This paper gives a foundation of log smooth deformation theory. We study the infinitesimal liftings of log smooth morphisms and show that the log smooth deformation functor has a representable hull. This deformation theory gives, for example, the following two types of deformations: (1) relative deformations of a certain kind of a pair of an algebraic variety and a divisor of it, and (2) global smoothings of normal crossing varieties. The former is a generalization of the relative deformation theory introduced by Makio, and the latter coincides with the logarithmic deformation theory introduced by Kawamata and Namikawa.

The aim of this paper is to define a reasonable moduli theory of log smooth curves which recovers the classical Deligne-Knudsen-Mumford moduli of pointed stable curves. IntroductionModuli theory in the framework of log geometry aims to construct good and natural compactifications of moduli of algebraic varieties. The motivating philosophy is that, since log smoothness includes some degenerating objects like semistable reductions, etc. the moduli space of log smooth objects should be already compactified, once its existence has been established. In this paper we will develop the theory of moduli of log smooth curves, which is expected to give a compactification of the classical moduli of algebraic curves. This theory should be a starting point and give a prototype of the future study of log moduli theory.Let us give a brief summary: in the next section we will define so-called log curves, which seems the most natural counterpart of smooth curves. Then, in Theorem 1.1, we will see that any geometric fiber of a log curve is a semistable curve, i.e. a curve with at most ordinary double points as singularities. Moreover, the log structures on log curves have more information similar to pointed structures. Corresponding to the n-pointed stable curves of genus g, we will have a natural concept of log stable curves of type (g, n) (cf. Definition 1.2). In log geometry, the natural way to set up the moduli problem is considering stacks in groupoids over the category of (fs) log schemes. We will see that the category LM g,n of log stable curves of type (g, n) is a stack in groupoids over the category of fs log schemes LSch fs with the strictétale topology.In general, for any stack S → Sch in groupoids over the category of schemes, we will have a reasonable definition of (fs) log structures on the stack S which generalizes the usual definition of (fs) log structures on schemes (cf. Definition 3.1). A stack over Sch with a log structure in this sense is called a log stack. For a stack

No abstract

No abstract

Abstract. We compute the dimension of the tangent space to, and the Krull dimension of the pro-representable hull of two deformation functors. The first one is the "algebraic" deformation functor of an ordinary curve X over a field of positive characteristic with prescribed action of a finite group G, and the data are computed in terms of the ramification behaviour of X → G\X. The second one is the "analytic" deformation functor of a fixed embedding of a finitely generated discrete group N in P GL(2, K) over a non-archimedean valued field K, and the data are computed in terms of the Bass-Serre representation of N via a graph of groups. Finally, if Γ is a free subgroup of N such that N is contained in the normalizer of Γ in P GL(2, K), then the Mumford curve associated to Γ becomes equipped with an action of N/Γ, and we show that the algebraic functor deforming the latter action coincides with the analytic functor deforming the embedding of N .Introduction.