A family of hemisystems on the parabolic quadrics

Relative $m$-ovoids of elliptic quadrics

Let ${\cal Q}^-(2n+1,q)$ be an elliptic quadric of ${\rm PG}(2n+1,q)$. A relative $m$-ovoid of ${\cal Q}^-(2n+1,q)$ (with respect to a parablic section ${\cal Q} := {\cal Q}(2n,q) \subset {\cal Q}^-(2n+1,q)$) is a subset $\cal R$ of points of ${\cal Q}^-(2n+1,q)\setminus {\cal Q}$ such that every generator of ${\cal Q}^-(2n+1,q)$ not contained in $\cal Q$ meets $\cal R$ in precisely $m$ points. A relative $m$-ovoid having the same size as its complement (in ${\cal Q}^-(2n+1,q) \setminus {\cal Q}$) is called a relative hemisystem. We show that a nontrivial relative $m$-ovoid of ${\cal Q}^-(2n+1…

Characterizing parabolic hyperplanes of the hyperbolic and elliptic quadrics in PG(2n + 1, q)

In this article, a combinatorial characterization of the family of parabolic hyperplanes of a hyperbolic (respectively, elliptic) quadric of PG(2n + 1, q), using their intersection properties with the points and subspaces of codimension 2, is given.

A family of $m$-ovoids of parabolic quadrics

We construct a family of $\frac{(q-1)}{2}$-ovoids of $Q(4,q)$, the parabolic quadric of $\textup{PG}(4,q)$, for $q\equiv 3\pmod 4$. The existence of $\frac{(q-1)}{2}$-ovoids of $Q(4,q)$ was only known for $q=3, 7,$ or $11$. Our construction provides the first infinite family of $\frac{(q-1)}{2}$-ovoids of $Q(4,q)$.Along the way, we also give a construction of $\frac{q+1}{2}$-ovoids in $Q(4,q)$ for $q\equiv 1\pmod 4$.

On large maximal partial ovoids of the parabolic quadric $\q(4,q)$

We use the representation $T_2(O)$ for $\q(4,q)$ to show that maximal partial ovoids of $\q(4,q)$ of size $q^2-1$, $q=p^h$, $p$ odd prime, $h > 1$, do not exist. Although this was known before, we give a slightly alternative proof, also resulting in more combinatorial information of the known examples for $q$ prime.

Characterising elliptic and hyperbolic hyperplanes of the parabolic quadric \Q(2n,q)

We provide a natural characterisation for the sets of elliptic and hyperbolic hyperplanes of the parabolic quadric Q(2n,q) when q is even. This characterisation is based on the number of elements of these sets through points and codimension 2 spaces and generalises [S. Barwick, A. Hui, and W-A. Jackson. Characterising elliptic solids of Q(4,q), q even. Discrete Math., 343 (6) (2020), 111857] and [S. Barwick, A. Hui, W-A. Jackson, and J. Schillewaert. Characterising hyperbolic solids of Q(4,q), q even. Des. Codes Cryptogr., 88 (1) (2020), 33--39.].

仿射向量空间划分与二次曲面的扩散

Affine vector space partitions and spreads of quadrics

本文研究了仿射扩散与二次曲面扩散的等价性，证明了双曲、抛物或椭圆仿射扩散分别等价于相应二次曲面的扩散。