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

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

On the Picard group scheme of the moduli stack of stable pointed curves

The aim of the present paper is to study the (abstract) Picard group and the Picard group scheme of the moduli stack of stable pointed curves over an arbitrary scheme. As a byproduct, we compute the Picard groups of the moduli stack of stable or smooth pointed curves over a field of characteristic different from two.Comment: 36 pages. v2: added a new section on the first Chern class and the divisor class group of the coarse moduli spac

Picard group of moduli of hyperelliptic curves

The main subject of this work is the difference between the coarse moduli space and the stack of hyperelliptic curves. We compute their Picard groups, giving explicit description of the generators. We get an application to the (non-)existence of a tautological family over the coarse moduli space.Comment: 13 pages. Section 2 has been shortened and the final appendix has been erased. Final version, to appear on Math. Zei