We compute the Mori cones of the moduli spaces M g,n of n pointed stable curves of genus g, when g and n are relatively small. For instance we show that for g < 14 every curve in M g is equivalent to an effective combination of the components of the locus of curves with 3g − 4 nodes. We completely describe the cone of nef divisors for the space M 0,6 , thus verifying Fulton's conjecture for this space. Using this description we obtain a classification of all the fibrations of M 0,6 .

Abstract. In this paper, we initiate our investigation of log canonical models for (M g , αδ) as we decrease α from 1 to 0. We prove that for the first critical value α = 9/11, the log canonical model is isomorphic to the moduli space of pseudostable curves, which have nodes and cusps as singularities. We also show that α = 7/10 is the next critical value, i.e., the log canonical model stays the same in the interval (7/10, 9/11]. In the appendix, we develop a theory of log canonical models of stacks that explains how these can be expressed in terms of the coarse moduli space.

A natural question is to determine which algebraic stacks are qoutient stacks. In this paper we give some partial answers and relate it to the old question of whether, for a scheme X, the natural map from the Brauer goup (equivalence classes of Azumaya algebras) to the cohomological Brauer group (the torsion subgroup of H 2 (X, G m ) is surjective.

We study rationality properties of quadric surface bundles over the projective plane. We exhibit families of smooth projective complex fourfolds of this type over connected bases, containing both rational and non-rational fibers. GeneralitiesWe recall implications of the "integral decomposition of the diagonal and specialization" method, following [CTP14], [Voi15b], and [Pir16].A projective variety X over a field k is universally CH 0 -trivial if for all field extensions k ′ /k the natural degree homomorphism from the Chow group of zero-cycles CH 0 (X k ′ ) → Z is an isomorphism. Examples include smooth k-rational varieties. More complicated examples arise as follows: