT. We study a moduli spaceASg for ArtinSchreier curves of genus g over an algebraically closed field k of characteristic p. We study the stratification ofASg by prank into strataASg.s of ArtinSchreier curves of genus g with prank exactly s. We enumerate the irreducible components ofASg,s and find their dimensions. As an application, when p = 2, we prove that every irreducible component of the moduli space of hyperelliptic kcurves with genus g and 2rank s has dimension g−1+s. We also determine all pairs (p,g) for whichASg is irreducible. Finally, we study deformations of ArtinSchreier curves with varying prank. La stratification de prang des courbes d’ArtinSchreier RÉSUMÉ. Nous étudions un espace de modulesASg des courbes d’Artin Schreier de genre g sur k, un corps algébriquement clos de caractéristique p. Nous étudions la stratification deASg par le prang, dont la strateASg,s décrit les courbes de genre g et de prang s. On énumère les composantes irréductibles deASg,s et on donne leurs dimensions. Une application, dans le cas p = 2, est que chaque composante irréductible de l’espace de modules des courbes hyperelliptiques sur k de genre g et de 2rang s est de dimension g−1+s. Finalement, nous déterminons toutes les paires (p,g) pour lesquellesASg est irréductible. 1.
Equiramified deformations of covers in positive characteristic
"... Suppose φ is a wildly ramified cover of germs of curves defined over an algebraically closed field of characteristic p. We study unobstructed deformations of φ in equal characteristic, which are equiramified in that the branch locus is constant and the ramification filtration is fixed. We show tha ..."
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Suppose φ is a wildly ramified cover of germs of curves defined over an algebraically closed field of characteristic p. We study unobstructed deformations of φ in equal characteristic, which are equiramified in that the branch locus is constant and the ramification filtration is fixed. We show that the moduli space Mφ parametrizing equiramified deformations of φ is a subscheme of an explicitly constructed scheme. This allows us to give an explicit upper and lower bound for the Krull dimension dφ of Mφ. These bounds depend only on the ramification filtration of φ. When φ is an abelian pgroup cover, we use class field theory to show that the upper bound for dφ is realized.
Rigidity, Reduction, and Ramification
"... In this paper we consider wildly ramified GGalois covers of curves f: Y → P 1 k branched at exactly one point over an algebraically closed field k of characteristic p. For G equal to Ap or PSL2(p), we prove Abhyankar’s Inertia Conjecture that all possible inertia groups occur over infinity for such ..."
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In this paper we consider wildly ramified GGalois covers of curves f: Y → P 1 k branched at exactly one point over an algebraically closed field k of characteristic p. For G equal to Ap or PSL2(p), we prove Abhyankar’s Inertia Conjecture that all possible inertia groups occur over infinity for such covers f. In addition, we prove that the set of conductors that can be realized depends on the group. The method we use is to compute the reduction of Galois covers of P 1 branched at 3 points. We observe that the existence of covers with given inertia Q in characteristic p is closely related to the arithmetic of covers in characteristic zero. 2000 Mathematical Subject Classification: 14H30, 14G32
Wild cyclicbytame extensions
, 807
"... Suppose G is a semidirect product of the form Z/p n ⋊ Z/m where p is prime and m is relatively prime to p. Suppose K is a complete local field of characteristic p> 0. The main result states necessary and sufficient conditions on the ramification filtrations that occur for wildly ramified GGalois e ..."
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Suppose G is a semidirect product of the form Z/p n ⋊ Z/m where p is prime and m is relatively prime to p. Suppose K is a complete local field of characteristic p> 0. The main result states necessary and sufficient conditions on the ramification filtrations that occur for wildly ramified GGalois extensions of K. In addition, we prove that there exists a parameter space for GGalois extensions of K with given ramification filtration whose dimension depends only on the ramification filtration. We provide explicit equations for wild cyclic extensions of K of degree p 3.
Wild cyclicbytame extensions
"... Suppose G is a semidirect product of the form Z/p n ⋊Z/m where p is prime and m is relatively prime to p. Suppose K is a complete local field of characteristic p> 0 with algebraically closed residue field. The main result states necessary and sufficient conditions on the ramification filtrations th ..."
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Suppose G is a semidirect product of the form Z/p n ⋊Z/m where p is prime and m is relatively prime to p. Suppose K is a complete local field of characteristic p> 0 with algebraically closed residue field. The main result states necessary and sufficient conditions on the ramification filtrations that occur for wildly ramified GGalois extensions of K. In addition, we prove that there exists a parameter space for GGalois extensions of K with given ramification filtration, and we calculate its dimension in terms of the ramification filtration. We provide explicit equations for wild cyclic extensions of K of degree p 3.
Supervisor of Dissertation
, 2009
"... Graduate Group Chairpersonsearch conversations. Rachel Pries has been a wonderful coauthor, and working with her has been an ideal introduction to mathematical collaboration. Mohamed Saïdi has been a great source of information and helped to steer me away from investing much effort into proving a fa ..."
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Graduate Group Chairpersonsearch conversations. Rachel Pries has been a wonderful coauthor, and working with her has been an ideal introduction to mathematical collaboration. Mohamed Saïdi has been a great source of information and helped to steer me away from investing much effort into proving a false theorem. Many of the ideas in §2.1 of this thesis came about from conversations with Bob Guralnick. Thanks especially to Stefan Wewers, whose fascinating work inspired my research. • To Dick Gross, who has been a mentor and role model to me for many years, and has encouraged me every step of the way. • To Janet Burns, Monica Pallanti, Paula Scarborough, Robin Toney, and Henry Benjamin for their help, their friendliness, and their patience with a certain math graduate student who can never seem to remember how to use the fax machine. Or the copier. • To all my friends in the department—you make me look forward to coming into DRL (seriously, it’s not the beautiful exterior). To Clay and Shea for being wonderful officemates and for their frequent technological help. To Asher, Shuvra, and Scott C. for many useful mathematical conversations. To John for being a great officemate for the last two years. To Wil, Andy, Dragos, Colin, Tobi, Dave F., other Dave F.,