Log Smooth Deformation and Moduli of Log Smooth Curves

Abstract

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

Stable logarithmic maps to Deligne--Faltings pairs I

“…Thus, we define a functor from X to LogSch S , by pulling back the log structure M X . The stack X associated with this functor is called a log stack in [Kat00]. A fine log scheme (X, M X ) can be naturally viewed as a log algebraic stack.…”

“…We will prove that the stack M pre g,n parameterizing log pre-stable curves of genus g and n marked points is an open substack of some Olsson's log stack as above, hence is algebraic in the sense of [Art74, 5.1]. We refer to [Kat00], [Moc95], and [Ols07] for more details of log structures on curves.…”

Toric degenerations of toric varieties and tropical curves

“…On this part toric transversality implies that ϕ * 0 M X 0 ,P → O C 0 ,P has precisely one smooth extension, namely the sum of (π • ϕ 0 ) * M O 0 and the log structure associated to the toric divisor X h(E) ⊂ X h(∂E) , see Proposition 1.1 in [KaF2].…”

“…From [KaF2] it is clear that C ∞ → O ∞ is log smooth. By construction the restriction ϕ 0 : C 0 → X 0 to the central fiber also fulfills the other properties listed in Proposition 7.1.…”

Untitled

“…where (M g,n , ᏹ M g,n ) denotes the log stack of (g, n)-prestable curves; see [Kato 2000] and [Olsson 2007, Theorem 1.10] for the definition and construction of this log stack. Since (M g,n , ᏹ M g,n ) is log smooth over (k, ᏻ * k ), it suffices to show that π is log smooth.…”