Infinitesimal deformations of restricted simple Lie algebras I

We compute the infinitesimal deformations of two families of restricted simple modular Lie algebras of Cartan type: the Witt-Jacobson and the Special Lie algebras.Comment: 27 pages, title slightly changed, references updated, typos corrected, final version to appear on J. Algebr

Restricted simple Lie algebras and their infinitesimal deformations

In this expository paper, we first review the classification of the restricted simple Lie algebras in characteristic different from 2 and 3 and then we describe their infinitesimal deformations. We conclude by indicating some possible application to the deformations of simple finite group schemes.Comment: 11 pages, An Introduction to the classification of restricted simple Lie algebras and their deformation

Ramification groups and Artin conductors of radical extensions of the rationals

We compute the higher ramification groups and the Artin conductors of radical extensions of the rationals. As an application, we give formulas for their discriminant (using the conductor-discriminant formula). The interest in such number fields is due to the fact that they are among the simplest non-abelian extensions of the rationals (and so not classified by Class Field Theory). We show that this extensions have non integer jumps in the superior ramification groups, contrarily to the case of abelian extensions (as prescribed by Hasse-Arf theorem).Comment: 29 pages, to be published on the Journal de Theorie de Nombres de Bourdeau

A minimal-variable symplectic method for isospectral flows

Isospectral flows are abundant in mathematical physics; the rigid body, the the Toda lattice, the Brockett flow, the Heisenberg spin chain, and point vortex dynamics, to mention but a few. Their connection on the one hand with integrable systems and, on the other, with Lie--Poisson systems motivates the research for optimal numerical schemes to solve them. Several works about numerical methods to integrate isospectral flows have produced a large varieties of solutions to this problem. However, many of these algorithms are not intrinsically defined in the space where the equations take place and/or rely on computationally heavy transformations. In the literature, only few examples of numerical methods avoiding these issues are known, for instance, the \textit{spherical midpoint method} on \SO(3). In this paper we introduce a new minimal-variable, second order, numerical integrator for isospectral flows intrinsically defined on quadratic Lie algebras and symmetric matrices. The algorithm is isospectral for general isospectral flows and Lie--Poisson preserving when the isospectral flow is Hamiltonian. The simplicity of the scheme, together with its structure-preserving properties, makes it a competitive alternative to those already present in literature.Comment: 17 pages, 9 figure

Moduli and Periods of Supersymmetric Curves

Supersymmetric curves are the analogue of Riemann surfaces in super geometry. We establish some foundational results about complex Deligne-Mumford superstacks, and we then prove that the moduli superstack of supersymmetric curves is a smooth complex Deligne-Mumford superstack. We then show that the superstack of supersymmetric curves admits a coarse complex superspace, which, in this case, is just an ordinary complex space. In the second part of this paper we discuss the period map. We remark that the period domain is the moduli space of ordinary abelian varieties endowed with a symmetric theta divisor, and we then show that the differential of the period map is surjective. In other words, we prove that any first order deformation of a classical Jacobian is the Jacobian of a supersymmetric curve.Comment: Minor revision, to appear on Advances in Theoretical and Mathematical Physic

Families of n-gonal curves with maximal variation of moduli

We study families of n-gonal curves with maximal variation of moduli, which have a rational section. Certain numerical results on the degree of the modular map are obtained for such families of hyperelliptic and trigonal curves. In the last case we use the description of the relative Picard group of the universal family of trigonal curves.Comment: 16 pages. Some modifications on the third section. References adde

The Chow ring of the stack of cyclic covers of the projective line

In this paper we compute the integral Chow ring of the stack of smooth uniform cyclic covers of the projective line and we give explicit generators.Comment: 20 pages; final version, to appear in Ann. Inst. Fourie