Hermitian Varieties over finite fields

In this work, submitted for the award of the title of Doctor of Philosophy, we have investigated some properties of the configurations arising from the intersection of Hermitian varieties in a finite projective space. \par {\bf Chapter 1} introduces some background material on the theory of finite fields, projective spaces, Hermitian varieties and classical groups. \par {\bf Chapter 2} deals with the $2$-dimensional case. In Section 2.1, we present the the point-line classification of the intersections, due to Kestenband. In Section 2.2, we determine the full linear collineation group stabilising any of the configurations of 2.1 and we prove that if two configurations have the same point-line structure, then they are in fact projectively equivalent. A new and simplified proof of the group theoretical characterization of the Hermitian curve as the unital stabilised by a Singer subgroup of order $q-\sqrt{q}+1$ closes the chapter in Section 2.3. \par In {\bf Chapter 3} we study the $3$-dimensional case. In Section 3.1 we determine what incidence configurations fulfill the combinatorial properties required in order to be the intersection of Hermitian surfaces. Section 3.2 presents some further general remarks on linear systems of Hermitian curves and extensive computations on $4\times 4$ Hermitian matrices. In Section 3.3, we produce models that realize all the possible intersection configurations in dimension $3$. \par {\bf Chapter 4} is organized in two independent sections. In Section 4.1 we provide a general formula to determine the list of possible sizes of Hermitian intersections in ${\rm PG}(n,q)$. The formula itself has been obtained by studying the geometry of the set ${\mathscr H}$ of all singular Hermitian hypersurfaces of ${\rm PG}(n,q)$. Such a set is endowed with the structure of an algebraic hypersurface of ${\rm PG}(n^2+2n,q)$ of degree $n+1$; the locus of the singular points of ${\mathscr H}$ is analyzed in detail. In Section 4.2 we introduce some computer code in order to explicitly compute the intersection configurations arising in $\PG(n,q)$.