Let X be a smooth projective curve of genus $g > 1$ over an algebraically closed field of positive characteristic. This paper is a study of a natural stratification, defined by the absolute Frobenius morphism of X, on the moduli space of vector bundles. In characteristic two, there is a complete classification of semi-stable bundles of rank 2 which are destabilized by Frobenius pull-back. We also show that these strata are irreducible and obtain their respective dimensions. In particular, the dimension of the locus of bundles of rank two which are destabilized by Frobenius is $3g-4$. These Frobenius destabilized bundles also exist in characteristics two, three and five with ranks 4, 3 and 5, respectively. Finally, there is a connection between (pre)-opers and Frobenius destabilized bundles. This allows an interpretation of some of the above results in terms of pre-opers and provides a mechanism for constructing Frobenius destabilized bundles in large characteristics.

We present an analogy of the famous formula that the square of the Pfaffian is equal to the determinant for an alternating matrix for the case where the entries are the generators of the orthogonal Lie algebras. This identity clarifies the relation between the two sets of central elements in the enveloping algebra of the orthogonal Lie algebras. We employ systematically the exterior calculus for the proof.

We study residues on a complete toric variety $X$, which are defined in terms of the homogeneous coordinate ring of $X$. We first prove a global transformation law for toric residues. When the fan of the toric variety has a simplicial cone of maximal dimension, we can produce an element with toric residue equal to 1. We also show that in certain situations, the toric residue is an isomorphism on an appropriate graded piece of the quotient ring. When $X$ is simplicial, we prove that the toric residue is a sum of local residues. In the case of equal degrees, we also show how to represent $X$ as a quotient $(Y\setminus\{0\})/C\ast$ such that the toric residue becomes the local residue at 0 in $Y$.

We deal with a divisorial contraction in dimension three which contracts its exceptional divisor to a compound A1 point. We prove that any such contraction is obtained by a suitable weighted blow-up.

We compute coherent presentations of Artin monoids, that is, presentations by generators, relations, and relations between the relations. For that, we use methods of higher-dimensional rewriting that extend Squier’s and Knuth–Bendix’s completions into a homotopical completion–reduction, applied to Artin’s and Garside’s presentations. The main result of the paper states that the so-called Tits–Zamolodchikov 3-cells extend Artin’s presentation into a coherent presentation. As a byproduct, we give a new constructive proof of a theorem of Deligne on the actions of an Artin monoid on a category.

Given a finite set A of integral vectors and a parameter vector, Gel'fand, Kapranov, and Zelevinskii defined a system of differential equations, called an A-hypergeometric (or a GKZ hypergeometric) system. Classifying the parameters according to the D-isomorphism classes of their corresponding A-hypergeometric systems is one of the most fundamental problems in the theory. In this paper we give a combinatorial answer for the problem under the assumption that the finite set A lies in a hyperplane off the origin, and illustrate it in two particularly simple cases: the normal case and the monomial curve case.

We prove the unirationality of the moduli space of complex curves of genus 14. The method essentially relies on linkage of curves. In particular it is shown that a general curve of genus 14 admits a projective model D, in a six-dimensional projective space, which is linked to a general curve C of degree 14 and genus 8 by a complete intersection of quadrics. Using this property we are able to obtain, after a reasonable amount of further work, the unirationality result in the case of genus 14. Moreover, some variations of the same method, involving the Hilbert schemes of curves of very low genus, are used to obtain the same result for the known cases of genus 11, 12, 13.

Let V be a vector space with a non-degenerate symmetric form and OG be the orthogonal Grassmannian which parametrizes maximal isotropic subspaces in V. We give a presentation for the (small) quantum cohomology ring QH*(OG) and show that its product structure is determined by the ring of $\widetilde{P}$-polynomials. A ‘quantum Schubert calculus’ is formulated, which includes quantum Pieri and Giambelli formulas, as well as algorithms for computing Gromov--Witten invariants. As an application, we show that the table of three-point, genus zero Gromov–Witten invariants for OG coincides with that for a corresponding Lagrangian Grassmannian LG, up to an involution.

We study automorphisms of a generic Jacobian Kummer surface. First we analyse the action of classically known automorphisms on the Picard lattice of the surface, then proceed to construct new automorphisms not generated by classical ones. We find 192 such automorphisms, all conjugate by the symmetry group of the (16,6)-configuration.

In this paper, we prove an analogue of the result known as Mazur‘s Principle concerning optimal levels of mod [ell ] Galois representations. The paper is divided into two parts. We begin with the study (following Katz–Mazur) of the integral model for certain Shimura curves and the structure of the special fibre. It is this study which allows us to generalise, in the second part of this paper, Mazur‘s result to totally real fields of odd degree.

In this paper, we prove that the log minimal model program in dimension d−1 implies the existence of log minimal models for effective lc pairs (e.g. of non-negative Kodaira dimension) in dimension d. In fact, we prove that the same conclusion follows from a weaker assumption, namely, the log minimal model program with scaling in dimension d−1. This enables us to prove that effective lc pairs in dimension five have log minimal models. We also give new proofs of the existence of log minimal models for effective lc pairs in dimension four and of the Shokurov reduction theorem.

The root discriminant of a number field of degree n is the nth root of the absolute value of its discriminant. Let R2m be the minimal root discriminant for totally complex number fields of degree 2m, and put α0 = lim infm R2m. One knows that α0 [ges ] 4πeγ ≈ 22.3, and, assuming the Generalized Riemann Hypothesis, α0 [ges ] 8πeγ ≈ 44.7. It is of great interest to know if the latter bound is sharp. In 1978, Martinet constructed an infinite unramified tower of totally complex number fields with small constant root discriminant, demonstrating that α0 < 92.4. For over twenty years, this estimate has not been improved. We introduce two new ideas for bounding asymptotically minimal root discriminants, namely, (1) we allow tame ramification in the tower, and (2) we allow the fields at the bottom of the tower to have large Galois closure. These new ideas allow us to obtain the better estimate α0 < 83.9.

Multivariate hypergeometric functions associated with toric varieties were introduced by Gel'fand, Kapranov and Zelevinsky. Singularities of such functions are discriminants, that is, divisors projectively dual to torus orbit closures. We show that most of these potential denominators never appear in rational hypergeometric functions. We conjecture that the denominator of any rational hypergeometric function is a product of resultants, that is, a product of special discriminants arising from Cayley configurations. This conjecture is proved for toric hypersurfaces and for toric varieties of dimension at most three. Toric residues are applied to show that every toric resultant appears in the denominator of some rational hypergeometric function.

We give the first examples over finite fields of rings of invariants that are not finitely generated. (The examples work over arbitrary fields, for example the rational numbers.) The group involved can be as small as three copies of the additive group. The failure of finite generation comes from certain elliptic fibrations or abelian surface fibrations having positive Mordell–Weil rank. Our work suggests a generalization of the Morrison–Kawamata cone conjecture on Calabi–Yau fiber spaces to klt Calabi–Yau pairs. We prove the conjecture in dimension two under the assumption that the anticanonical bundle is semi-ample.

Let k be a perfect field of characteristic p > 0, K$_0$ = Frac(W(k)), π a uniformizer in K$_0$ and π$_n$ ∈ K $_0$ (n∈ N) such that π$_0$ = π and π$^p$$_n$ $_ + 1$ = π$_n$. We write K$_∞$ = ∪$_n$ $_∈N$ K$_0$ (π$_n$), H$_∞$ = Gal (K$_0$/ K$_∞$ and G = Gal(K$_0$/ K$_0$). The main result of this paper is that the functor ’restriction of the Galois action‘ from the category of crystalline representations of G with Hodge–Tate weights in an interval of length [les ] p − 2 to the category of p-adic representations of H$_∞$ is fully faithful and its essential image is stable by sub-object and quotient. The proof uses the comparison between two ways of building mod. p representations of H$_∞$: one thanks to the norm field of K$_∞$, the other thanks to some categories of ’filtered‘ modules with divided powers previously introduced by the author.

We introduce a new method for computing triply graded link homology, which is particularly well adapted to torus links. Our main application is to the $(n,n)$-torus links, for which we give an exact answer for all $n$. In several cases, our computations verify conjectures of Gorsky et al. relating homology of torus links with Hilbert schemes.

We prove the finiteness of log pluricanonical representations for projective log canonical pairs with semi-ample log canonical divisor. As a corollary, we obtain that the log canonical divisor of a projective semi log canonical pair is semi-ample if and only if the log canonical divisor of its normalization is semi-ample. We also treat many other applications.

We prove the Bertram–Feinberg–Mukai conjecture for a generic curve C of genus g and a semistable vector bundle E of rank two and determinant K on C, namely we prove the injectivity of the Petri-canonical map S2(H0(E))→H0(S2(E)).

We determine the set of connected components of minuscule affine Deligne–Lusztig varieties for hyperspecial maximal compact subgroups of unramified connected reductive groups. Partial results are also obtained for non-minuscule closed affine Deligne–Lusztig varieties. We consider both the function field case and its analog in mixed characteristic. In particular, we determine the set of connected components of unramified Rapoport–Zink spaces.

We show how some of the refined tropical counts of Block and Göttsche emerge from the wall-crossing formalism. This leads naturally to a definition of a class of putative $q$-deformed Gromov–Witten invariants. We prove that this coincides with another natural $q$-deformation, provided by a result of Reineke and Weist in the context of quiver representations, when the latter is well defined.