On the number of Galois points for a plane curve in positive characteristic,II

For a smooth plane curve , we call a point a Galois point if the point projection at P is a Galois covering. We study Galois points in positive characteristic. We give a complete classification of the Galois group given by a Galois point and estimate the number of Galois points for C in most cases.      

Computing invariants of algebraic groups in arbitrary characteristic

Let G be an affine algebraic group acting on an affine variety X. We present an algorithm for computing generators of the invariant ring KG[X] in the case where G is reductive. Furthermore, we address the case where G is connected and unipotent, so the invariant ring need not be finitely generated. For this case, we develop an algorithm which computes KG[X] in terms of a so-called colon-operation. From this, generators of KG[X] can be obtained in finite time if it is finitely generated. Under the additional hypothesis that K[X] is factorial, we present an algorithm that finds a quasi-affine variety whose coordinate ring is KG[X]. Along the way, we develop some techniques for dealing with nonfinitely generated algebras. In particular, we introduce the finite generation ideal.     

On the plane section of an integral curve in positive characteristic

If is an integral curve and an algebraically closed field of characteristic 0, it is known that the points of the general plane section of are in uniform position. From this it follows easily that the general minimal curve containing is irreducible. If char, the points of may not be in uniform position. However, we prove that the general minimal curve containing is still irreducible.

     

Cohomology of classical algebraic groups from the functorial viewpoint

We prove that extension groups in strict polynomial functor categories compute the rational cohomology of classical algebraic groups. This result was previously known only for general linear groups. We give several applications to the study of classical algebraic groups, such as a cohomological stabilization property, the injectivity of external cup products, and the existence of Hopf algebra structures on the (stable) cohomology of a classical algebraic group with coefficients in a Hopf algebra. Our result also opens the way to new explicit cohomology computations. We give an example inspired by recent computations of Djament and Vespa.     

Complete reducibility of subgroups of reductive algebraic groups over nonperfect fields II

Let k be a separably closed field. Let G be a reductive algebraic k-group. We study Serre’s notion of complete reducibility of subgroups of G over k. In particular, using the recently proved center conjecture of Tits, we show that the centralizer of a k-subgroup H of G is G-completely reducible over k if it is reductive and H is G-completely reducible over k. We show that a regular reductive k-subgroup of G is G-completely reducible over k. We present examples where the number of overgroups of irreducible subgroups and the number of G(k)-conjugacy classes of k-anisotropic unipotent elements are infinite.     

Even points on an algebraic curve

We show that the points of a global function field, whose classes are 2-divisible in the Picard group, form a connected regular infinite graph, with the incidence relation generalizing the well known quadratic reciprocity law. We prove that for every global function field the diameter of this graph is precisely 2. In addition we develop an analog of global square theorem that concerns points 2-divisible in the Picard group.     

Generic positivity and foliations in positive characteristic

We prove generic semipositivity of the tangent bundle of a non-uniruled Calabi–Yau variety in positive characteristic. We also construct an example of a nef line bundle in characteristic zero, whose each reduction to positive characteristic is not nef.     

Remark on convergence of algebraic multigrid in the form of matrix decomposition

We introduce the convergence of algebraic multigrid in the form of matrix decomposition. The convergence is proved in block versions of the multi-elimination incomplete LU (BILUM) factorization technique and the approximation of their inverses to preserve sparsity. The convergence theorem can be applied to general interpolation operator. Furthermore, we discuss the error caused by the error matrix.     

On abstract representations of the groups of rational points of algebraic groups in positive characteristic

We analyze the structure of a large class of connected algebraic rings over an algebraically closed field of positive characteristic using Greenberg’s perfectization functor. We then give applications to rigidity problems for representations of Chevalley groups.     

k-Involutions of SL(n,k) over fields of characteristic 2

Symmetric k-varieties generalize Riemannian symmetric spaces to reductive groups defined over arbitrary fields. For most perfect fields, it is known that symmetric k-varieties are in one-to-one correspondence with isomorphy classes of k-involutions. Therefore, it is useful to have representatives of each isomorphy class in order to describe the k-varieties. Here we give matrix representatives for each isomorphy class of k-involutions of SL(n,k) in the case that k is any field of characteristic 2; we also describe fixed-point groups of each type of involution.     

The maximum element order in the groups related to the linear groups which is a multiple of the defining characteristic

Let n be a natural number and q be the power of a prime p. The general, special and projective special linear groups are denoted by GL_n(q), SL_n(q) and PSL_n(q), respectively. In this paper we find the maximum order of an element of the above groups which is a multiple of p.     

On disjunctions of algebraic sets in completely simple semigroups

A semigroup S is called an equational domain if any finite union of algebraic sets over S is algebraic. We give some necessary and su?cient conditions for a completely simple semigroup to be an equational domain.     

Complete reducibility of subgroups of reductive algebraic groups over nonperfect fields III

Let G be a reductive group over a nonperfect field k. We study rationality problems for Serre’s notion of complete reducibility of subgroups of G. In our previous work, we constructed examples of subgroups H of G that are G-completely reducible but not G-completely reducible over k (and vice versa). In this article, we give a theoretical underpinning of those constructions. Then using Geometric Invariant Theory, we obtain a new result on the structure of G(k)-(and G-) orbits in an arbitrary affine G-variety. We discuss several related problems to complement the main results.     

Discrete subgroups of algebraic groups over local fields of positive characteristics

It is shown in this paper that ifG is the group ofk-points of a semisimple algebraic groupG over a local fieldk of positive characteristic such that all itsk-simple factors are ofk-rank 1 and Γ ⊂G is a non-cocompact irreducible lattice then Γ admits a fundamental domain which is a union of translates of Siegel domains. As a consequence we deduce that ifG has more than one simple factor, then Γ is finitely generated and by a theorem due to Venkataramana, it is arithmetic.