A linear degree-bound for generating invariants of n points in the projective plane

Tyrrell McAllister

Let R be the ring of projective invariants of n labeled points in the projective plane under the action of SL3(C). We give an upper bound, linear in n, on the degree in which R is generated. This result follows as a corollary of a more general result bounding the degrees in which type-A weight rings are generated. These rings arise as follows: Given a pair λ, μ of weights for SL3(C) with λ dominant, let Vλ[μ] denote the μ-isotypic component of the irreducible representation Vλ with highest weight λ. A type-A weight ring R(λ, μ) is a graded ring with graded pieces VN λ[N μ], where N ranges over the positive integers. Such a ring is the projective coordinate ring of a weight variety W(λ, μ). Using results of H. Derksen and V. L. Popov, we show that R(λ, μ) is generated in degree at most dimW(λ, μ).

back to EIDMA Seminar Combinatorial Theory announcements