Intersections of the Hermitian surface with irreducible quadrics in $PG(3,q^2)$, $q$ odd

TL;DR

This paper investigates the possible intersection sizes between a Hermitian surface and an irreducible quadric in a projective 3-space over a finite field, focusing on cases where they share a tangent plane at a common point.

Contribution

It characterizes the intersection sizes of Hermitian surfaces and irreducible quadrics in $PG(3,q^2)$ for odd q, a problem not previously fully explored.

Findings

Identifies all possible intersection sizes under given conditions.

Provides a classification of intersection configurations.

Advances understanding of algebraic varieties in finite projective spaces.

Abstract

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 $P\in{\mathcal Q}\cap{\mathcal H}$.