Defining Equations of Nilpotent Orbits for Borel Subgroups of Modality Zero in Type $A_{n}$

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.