Related papers: F4 and desmic quartic surfaces

In this paper, the problem of bounding the number of reducible curves in a pencil of algebraic plane curves is addressed. Unlike most of the previous related works, each reducible curve of the pencil is here counted with its appropriate…

We study the Frobenius-Lusztig kernel for quantum affine algebras at root of unity of small orders that are usually excluded in literature. These cases are somewhat degenerate and we find that the kernel is in fact mostly related to…

In this work we study the connection between the existence of finite dihedral covers of the projective plane ramified along an algebraic curve C, infinite dihedral covers, and pencils of curves containing C.

Let $G\subset SO(4)$ denote a finite subgroup containing the Heisenberg group. In these notes we classify all these groups, we find the dimension of the spaces of $G$-invariant polynomials and we give equations for the generators whenever…

We consider lattice equations on ${\mathds{Z}}^2$ which are autonomous, affine linear and possess the symmetries of the square. Some basic properties of equations of this type are derived, as well as a sufficient linearization condition and…

We generalize Iskovskih's theorem about surfaces without irregularity and bigenus from the smooth case to regular surfaces over arbitrary fields, with special focus on the case of imperfect fields. This includes surfaces that are…

Let $C/K$ be a smooth plane quartic over a discrete valuation field. We characterize the type of reduction (i.e. smooth plane quartic, hyperelliptic genus 3 curve or bad) over $K$ in terms of the existence of a special plane quartic model…

Two families of surfaces arise from considering cyclic branched covers of $\mathbb{P}^{2}$ over smooth quartic curves. These consist of degree 2 del Pezzo surfaces with a $\mathbb{Z}/2\mathbb{Z}$ action and $K3$ surfaces with a…

Let T -> S be a finite flat morphism of degree two between regular integral schemes of dimension at most two (and with 2 invertible), having regular branch divisor D. We establish a bijection between Azumaya quaternion algebras on T and…

We describe smooth rational projective algebraic surfaces X, over an algebraically closed field of characteristic different from 2, having an even set of four disjoint (-2)-curves N_1,...,N_4, i.e. such that N_1+...+N_4 is divisible by 2 in…

We introduce the notion of radical parametrization of a surface, and we provide algorithms to compute such type of parametrizations for families of surfaces, like: Fermat surfaces, surfaces with a high multiplicity (at least the degree…

We show that the irreducible holomorphic symplectic eightfold Z associated to a cubic fourfold Y not containing a plane is deformation-equivalent to the Hilbert scheme of four points on a K3 surface. We do this by constructing for a generic…

We study quartic surfaces that admit a group of projective automorphisms isomorphic to icosahedron group.

A K3-surface is a (smooth) surface which is simply connected and has trivial canonical bundle. In these notes we investigate three particular pencils of K3-surfaces with maximal Picard number. More precisely the general member in each…

We study certain pencils of del Pezzo surfaces generated by a smooth del Pezzo surface S of degree less or equal to 3 anti-canonically embedded into a weighted projective space P and an appropriate multiple of a hyperplane H. Our main…

We show that every member of an infinite family of symplectic manifolds constructed by R. Inanc Baykur, Kenta Hayano, and Naoyuki Monden (arXiv:1903:02906) is diffeomorphic to an elliptic surface. As a result: (1) the symplectic Calabi-Yau…

We prove that there is a pencil of hypersurfaces in $\mathbb{P}^n$ of any given degree over a finite field $\mathbb{F}_q$ such that every $\mathbb{F}_q$-member of the pencil is not blocking with respect to $\mathbb{F}_q$-lines.

Recently, de Thanhoffer de Volcsey and Van den Bergh showed that Grothendieck groups of "noncommutative Del Pezzo surfaces" with an exceptional sequence of length 4 are isomorphic to one of three types, the third one not coming from a…

An algebraic condition for the singularity of certain T\"oplitz matrix pencils is derived which involves only the principal minors of the constant parts of the pencils. This leads to an algebraic conjecture which is equivalent to the…

This paper introduces and studies a class of Weyl-type algebras \(A_{p,t,\cA} = \Weyl{e^{\pm x^{p} e^{t x}},\; e^{\cA x},\; x^{\cA}}\) constructed over exponential-polynomial rings, where \(\FF\) is a field of characteristic zero, \(\cA\)…