For a local field K with positive residue characteristic p, we introduce, in the first part of this paper, a refinement bAr K of the classical Artin distribution Ar K .It takes values in cyclotomic extensions of Q which are unramified at p, and it bisects Ar K in the sense that Ar K is equal to the sum of bAr K and its conjugate distribution.Compared with 1 2 Ar K , the bisection bAr K provides a higher resolution on the level of tame ramification.In the second part of this article, we prove that the base change conductor c(T ) of an analytic K-torus T is equal to the value of bAr K on the Q p -rational Galois representation X * (T ) ⊗ Zp Q p that is given by the character module X * (T ) of T .We hereby generalize a formula for the base change conductor of an algebraic K-torus, and we obtain a formula for the base change conductor of a semiabelian K-variety with potentially ordinary reduction.

We associate a weight function to pairs consisting of a smooth and proper variety X over a complete discretely valued field and a differential form on X of maximal degree. This weight function is a real-valued function on the non-archimedean analytification of X. It is piecewise affine on the skeleton of any regular model with strict normal crossings of X, and strictly ascending as one moves away from the skeleton. We apply these properties to the study of the Kontsevich-Soibelman skeleton of such a pair, and we prove that this skeleton is connected when X has geometric genus one. This result can be viewed as an analog of the Shokurov-Kollar connectedness theorem in birational geometry.

We determine the Chow ring (with Q-coecients) of M6 by showing that all Chow classes are tautological. (In particular, all algebraic cohomology is tautological, and the natural map from Chow to cohomology is injective.) We stratify the moduli space into locally closed strata which are group quotients, and use a theorem of Vistoli to show that their Chow rings are generated by Chern classes of natural vector bundles. To demonstrate the utility of these methods, we also give quick derivations of the Chow groups of moduli spaces of curves of lower genus. The genus six case relies in addition on the particularly beautiful Brill{Noether theory in this case, and in particular on a rank ve vector bundle \relativizing a baby case of a celebrated construction of Mukai, which we interpret as a subbundle of the rank six vector bundle of quadrics cutting out the canonical curve.

Let K be the field of fractions of a henselian valuation ring A. Assume that the completion K is a separable extension of K. Let Y be a K-variety, let G be an algebraic group over K, and let f : X Y be a G-torsor over Y . We consider the induced map X(K) Y (K), which is continuous for the topologies deduced from the valuation. If I denotes the image of this map, we prove that I is locally closed in Y (K); moreover, the induced surjection X(K) I is a principal bundle with group G(K) (also topologized by the valuation).

We introduce the notion of Gepner type Bridgeland stability conditions on triangulated categories, which depends on a choice of an autoequivalence and a complex number.We conjecture the existence of Gepner type stability conditions on the triangulated categories of graded matrix factorizations of weighted homogeneous polynomials.Such a stability condition may give a natural stability condition for Landau-Ginzburg Bbranes, and correspond to the Gepner point of the stringy Kähler moduli space of a quintic threefold.The main result is to show our conjecture when the variety defined by the weighted homogeneous polynomial is a complete intersection of hyperplanes in a Calabi-Yau manifold with dimension less than or equal to two.

At every point on the rational normal curve, there is an osculating flag. Consider Schubert problems with respect to r such flags, corresponding to r distinct points of the curve. It is natural to consider the r points as parametrized by the moduli space of r distinct points in P 1 -the famous M 0,r . One can then ask to extend the results on Schubert intersections to the compactification M 0,r .