Related papers: On hyperovals in $Q^+(6,4)$
Let $ex(Q_n, H)$ be the largest number of edges in a subgraph $G$ of a hypercube $Q_n$ such that there is no subgraph of $G$ isomorphic to $H$. We show that for any integer $k\geq 3$, $$ex(Q_n, C_{4k+2})= O(n^{\frac{5}{6} +…
We study the number of planes for four dimensional projective hypersurfaces which has so-called inductive structure. We also determine transcendental lattices for cubic fourfolds of this type.
We prove there are exactly 16 arithmetic lattices of hyperbolic 3-space which are generated by two elements of finite orders p and q with p,q at least six. We also verify a conjecture of H.M. Hilden, M.T. Lozano, and J.M. Montesinos…
There is a modular curve X'(6) of level 6 defined over Q whose Q-rational points correspond to j-invariants of elliptic curves E over Q for which Q(E[2]) is a subfield of Q(E[3]). In this note we characterize the j-invariants of elliptic…
We give upper and lower bounds on the number of points on abelian varieties over finite fields, and lower bounds specific to Jacobian varieties. We also determine exact formulas for the maximum and minimum number of points on Jacobian…
We study transformations as in the title with emphasis on those having smooth connected base locus, called "special". In particular, we classify all special quadratic birational maps into a quadric hypersurface whose inverse is given by…
The classification of all semiovals and blocking semiovals in $\mathrm{PG}(2,8)$ and in $\mathrm{PG}(2,9)$ of size less than $17$ is determined. Also, some information on the stabilizer groups and the intersection sizes with lines is given.
In this survey article we describe the geometry of toric hyperk\"ahler varieties, which are hyperk\"ahler quotients of the quaternionic vector spaces by tori. In particular, we discuss the Betti numbers, the cohomology ring, and variation…
A hypersurface $M$ in $\mathbb{R}^n$, $n \geq 4$, has central ovaloid property if $M$ intersects some hyperplane transversally along an ovaloid and every such ovaloid on $M$ has central symmetry. We show that a complete, connected, smooth…
We investigate proper biharmonic hypersurfaces with at most three distinct principal curvatures in space forms. We obtain the full classification of proper biharmonic hypersurfaces in 4-dimensional space forms.
The structure of on-shell and off-shell 2D, (4,4) supersymmetric scalar multiplets is investigated, in components and in superspace. We reach the surprising result that there exist eight {\underline {distinct}} on-shell versions and an even…
Some constructions and bounds on the sizes of semiovals contained in the Hermitian curve are given. A construction of an infinite family of 2-blocking sets of the Hermitian curve is also presented.
We introduce two variants of $q$-hypergeometric equation. We obtain several explicit solutions of variants of $q$-hypergeometric equation. We show that a variant of $q$-hypergeometric equation can be obtained by a restriction of $q$-Appell…
We point out a link between two surfaces which have appeared recently in the literature: the surface of cuboids and the Schoen surface. Both surfaces give rise to a surface with q=4, whose canonical map is 2-to-1 onto an intersection of 4…
An old theorem, due to Graustein, asserts that the average curvature of a plane oval is attained at least at four points. We present a proof by way of wave propagation and extend this result to the spherical and hyperbolic geometries - in…
We discuss several ways of packing a hyperbolic surface with circles (of either varying radii or all being congruent) or horocycles, and note down some observations related to their symmetries (or the absence thereof).
An ordinary hypersphere of a set of points in real $d$-space, where no $d+1$ points lie on a $(d-2)$-sphere or a $(d-2)$-flat, is a hypersphere (including the degenerate case of a hyperplane) that contains exactly $d+1$ points of the set.…
Given an elliptic quartic of type $Y^2=f(X)$ representing an elliptic curve of positive rank over $\Q$, we investigate the question of when the $Y$-coordinate can be represented by a quadratic form of type $ap^2+bq^2$. In particular, we…
We establish new lower bounds ex(Q_7,C_4)>=304 and ex(Q_8,C_4)>=680 for the maximum number of edges in a C_4-free subgraph of the 7- and 8-dimensional hypercubes, and give a modern computational reproduction of ex(Q_6,C_4)=132. All bounds…
It is shown that the Veldkamp space of the unique generalized quadrangle GQ(2,4) is isomorphic to PG(5,2). Since the GQ(2,4) features only two kinds of geometric hyperplanes, namely point's perp-sets and GQ(2,2)s, the 63 points of PG(5,2)…