On the tangent space of the deformation functor of curves with automorphisms (original) (raw)
2003
We give an alternative approach to the computation of the dimension of the tangent space of the deformation space of curves with automorphisms. A homological version of the local-global principle similar to the one of J.Bertin, A. Mézard is proved, and a computation in the case of ordinary curves is obtained, by application of the results of S. Nakajima for the Galois module structure of the space of 2-holomorphic differentials on them.
Polydifferentials and the deformation functor of curves with automorphisms
Journal of Pure and Applied Algebra, 2007
We apply the known results on the Galois module structure of the sheaf of polydifferentials in order to study the dimension of the tangent space of the deformation functor of curves with automorphisms. We are able to find the dimension for the case of weakly ramified covers and for the case of the action of a cyclic group of order p v .
Quadratic differentials and equivariant deformation theory of curves
2011
Given a finite p-group G acting on a smooth projective curve X over an algebraically closed field k of characteristic p, the dimension of the tangent space of the associated equivariant deformation functor is equal to the dimension of the space of coinvariants of G acting on the space V of global holomorphic quadratic differentials on X. We apply known
On deformations of maps and curve singularities
manuscripta mathematica, 2008
We study several deformation functors associated to the normalization of a reduced curve singularity (X, 0) ⊂ (C n , 0). The main new results are explicit formulas, in terms of classical invariants of (X, 0), for the cotangent cohomology groups T i , i = 0, 1, 2, of these functors. Thus we obtain precise statements about smoothness and dimension of the corresponding local moduli spaces. We apply the results to obtain explicit formulas resp. estimates for the Ae-codimension of a parametrized curve singularity, where Ae denotes the Mather-Wall group of left-right equivalence.
Realizing deformations of curves using Lubin-Tate formal groups
Israel Journal of Mathematics, 2004
Let k be an algebraically closed field of chm'acteristic p > 0 and R be a suitable valuation ring of characteristic 0~ dominating the Witt vectors W(k). We show how Lubin-Tate formal groups can be used to lift those order p" automorphisms of k[Z] to R[Z'~, which occur as endolnorphisms of a formal group over k of suitable height. We apply this result to prove the existence of smooth liftings of galois covers of smooth curves from characteristic p to characteristic 0, provided the p-part of the inertia groups acting on the completion of the local rings at the points of the cover over k are p-power cyclic and determined by an endomorphism of a suitable formal group over k. * The author would like to express his thanks to the Max-Planck-Institut fiir Mathematik, Bonn, for its hospitality and support, where this research was done during a visit from June-September 2001.
Mathematische Zeitschrift, 2008
For nonsingular projective curves defined over algebraically closed fields of positive characteristic the dependence of the ramification filtration of decomposition groups of the automorphism group with Weierstrass semigroups attached at wild ramification points is studied. A faithful representation of the p-part of the decomposition group at each wild ramified point to a Riemann-Roch space is defined.
The local lifting problem for actions of finite groups on curves. Annales scientifiques de l’ENS
2016
Let k be an algebraically closed field of characteristic p > 0. We study obstructions to lifting to characteristic 0 the faithful continuous action φ of a finite group G on k[[t]]. To each such φ a theorem of Katz and Gabber associates an action of G on a smooth projective curve Y over k. We say that the KGB obstruction of φ vanishes if G acts on a smooth projective curve X in characteristic 0 in such a way that X/H and Y /H have the same genus for all subgroups H ⊂ G. We determine for which G the KGB obstruction of every φ vanishes. We also consider analogous problems in which one requires only that an obstruction to lifting φ due to Bertin vanishes for some φ, or for all sufficiently ramified φ. These results provide evidence for the strengthening of Oort's lifting conjecture which is discussed in [8, Conj. 1.2]. Résumé. Soit k un corps algébriquement clos de caractéristique p > 0. Nouś etudions les obstructions au relèvement en caractéristique 0 d'une action fidèle et continue φ d'un groupe fini G sur k[[t]]. Le théorème de Katz-Gabber associeà φ, une action du groupe G sur une courbe projective Y lisse sur k. La KGBobstruction de φ est dite nulle si G agit sur une courbe projective lisse X de caractéristique 0 avecégalité des genres de X/H et Y /H pour tout sous-groupe H ⊂ G. Nous déterminons les groupes G pour lesquels la KGB-obstruction s'annule pour toute action φ. Nous considéronségalement des situations analogues pour lesquelles il suffit d'annuler l'obstruction de Bertinà relever une action φ ou toutes actions φ suffisamment ramifiées. Ces résultats renforcent les convictions en faveur de la conjecture de Oort généralisée aux relèvements d'une action fidèle sur une courbe projective lisse ([8], Conj. 1.2). Contents 1. Introduction. 2 2. The Bertin obstruction. 7 3. Constants associated to cyclic subgroups which are not p-groups. 11 4. The Katz-Gabber-Bertin obstruction 12 5. Functorality. 15 6. The reduction to p-groups.
Deformation of Outer Representations of Galois Group II
2006
This paper is devoted to deformation theory of "anabelian" representations of the absolute Galois group landing in outer automorphism group of the algebraic fundamental group of a hyperbolic smooth curve defined over a number-field. In the first part of this paper, we obtained universal deformations for Lie-algebra versions of the above representation using the Schlessinger criteria for functors on Artin local rings. In the second part, we use a version of Schlessinger criteria for functors on the Artinian category of nilpotent Lie algebras which is formulated by Pridham, and explore arithmetic applications.
Deformation of Curves with Automorphisms and representations on Riemann-Roch spaces
2008
We study the deformation theory of nonsigular projective curves defined over algebraic closed fields of positive characteristic. We show that under some assumptions the local deformation problem for automorphisms of powerseries can be reduced to a deformation problem for matrix representations. We study both equicharacteristic and mixed deformations in the case of two dimensional representations.
Israel Journal of Mathematics, 2010
We consider three examples of families of curves over a non-archimedean valued field which admit a non-trivial group action. These equivariant deformation spaces can be described by algebraic parameters (in the equation of the curve), or by rigid-analytic parameters (in the Schottky group of the curve). We study the relation between these parameters as rigid-analytic self-maps of the disk.
Deformation of Outer Representations of Galois Group
2004
To a hyperbolic smooth curve defined over a number-field one naturally associates an "anabelian" representation of the absolute Galois group of the base field landing in outer automorphism group of the algebraic fundamental group. In this paper, we introduce several deformation problems for Lie-algebra versions of the above representation and show that, this way we get a richer structure than those coming from deformations of "abelian" Galois representations induced by the Tate module of associated Jacobian variety. We develop an arithmetic deformation theory of graded Lie algebras with finite dimensional graded components to serve our purpose.
Deformations and elements of deformation theory
2016
This article consisted of an elementary introduction to deformation theory of varieties, schemes and manifolds, with some applications to local and global shtukas and fever to Newton polygons of ppp-divisible groups . Soft problems and results mainly are considered. In the framework we give review of some novel results in the theory of local shtukas, Anderson-modules, global shtukas, Newton polygons of ppp-divisible groups and on deformations of ppp-divisible groups with given Newton polygons.
The local lifting problem for actions of finite groups on curves
Annales scientifiques de l'École normale supérieure, 2011
Aʙʀ.-Let k be an algebraically closed field of characteristic p > 0. We study obstructions to lifting to characteristic 0 the faithful continuous action φ of a finite group G on k[[t]]. To each such φ a theorem of Katz and Gabber associates an action of G on a smooth projective curve Y over k. We say that the KGB obstruction of φ vanishes if G acts on a smooth projective curve X in characteristic 0 in such a way that X/H and Y /H have the same genus for all subgroups H ⊂ G. We determine for which G the KGB obstruction of every φ vanishes. We also consider analogous problems in which one requires only that an obstruction to lifting φ due to Bertin vanishes for some φ, or for all sufficiently ramified φ. These results provide evidence for the strengthening of Oort's lifting conjecture which is discussed in [8, Conj. 1.2]. R.-Soit k un corps algébriquement clos de caractéristique p > 0. Nous étudions les obstructions au relèvement en caractéristique 0 d'une action fidèle et continue φ d'un groupe fini G sur k[[t]]. Le théorème de Katz-Gabber associe à φ une action du groupe G sur une courbe projective Y lisse sur k. La KGB-obstruction de φ est dite nulle si G agit sur une courbe projective lisse X de caractéristique 0 avec égalité des genres de X/H et Y /H pour tout sous-groupe H ⊂ G. Nous déterminons les groupes G pour lesquels la KGB-obstruction s'annule pour toute action φ. Nous considérons également des situations analogues pour lesquelles il suffit d'annuler l'obstruction de Bertin à relever une action φ ou toutes actions φ suffisamment ramifiées. Ces résultats renforcent les convictions en faveur de la conjecture de Oort généralisée aux relèvements d'une action fidèle sur une courbe projective lisse ([8], Conj. 1.2).
A Superficial Working Guide to Deformations and Moduli
This is the first part of a guide to deformations and moduli, especially viewed from the perspective of algebraic surfaces (the simplest higher dimensional varieties). It contains also new results, regarding the question of local homeomorphism between Kuranishi and Teichmueller space, and a survey of new results with Ingrid Bauer, concerning the discrepancy between the deformation of the action of a group G on a minimal models S, respectively the deformation of the action of G on the canonical model X. Here Def(S) maps properly onto Def(X), but the same does not hold for pairs: Def(S,G) does not map properly onto Def(X,G). Indeed the connected components of Def(S), in the case of tertiary Burniat surfaces, only map to locally closed sets. The last section contains anew result on some surfaces whise Albanese map has generic degree equal to 2.
$p$-group Galois covers of curves in characteristic ppp
2022
We study cohomologies of a curve with an action of a finite ppp-group over a field of characteristic ppp. Assuming the existence of a certain 'magical element' in the function field of the curve, we compute the equivariant structure of the module of holomorphic differentials and the de Rham cohomology, up to certain local terms. We show that a generic ppp-group cover has a 'magical element'. As an application we compute the de Rham cohomology of a curve with an action of a finite cyclic group of prime order.
Level mmm stratifications of versal deformations of ppp-divisible groups
Journal of Algebraic Geometry, 2008
Let k be an algebraically closed field of characteristic p > 0. Let c, d, m be positive integers. Let D be a p-divisible group of codimension c and dimension d over k. Let D be a versal deformation of D over a smooth k-scheme A which is equidimensional of dimension cd. We show that there exists a reduced, locally closed subscheme s D (m) of A that has the following property: a point y ∈ A(k) belongs to s D (m)(k) if and only if y * (D)[p m ] is isomorphic to D[p m ]. We prove that s D (m) is regular and equidimensional of dimension cd−dim(Aut Aut Aut(D[p m ])). We give a proof of Traverso's formula which for m >> 0 computes the codimension of s D (m) in A (i.e., dim(Aut Aut Aut(D[p m ]))) in terms of the Newton polygon of D. We also provide a criterion of when s D (m) satisfies the purity property (i.e., it is an affine A-scheme). Similar results are proved for quasi Shimura p-varieties of Hodge type that generalize the special fibres of good integral models of Shimura varieties of Hodge type in unramified mixed characteristic (0, p).
Connectedness of the moduli space of Artin-Schreier curves of fixed genus
Journal of Algebra, 2020
We study the moduli space AS g of Artin-Schreier curves of genus g over an algebraically closed field k of positive characteristic p. The moduli space is partitioned into strata, which are irreducible. Each stratum parameterizes Artin-Schreier curves whose ramification divisors have the same coefficients. We construct deformations of these curves to study the relations between those strata. As an application, when p = 3, we prove that AS g is connected for all g. When p > 3, it turns out that AS g is connected for a sufficiently large value of g. In the course of our work, we answer a question of Pries and Zhu about how a combinatorial graph determines the geometry of AS g. 2 Moduli space of Artin-Schreier curves 2.1 Artin-Schreier Theory Let K be a field of characteristic p. If L is a separable extension of K of degree p, then Artin-Schreier theory says that L = K(α) where α is a root of the polynomial y p − y = a for some element a ∈ K (detailed in [Lan02]). Let Y be an Artin-Schreier k-curve. Then there is a Z/p-cover φ: Y → P 1 k with an affine equation of the form y p − y = f (x) for some non-constant rational function f (x) ∈ k(x). A cover φ ′ : Y ′ → P 1 k defined by an affine equation y p − y = g(x) is isomorphic to φ if g(x) = f (x)+h(x) p −h(x) for some h(x) ∈ K(x). At each ramification point, there is a filtration
On the Automorphisms of Moduli Spaces of Curves
Springer Proceedings in Mathematics & Statistics, 2014
In the last years the biregular automorphisms of the Deligne-Mumford's and Hassett's compactifications of the moduli space of n-pointed genus g smooth curves have been extensively studied by A. Bruno and the authors. In this paper we give a survey of these recent results and extend our techniques to some moduli spaces appearing as intermediate steps of the Kapranov's and Keel's realizations of M 0,n, and to the degenerations of Hassett's spaces obtained by allowing zero weights.