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.

No abstract

The locus of genus-two curves with n marked Weierstrass points has codimension n inside the moduli space of genus-two curves with n marked points, for n ≤ 6. It is well known that the class of the closure of the divisor obtained for n = 1 spans an extremal ray of the cone of effective divisor classes. We generalize this result for all n: we show that the class of the closure of the locus of genustwo curves with n marked Weierstrass points spans an extremal ray of the cone of effective classes of codimension n, for n ≤ 6. A related construction produces extremal nef curve classes in moduli spaces of pointed elliptic curves.Every smooth curve of genus two has a unique map of degree two to the projective line, ramified at six points, called Weierstrass points. It follows that the locus Hyp 2,n of curves of genus two with n marked Weierstrass points has codimension n inside the moduli space M 2,n of smooth curves of genus two with n marked points, for 1 ≤ n ≤ 6. In this paper, we study the classes of the closures of the loci Hyp 2,n inside the moduli space of stable curves M 2,n .The cone of effective codimension-one classes on M 2,1 is explicitly described in [Rul01] and [Rul06], and encodes the rational contractions of M 2,1 . It is thus natural to study cones of effective classes of higher codimension. The following is one of the first results in this direction.Theorem 1. For 1 ≤ n ≤ 6, the class of Hyp 2,n is rigid and extremal in the cone of effective classes of codimension n in M 2,n .

We compute the Euler characteristic of the structure sheaf of the Brill–Noether locus of linear series with special vanishing at up to two marked points. When the Brill–Noether number $\rho $ is zero, we recover the Castelnuovo formula for the number of special linear series on a general curve; when $\rho =1$, we recover the formulas of Eisenbud-Harris, Pirola, and Chan–Martín–Pflueger–Teixidor for the arithmetic genus of a Brill–Noether curve of special divisors. These computations are obtained as applications of a new determinantal formula for the $K$-theory class of certain degeneracy loci. Our degeneracy locus formula also specializes to new determinantal expressions for the double Grothendieck polynomials corresponding to 321-avoiding permutations and gives double versions of the flagged skew Grothendieck polynomials recently introduced by Matsumura. Our result extends the formula of Billey–Jockusch–Stanley expressing Schubert polynomials for 321-avoiding permutations as generating functions for flagged skew tableaux.