Title: Defining Equations of Nilpotent Orbits for Borel Subgroups of Modality Zero in Type $A_{n}$,
Abstract: Let $G$ be a quasi-simple algebraic group defined over an algebraically
closed field $k$ and $B$ a Borel subgroup of $G$ acting on the nilradical
$\mathfrak{n}$ of its Lie algebra $\mathfrak{b}$ via the Adjoint
representation. It is known that $B$ has only finitely many orbits in only five
cases: when $G$ is of type $A_{n}$ for $n \leq 4$, and when $G$ is type
$B_{2}$. In this paper, we elaborate on this work in the case when $G =SL_{n
+1}(k)$ (type $A_{n})$, for $n \leq 4$, by finding the polynomial defining
equations of each orbit. Consequences of these equations include the dimension
of the orbits and the closure ordering on the set of orbits, although these
facts are already known. The other case, when $G$ is type $B_{2}$, can be
approached the same way and is treated in a separate paper, where we believe
the determination of the closure order is new.