  Let $V$ be a finite-dimensional vector space over a finite field, and suppose
$G \leq \Gamma \mathrm{L}(V)$ is a group with a unique subnormal quasisimple
subgroup $E(G)$ that is absolutely irreducible on $V$. A base for $G$ is a set
of vectors $B\subseteq V$ with pointwise stabiliser $G_B=1$. If $G$ has a base
of size 1, we say that it has a regular orbit on $V$. In this paper we
investigate the minimal base size of groups $G$ with $E(G)/Z(E(G)) \cong
\mathrm{PSL}_n(q)$ in defining characteristic, with an aim of classifying those
with a regular orbit on $V$.
