SL2-modules of small homological dimension

Let $V_n$ be the $\rm{S}\rm{L}_2$-module of binary forms of degree $n$ and let $V = V_{n_1} \oplus ... \oplus V_{n_p}$. We consider the algebra $R = O(V)^{\rm{S}\rm{L}_2}$ of polynomial functions on $V$ invariant under the action of $\rm{S}\rm{L}_2$. The measure of the intricacy of these algebras is the length of their chains of syzygies, called homological dimension hd $R$. Popov gave in 1983 a classification of the cases in which hd $R \leq 10$ for a single binary form $(p=1)$ or hd $R \leq 3$ for a system of two or more binary forms $(p>1)$. We extend Popov’s result and determine for $p=1$ the cases with hd $R \leq 100$, and for $p>1$ those with hd $R \leq 15$. In these cases we give a set of homogeneous parameters and a set of generators for the algebra $R$.