SL2-modules of small homological dimension

Let Vn be the SL2-module of binary forms of degree n and let V = Vn1 ⊕ . . . ⊕ Vnp . We consider the algebra R = O(V )2 of polynomial functions on V invariant under the action of SL2. The measure of the intricacy of these algebras is the length of their chains of syzygies, called homological dimension hdR. Popov gave in 1983 a classi cation of the cases in which hdR ≤ 10 for a single binary form (p = 1) or hdR ≤ 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 hdR ≤ 100, and for p > 1 those with hdR ≤ 15. In these cases we give a set of homogeneous parameters and a set of generators for the algebra R.