New hemisystems of the Hermitian surface

A note on relative hemisystems of Hermitian generalised quadrangles

In this paper we introduce a set of sufficient criteria for the construction of relative hemisystems of the Hermitian space $\mathrm{H}(3,q^2)$, unifying all known infinite families. We use these conditions to provide new proofs of the existence of the known infinite families of relative hemisystems. Reproving these results has allowed us to find new relative hemisystems closely related to an infinite family of Cossidente's, and develop techniques that are likely to be useful in finding relative hemisystems in future.

A New Infinite Family of Hemisystems of the Hermitian Surface

In this paper, we construct an infinite family of hemisystems of the Hermitian surface $\mathsf{H}(3,q^2)$. In particular, we show that for every odd prime power $q$ congruent to $3$ modulo $4$, there exists a hemisystem of $\mathsf{H}(3,q^2)$ admitting $C_{(q^3+1)/4} : C_3$.

Hemisystems of the Hermitian Surface

We present a new method for the study of hemisystems of the Hermitian surface $\mathcal{U}_3$ of $PG(3,q^2)$. The basic idea is to represent generator-sets of $\mathcal{U}_3$ by means of a maximal curve naturally embedded in $\mathcal{U}_3$ so that a sufficient condition for the existence of hemisystems may follow from results about maximal curves and their automorphism groups. In this paper we obtain a hemisystem in $\ PG(3,p^2)$ for each $p$ prime of the form $p=1+16n^2$ with an integer $n$. Since the famous Landau's conjecture dating back to 1904 is still to be proved (or disproved), it is …

A relative $m$-cover of a Hermitian surface is a relative hemisystem

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 odd, then $m=(q+1)/2$, and called such a set $\mathcal{S}$ of lines a hemisystem. Penttila and Williford (2011) introduced the notion of a relative hemisystem: a set of lines $\mathcal{R}$ of $H(3,q^2)$, $q$ even, disjoint from a symplectic subgeometry $W(3,q)$ such that every point of $H(3,q^2)\setminus W(3,q)$ lies on exactly $q/2$ elements of $\mathcal{R}…

关于经典负型距离正则图分类及hemisystems不存在性的注记

A note on the classification of classical distance-regular graphs of negative type and the non-existence of hemisystems

本文通过连接代数图论和有限几何领域结果，证明了经典负型距离正则图及Hermitian极空间中hemisystems的不存在性。

On the geometry of the Hermitian Veronese curve and its quasi-Hermitian surfaces

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 $\PGL(2,q^2)$ (and $\PSL(2,q^2)$) arising from the Hermitian Veronese curve in $\PG(3, q^2)$, a maximal rational curve embedded on a smooth Hermitian surface with some fascinating properties. The study of its orbits leads to a new construction of quasi-Hermitian surfaces: sets of points with the same combinatorial and geometric properties as a non-degenerate Hermiti…