Related papers: On Hermitian varieties in $\mathrm{PG}(6,q^2)$

In this paper, we present a combinatorial characterization of the hyperplanes associated with non-singular hermitian varieties ${H}\left(s, q^2\right)$ in the projective space $\mathrm{PG}\left(s,q^2\right)$ where $s\geq3$ and $q>2$. By…

In $PG(3,q^2)$, with $q$ odd, we determine the possible intersection sizes of a Hermitian surface $\mathcal{H}$ and an irreducible quadric $\mathcal{Q}$ having the same tangent plane $\pi$ at a common point $P\in{\mathcal Q}\cap{\mathcal…

It is known that the Hermitian varieties are codewords in the code defined by the points and hyperplanes of the projective spaces $PG(r,q^2)$. In finite geometry, also quasi-Hermitian varieties are defined. These are sets of points of…

The nonsingular Hermitian surface of degree $\sqrt{q} +1$ is characterized by its number of $\Bbb{F}_q$-points among the irreducible surfaces over $\Bbb{F}_q$ of degree $\sqrt{q} +1$ in the projective 3-space.

We determine the possible intersection sizes of a Hermitian surface $\mathcal H$ with an irreducible quadric of ${\mathrm PG}(3,q^2)$ sharing at least a tangent plane at a common non-singular point when $q$ is even.

A spread of a Hermitian unital in PG(2,q^2) is a set of q^2+q+1 pairwise disjoint blocks that partition the points of the unital. In this paper, we discuss the results of an exhaustive computer search for spreads of Hermitian unitals of…

In [1], a new quasi-Hermitian variety $\mathcal{H}_\varepsilon^r$ in $\mathrm{PG}(r, q^2)$, with $q = 2^e$ and $e \geq 3$ an odd integer, was constructed. The variety depends on a primitive element $\varepsilon$ of the underlying field…

In this paper we deal with the problem of classifying the genera of quotient curves $\mathcal{H}_q/G$, where $\mathcal{H}_q$ is the $\mathbb{F}_{q^2}$-maximal Hermitian curve and $G$ is an automorphism group of $\mathcal{H}_q$. The groups…

Let $E$ be a set of solids (hyperplanes) in $PG(4,q)$, $q$ even, $q>2$, such that every point of $PG(4,q)$ lies in either $0$, $\frac12q^3$ or $\frac12(q^3-q^2)$ solids of $E$, and every plane of $PG(4,q)$ lies in either $0$, $\frac12q$ or…

We prove that a non-empty set L of at most q^5+q^4+q^3+q^2+q+1 lines of PG(n, q) with the properties that (1) every point of PG(n,q) is incident with either 0 or q+1 elements of L, (2) every plane plane of PG(n, q) is incident with either…

An $m$-cover of the Hermitian surface $H(3,q^2)$ of $PG(3,q^2)$ is a set $\mathcal{S}$ of lines of $H(3,q^2)$ such that every point of $H(3,q^2)$ lies on exactly $m$ lines of $\mathcal{S}$, and $0<m<q+1$. Segre (1965) proved that if $q$ is…

In [A. Aguglia, A. Cossidente, G. Korchmaros, "On quasi-Hermitian varieties", J. Comb. Des. 20 (2012), 433-447] new quasi-Hermitian varieties ${\mathcal M}_{\alpha,\beta}$ in $\mathrm{PG}(r,q^2)$ depending on a pair of parameters…

The Hermitian Veronesean in $PG(3,q^2)$, given by $\mathcal{V}:=\{ (1,x,x^q,x^{q+1}):x\in\mathbb{F}_q\}\cup\{(0,0,0,1)\}$, is a well-studied rational curve, and forms a {\em special} set of the Hermitian surface $H(3,q^2)$. In this paper,…

The Hermite invariant H is the defining equation for the hypersurface of binary quintics in involution. This paper analyses the geometry and invariant theory of H. We determine the singular locus of this hypersurface and show that it is a…

The complete classification of the orbits on subspaces under the action of the projective stabiliser of (classical) algebraic varieties is a challenging task, and few classifications are complete. We focus on a particular action of…

Let $\mathcal W$ be a non-empty set of points of a finite Desarguesian projective space $\mathrm{PG}(n,q)$. A collection of varieties of $\mathrm{PG}(n,q)$ is \emph{mutually $\mu$-intersecting (relatively to $\mathcal W$)} if its elements…

We study the dual linear code of points and generators on a non-singular Hermitian variety $\mathcal{H}(2n+1,q^2)$. We improve the earlier results for $n=2$, we solve the minimum distance problem for general $n$, we classify the $n$…

Let $(G,X)$ be a Shimura variety of PEL type such that $G_{{\bf Q}_2}$ is a split ${\bf GSO}_{2n}$ group with $n\ge 2$. We prove the existence of the integral canonical models of ${\rm Sh}(G,X)/H_2$ in unramified mixed characteristic…

We investigate birational properties of hypersurfaces of degree $6$ in the weighted projective space $\mathbf{P}(1,1,2,2,3)$. In particular, we prove that any such quasi-smooth hypersurface is not rational.

Edoukou, Ling and Xing in 2010, conjectured that in \mathbb{P}^n(\mathbb{F}_{q^2}), n \geq 3, the maximum number of common points of a non-degenerate Hermitian variety \mathcal{U}_n and a hypersurface of degree d is achieved only when the…