Automorphisms of the subspace sum graphs on a vector space

The subspace sum graph $\mathcal{G}(\mathbb{V})$ on a finite dimensional vector space $\mathbb{V}$ was introduced by Das [Subspace Sum Graph of a Vector Space, arXiv:1702.08245], recently. The vertex set of $\mathcal{G}(\mathbb{V})$ consists of all the nontrivial proper subspaces of $\mathbb{V}$ and two distinct vertices $W_1$ and $W_2$ are adjacent if and only if $W_1+W_2=\mathbb{V}$. In that paper, some structural indices (e.g., diameter, girth, connectivity, domination number, clique number and chromatic number) were studied, but the characterization of automorphisms of $\mathcal{G}(\mathbb{V})$ was left as one of further research topics. Motivated by this, we in this paper characterize the automorphisms of $\mathcal{G}(\mathbb{V})$ completely

Automorphism group of the subspace inclusion graph of a vector space

In a recent paper [Comm. Algebra, 44(2016) 4724-4731], Das introduced the graph $\mathcal{I}n(\mathbb{V})$, called subspace inclusion graph on a finite dimensional vector space $\mathbb{V}$, where the vertex set is the collection of nontrivial proper subspaces of $\mathbb{V}$ and two vertices are adjacent if one is properly contained in another. Das studied the diameter, girth, clique number, and chromatic number of $\mathcal{I}n(\mathbb{V})$ when the base field is arbitrary, and he also studied some other properties of $\mathcal{I}n(\mathbb{V})$ when the base field is finite. In this paper, the automorphisms of $\mathcal{I}n(\mathbb{V})$ are determined when the base field is finite.Comment: 10 page

Functions realising as abelian group automorphisms

Let $A$ be a set and $f:A\rightarrow A$ a bijective function. Necessary and sufficient conditions on $f$ are determined which makes it possible to endow $A$ with a binary operation $*$ such that $(A,*)$ is a cyclic group and f\in \mbox{Aut}(A). This result is extended to all abelian groups in case $|A|=p^2, \ p$ a prime. Finally, in case $A$ is countably infinite, those $f$ for which it is possible to turn $A$ into a group $(A,*)$ isomorphic to ${\Bbb Z}^n$ for some $n\ge 1$, and with f\in \mbox{Aut} (A), are completely characterised.Comment: 17 page

Orbit Parametrizations for K3 Surfaces

We study moduli spaces of lattice-polarized K3 surfaces in terms of orbits of representations of algebraic groups. In particular, over an algebraically closed field of characteristic 0, we show that in many cases, the nondegenerate orbits of a representation are in bijection with K3 surfaces (up to suitable equivalence) whose N\'eron-Severi lattice contains a given lattice. An immediate consequence is that the corresponding moduli spaces of these lattice-polarized K3 surfaces are all unirational. Our constructions also produce many fixed-point-free automorphisms of positive entropy on K3 surfaces in various families associated to these representations, giving a natural extension of recent work of Oguiso.Comment: 83 pages; to appear in Forum of Mathematics, Sigm

Holonomic D-modules and positive characteristic

This article is based on the 5th Takagi Lectures delivered at the University of Tokyo in 2008. We discuss a hypothetical correspondence between holonomic D-modules on an algebraic variety X defined over a field of zero characteristic, and certain families of Lagrangian subvarieties in the cotangent bundle to X. The correspondence is based on the reduction to positive characteristic.Comment: 29 page

Integral point sets over finite fields

We consider point sets in the affine plane $\mathbb{F}_q^2$ where each Euclidean distance of two points is an element of $\mathbb{F}_q$. These sets are called integral point sets and were originally defined in $m$-dimensional Euclidean spaces $\mathbb{E}^m$. We determine their maximal cardinality $\mathcal{I}(\mathbb{F}_q,2)$. For arbitrary commutative rings $\mathcal{R}$ instead of $\mathbb{F}_q$ or for further restrictions as no three points on a line or no four points on a circle we give partial results. Additionally we study the geometric structure of the examples with maximum cardinality.Comment: 22 pages, 4 figure

On the Krull Dimension of the deformation ring of curves with automorphisms

We reduce the study of the Krull dimension d of the deformation ring of the functor of deformations of curves with automorphisms to the study of the tangent space of the deformation functor of a class of matrix representations of the p-part of the decomposition groups at wild ramified points, and we give a method in order to compute d.Comment: New Revised Versio

Notes on motives in finite characteristic

Motivic local systems over a curve in finite characteristic form a countable set endowed with an action of the absolute Galois group of rational numbers commuting with the Frobenius map. I will discuss three series of conjectures about such sets, based on an analogy with algebraic dynamics, on a formalism of commutative algebras of motivic integral operators, and on an analogy with 2-dimensional lattice models.Comment: 33 page

On the birational geometry of spaces of complete forms I: collineations and quadrics

Moduli spaces of complete collineations are wonderful compactifications of spaces of linear maps of maximal rank between two fixed vector spaces. We investigate the birational geometry of moduli spaces of complete collineations and quadrics from the point of view of Mori theory. We compute their effective, nef and movable cones, the generators of their Cox rings, and their groups of pseudo-automorphisms. Furthermore, we give a complete description of both the Mori chamber and stable base locus decompositions of the effective cone of the space of complete collineations of the 3-dimensional projective space.Comment: 38 page

Alterations and resolution of singularities

On July 26, 1995, at the University of California, Santa Cruz, a young Dutch mathematician by the name Aise Johan de Jong made a revolution in the study of the arithmetic, geometry and cohomology theory of varieties in positive or mixed characteristic. The talk he delivered, first in a series of three entitled "Dominating Varieties by Smooth Varieties", had a central theme: a systematic application of fibrations by nodal curves. Among the hundreds of awe struck members of the audience, participants of the American Mathematical Society Summer Research Institute on Algebraic Geometry, many recognized the great potential of Johan de Jong's ideas even for complex algebraic varieties, and indeed soon more results along these lines began to form. This paper is an outgrowth of our course material prepared for the Working Week on Resolution of Singularities, which was held during September 7-14, 1997 in Obergurgl, Tirol, Austria. As we did in the workshop, we intend to explain Johan de Jong's results in some detail, and give some other results following the same paradigm, as well as a few applications, both arithmetic and in characteristic zero. We hope that the reader will come to share some of the excitement we felt on that beautiful July day in Santa Cruz.Comment: 66 pages, latex2