Let F be an arbitrary field. Let A be a central simple F -algebra. Let G be the algebraic group AutA of automorphisms of A. Let XA be the class of finite direct products of projective G-homogeneous F -varieties (the class XA includes the generalized Severi-Brauer varieties of the algebra A). Let p be a positive prime integer. For any variety in XA, we determine its canonical dimension at p. In ...

We say an algebraic object or property over a field k is arithmetic if it becomes trivial or vanishes after finite separable base extension. Since such objects or properties owe their existence to the presence of “arithmetic gaps” in k, i.e., the failure of k to be algebraically closed, we view them as responses to specific arithmetic properties of k, and we study them in order to gain insight ...

Fixing an arbitrary point p ∈ CP and a triple (g, d, `) of nonnegative integers satisfying the inequality g ≤ (d+l−1 2 ) − (l 2 ) , we associate a natural Hurwitz number to the (open) Severi-type varietyWg,d,` consisting of all reduced irreducibke plane curves of degree d + l with genus g and having an ordinary singularity of order l at p (the remaining singular points of such curves being usua...

In this paper we show that the problem of checking consistency of a knowledge base in the Description Logic ALCM is ExpTime-complete. The M stands for meta-modelling as defined by Motz, Rohrer and Severi. To show our main result, we define an ExpTime Tableau algorithm as an extension of an algorithm for checking consistency of a knowledge base in ALC by Nguyen and Szalas.

We prove that certain Severi varieties of nodal curves positive genus on general blow-ups the twofold symmetric product a elliptic curve are non-empty and smooth expected dimension. This result, besides its intrinsic value, is an important preliminary step for proof nonemptiness Enriques surfaces in arXiv:2109.10735.

Contents 1. Introduction 2 1.1. The general strategy: the cross-ratio method 2 1.2. Two sample calculations 7 1.3. Notation and Terminology 14 1.4. Summary of results 15 2. Degenerations of rational curves 17 2.1. The basic setup 17 2.2. The main results from deformation theory 20 2.3. The geometry of the Severi varieties 21 2.4. Singularities of the total space 37 3. Formulas 56 3.

Suppose π : S → P is an elliptic K3-surface over C with a section, such that all its singular fibres are multiplicative. Suppose moreover that π is extremal, i.e., the group of sections of π is finite and the rank ρ of the Néron-Severi group NS(S) of S is maximal, so ρ = 20. A formula of Shioda and Tate [Sh90, Cor. 5.3] then implies that π has precisely 6 singular fibers, consisting of ni compo...

In the present paper, we provide general bounds for the torsion in the codimension 2 Chow groups of the products of projective homogeneous surfaces. In particular, we determine the torsion for the product of four Pfister quadric surfaces and the maximal torsion for the product of three Severi-Brauer surfaces. We also find an upper bound for the torsion of the product of three quadric surfaces w...

We prove a sufficient condition for a time-dependent closed set to be viable with respect to a delay evolution inclusion governed by a strongly-weakly u.s.c. perturbation of an infinitesimal generator of a C0-semigroup. This condition is expressed in terms of a natural concept involving tangent sets, generalizing tangent vectors in the sense of Bouligand and Severi. Mathematics Subject Classifi...