To Bill Fulton on his sixtieth birthday §0. Introduction and statement of resultsThe moduli space of stable curves is among the most studied objects in algebraic geometry. Nonetheless, its birational geometry remains largely a mystery and most Mori theoretic problems in the area are entirely open. Here we consider one of the most basic:There is a stratification of M g,n by topological type where the codimension k strata are the irreducible components of the locus parameterizing pointed curves with at least k singular points. An ample divisor must intersect any one dimensional stratum positively. It is thus natural to consider the following conjecture. (M g,n ) is generated by a one dimensional stratum. F 1 (M 0,n ) was previously conjectured by Fulton, whence our choice of notation. Special cases of F 1 (M 0,n ) are known (see [KeelMcKernan96], §7) but the general case remains open. The central observation of this work is that the case of higher genus is no harder. Indeed, N E 1 (M g,n ) is naturally a quotient of N E 1 (M 0,2g+n ) × R ≥0 and Conjecture (0.2) is true for all g iff it is true for g = 0; see (0.3)-(0.5). Further, the second formulation of Conjecture (0.2) holds except possibly for very degenerate families in M g,n ; see (0.6). Moreover, we are able to give strong results on the contractions of M g,n (i.e. morphisms with geometrically connected fibers from it to other projective varieties). For example we show that for g ≥ 2 the only fibrations of M g,n are compositions of a tautological fibration, given by dropping
Abstract. We introduce a smooth projective variety T d,n which compactifies the space of configurations of n distinct points on affine d-space modulo translation and homothety. The points in the boundary correspond to n-pointed stable rooted trees of ddimensional projective spaces, which for d = 1, are (n + 1)-pointed stable rational curves. In particular, T 1,n is isomorphic to M 0,n+1 , the moduli space of such curves. The variety T d,n shares many properties with M 0,n . For example, as we prove, the boundary is a smooth normal crossings divisor whose components are products of T d,i for i < n, it has an inductive construction analogous to but differing from Keel's for M 0,n which can be used to describe its Chow groups, Chow motive and Poincaré polynomials, generalizing [Kee92,Man95]. We give a presentation of the Chow rings of T d,n , exhibit explicit dual bases for the dimension 1 and codimension 1 cycles. The variety T d,n is embedded in the Fulton-MacPherson spaces X[n] for any smooth variety X and we use this connection in a number of ways. For example, to give a family of ample divisors on T d,n and to give an inductive presentation of the Chow groups and the Chow motive of X[n] analogous to Keel's presentation for M 0,n , solving a problem posed by Fulton and MacPherson.
Modules over conformal vertex algebras give rise to sheaves of coinvariants and conformal blocks on moduli of stable pointed curves.Here we prove the factorization conjecture for these sheaves. Our results apply in arbitrary genus and for a large class of vertex algebras. As an application, sheaves defined by finitely generated admissible modules over vertex algebras satisfying natural hypotheses are shown to be vector bundles. Factorization is essential to a recursive formulation of invariants, like ranks and Chern classes, and to produce new constructions of rational conformal field theories and cohomological field theories. MSC2010. 14H10, 17B69 (primary), 81R10, 81T40, 14D21 (secondary). Key words and phrases. Vertex algebras, conformal blocks and coinvariants, vector bundles on moduli of curves, factorization and sewing.2.1. The group scheme Aut O. Consider the functor which assigns to a C-algebra R the Lie group:of continuous automorphisms of the algebra R z preserving the ideal zR z . The group law is given by composition of series: ρ 1 •ρ 2 := ρ 2 •ρ 1 . This functor is represented by a group scheme, denoted Aut O.
We give explicit equations for the Chow and Hilbert quotients of a projective scheme X by the action of an algebraic torus T in an auxiliary toric variety. As a consequence we provide geometric invariant theory descriptions of these canonical quotients, and obtain other GIT quotients of X by variation of GIT quotient. We apply these results to find equations for the moduli space M 0,n of stable genus-zero n-pointed curves as a subvariety of a smooth toric variety defined via tropical methods.
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