Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the Hermitian Veronesean

John Bamberg, Geertrui Van de Voorde

TL;DR

This paper addresses the classification of special sets on the Hermitian surface $\\mathsf{H}(3,q^2)$ by introducing two local characterisations of the Hermitian Veronesean $\\mathcal{V}$. The authors first prove a subline-based criterion: if a set of noncollinear points contains four non-coplanar points and every triangle formed with three fixed vertices is in perspective, then the set has size at most $q^2+1$ and equality forces the set to be $\\mathcal{V}$ up to collineation. A second, Baer-subline based characterisation is established, using a fixed isotropic line $\\ell$ through a point $P$ and a correspondence $F_\\ell$ from sublines of $\\ell$ to planes, plus a perspectivity condition for triples involving a distinguished point $Q$ and a nondegenerate plane spanned by any three points of the set; under these hypotheses the set is again equivalent to $\\mathcal{V}$. Together with prior work on perspective triples and pseudo-ovoids, these results advance the local-to-global understanding of special sets, supporting the view that all such sets may be classical, and they refine the conditions under which the Hermitian Veronesean is uniquely determined by local geometric data.

Abstract

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, we give two local characterisations of the Hermitian Veronesean, based on sublines and triples of points in perspective.

On the Hermitian Veronesean

TL;DR

This paper addresses the classification of special sets on the Hermitian surface by introducing two local characterisations of the Hermitian Veronesean . The authors first prove a subline-based criterion: if a set of noncollinear points contains four non-coplanar points and every triangle formed with three fixed vertices is in perspective, then the set has size at most and equality forces the set to be up to collineation. A second, Baer-subline based characterisation is established, using a fixed isotropic line through a point and a correspondence from sublines of to planes, plus a perspectivity condition for triples involving a distinguished point and a nondegenerate plane spanned by any three points of the set; under these hypotheses the set is again equivalent to . Together with prior work on perspective triples and pseudo-ovoids, these results advance the local-to-global understanding of special sets, supporting the view that all such sets may be classical, and they refine the conditions under which the Hermitian Veronesean is uniquely determined by local geometric data.

Abstract

The Hermitian Veronesean in , given by , is a well-studied rational curve, and forms a {\em special} set of the Hermitian surface . In this paper, we give two local characterisations of the Hermitian Veronesean, based on sublines and triples of points in perspective.
Paper Structure (3 sections, 24 theorems, 39 equations)

This paper contains 3 sections, 24 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Setup
  3. A characterisation via sublines

Key Result

Theorem 1.1

Let $\mathcal{S}$ be a set of mutually noncollinear points on $\mathsf{H}(3,q^2)$, $q$ odd. Suppose there are four non-coplanar points $P,Q_1,Q_2,Q_3\in \mathcal{S}$, such that all triangles $PQ_iR$, $i=1,2,3$, $R\in \mathcal{S}\setminus \{P,Q_i\}$, are in perspective. Then $|\mathcal{S}|\leqslant q

Theorems & Definitions (44)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 2.1
  • Lemma 2.2: Shult:2005aa, Bamberg:2021aa
  • Lemma 2.3: Bamberg:2021aa
  • Theorem 2.4: cf. Thas:2011aa
  • Corollary 2.5
  • proof
  • Lemma 2.6
  • proof
  • ...and 34 more