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Handbook of Moduli Alternate Compactifications of Moduli Spaces of Curves Maksym Fedorchuk and David Ishii Smyth Abstract. We give an informal survey, emphasizing examples and open problems, of two interconnected research programs in moduli of curves: the systematic classification of modular compactifications of Mg;n, and the study of Mori chamber decompositions of M g;n. Contents 1 Introduction1 1.1 Notation4 2 Compactifying the moduli space of curves6 2.1 Modular and weakly modular birational models of Mg;n 6 2.2 Modular birational models via MMP 10 2.3 Modular birational models via combinatorics 16 2.4 Modular birational models via GIT 25 2.5 Looking forward 39 3 Birational geometry of moduli spaces of curves 40 3.1 Birational geometry in a nutshell 40 3.2 Effective and nef cone of M g;n 43 4 Log minimal model program for moduli spaces of curves 56 4.1 Log minimal model program for M g 57 4.2 Log minimal model program for M 0;n 67 4.3 Log minimal model program for M 1;n 70 4.4 Heuristics and predictions 71 arXiv:1012.0329v2 [math.AG] 9 Jun 2011 1. Introduction The purpose of this article is to survey recent developments in two inter- connected research programs within the study of moduli of curves: the systematic construction and classification of modular compactifications of Mg;n, and the study of the Mori theory of M g;n. In this section, we will informally describe the overall goal of each of these programs as well as outline the contents of this article. 2000 Mathematics Subject Classification. Primary 14H10; Secondary 14E30. Key words and phrases. moduli of curves, singularities, minimal model program. 2 Moduli of Curves One of the most fundamental and influential theorems in modern algebraic geometry is the construction of a modular compactification M M for the g ⊂ g moduli space of smooth complete curves of genus g [DM69]. In this context, the word modular indicates that the compactification M g is itself a parameter space for a suitable class of geometric objects, namely stable curves of genus g. Definition 1.1 (Stable curve). A complete curve C is stable if (1) C has only nodes (y2 = x2) as singularities. (2) !C is ample. The class of stable curves satisfies two essential properties, namely deforma- tion openness (Definition 2.2) and the unique limit property (Definition 2.3). As we shall see in Theorem 2.4, any class of curves satisfying these two properties gives rise to a modular compactification of Mg. While the class of stable curves gives a natural modular compactification of the space of smooth curves, it is not unique in this respect. The simplest example of an alternate class of curves satisfying these two properties is the class of pseudostable curves, introduced by Schubert [Sch91]. Definition 1.2 (Pseudostable curve). A curve C is pseudostable if (1) C has only nodes (y2 = x2) and cusps (y2 = x3) as singularities. (2) !C ample. (3) If E C is any connected subcurve of arithmetic genus one, ⊂ then E C E 2: j \ n j ≥ Notice that the definition of pseudostability involves a trade-off: cusps are exchanged for elliptic tails. It is easy to see how this trade-off comes about: As one ranges over all one-parameter smoothings of a cuspidal curve C, the associated limits in M are precisely curves of the form C~ E, where C~ is the normalization g [ of C at the cusp and E is an elliptic curve (of arbitrary j-invariant) attached to C~ at the point lying above the cusp (see Example 2.16). Thus, any separated moduli problem must exclude either elliptic tails or cusps. Schubert's construction naturally raises the question: Problem 1.3. Classify all possible stability conditions for curves, i.e. classes of possibly singular curves which are deformation open and satisfy the property that any one-parameter family of smooth curves contains a unique limit contained in that class. Of course, we can ask the same question with respect to classes of marked curves, i.e. modular compactifications of Mg;n, and the first goal of this article is to survey the various methods used to construct alternate modular compactifications of Mg;n. In Section2, we describe constructions using Mori theory, the combina- torics of curves on surfaces, and geometric invariant theory, and, in Section 2.5, we Maksym Fedorchuk and David Ishii Smyth 3 assemble a (nearly) comprehensive list of the alternate modular compactifications that have so far appeared in the literature. Since each stability condition for n-pointed curves of genus g gives rise to a proper birational model of M g;n, it is natural to wonder whether these models can be exploited to study the divisor theory of M g;n. This brings us to the second subject of this survey, namely the study of the birational geometry of M g;n from the perspective of Mori theory. In order to explain the overall goal of this program, let us recall some definitions from birational geometry. For any normal, Q-factorial, projective variety X, let Amp(X), Nef(X), and Eff(X) denote the cones of ample, nef, and effective divisors respectively (these are defined in Section 3.1). Recall that a birational contraction φ: X 99K Y is simply a birational map 1 to a normal, projective, Q-factorial variety such that codim(Exc(φ− )) 2. If φ ≥ is any birational contraction with exceptional divisors E1;:::;Ek, then the Mori chamber of φ is defined as: X Mor(φ) := φ∗ Amp(Y ) + a E Eff(X): i i ⊂ a 0 i≥ The birational contraction φ can be recovered from any divisor D in Mor(φ) as the L 0 contraction associated to the section ring of D, i.e. X 99K Proj H (X; mD). m 0 In good cases, e.g. when X is a toric or Fano variety, these Mori≥ chambers are polytopes and there exist finitely many contractions φi which partition the entire effective cone, i.e. one has Eff(X) = Mor(φ ) ::: Mor(φ ): 1 [ [ k Understanding the Mori chamber decomposition of the effective cone of a variety X is therefore tantamount to understanding the classification of birational con- tractions of X. This collection of ideas, which will be amplified in Section 3.1, suggests the following problems concerning M g;n. Problem 1.4. (1) Describe the nef and effective cone of M g;n. (2) Describe the Mori chamber decomposition of Eff(M g;n). These questions merit study for two reasons: First, the attempt to prove results concerning the geometry of M g;n has often enriched our understanding of algebraic curves in fundamental ways. Second, even partial results obtained along these lines may provide an interesting collection of examples of phenomena in higher-dimensional geometry. In Section 3.2, we will briefly survey the present state of knowledge concerning the nef and effective cones of M g;n, and many examples of partial Mori chamber decompositions will be given in Section4. We should emphasize that Problem 1.4 (1) has generated a large body of work over the past 30 years. While we hope that Section 3.2 can serve as a reasonable guide to the literature, we will not cover these results in the detail they merit. The interested 4 Moduli of Curves reader who wishes to push further in this direction is strongly encouraged to read the recent surveys of Farkas [Far09a, Far10] and Morrison [Mor07]. A major focus of the present survey is the program initiated by Hassett and Keel to study certain log canonical models of M g and M g;n. For any rational number α such that K + αδ is effective, they define Mg;n M 0 M g;n(α) = Proj H g;n; m(K + αδ) ; M Mg;n m 0 ≥ where the sum is taken over m sufficiently divisible, and ask whether the spaces M g;n(α) admit a modular description. When g and n are sufficiently large so that M g;n is of general type, M g;n(1) is simply M g;n, while M g;n(0) is the canonical model of M g;n. Thus, the series of birational transformations accomplished as α decreases from 1 to 0 constitute a complete minimal model program for M g;n. In terms of Problem 1.4 above, this program can be understood as asking: (1) Describe the Mori chamber decomposition within the two dimensional sub- space of Eff(M ) spanned by K and δ (where it is known to exist g;n g;n by theorems of higher-dimensionalM geometry [BCHM06]). (2) Find modular interpretations for the resulting birational contractions. Though a complete description of the log minimal model program for M (g 0) g is still out of reach, the first two steps have been carried out by Hassett and Hyeon [HH09, HH08]. Furthermore, the program has been completely carried out for M 2, M 3, M 0;n, and M 1;n [Has05, HL10, FS11, AS08, Smy11, Smy10], and we will describe the resulting chamber decompositions in Section4. Finally, in Section 4.4 we will present some informal heuristics for making predictions about future stages of the program. In closing, let us note that there are many fascinating topics in the study of birational geometry of M g;n and related spaces which receive little or no attention whatsoever in the present survey. These include rationality, unirationality and rational connectedness of M g;n [SB87, SB89, Kat91, Kat96, MM83, Ver05, BV05], complete subvarieties of Mg;n [Dia84, FvdG04, GK08], birational geometry of Prym varieties [Don84, Cat83, Ver84, Ver08, Kat94, Dol08, ILGS09, FL10], Ko- daira dimension of M for n 1 [Log03, Far09b] and of the universal Jaco- g;n ≥ bian over M g [FV10], moduli spaces birational to complex ball quotients [Kon00, Kon02], etc.
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