Poincaré Series and Deformations of Gorenstein Local Algebras

Abstract

Let (A, 𝔪, K) be an Artinian Gorenstein local ring with K an algebraically closed field of characteristic 0. In the present article, we prove a structure theorem describing the Artinian Gorenstein local K-algebras satisfying 𝔪4 = 0. We use this result in order to prove that such a K-algebra has rational Poincaré series and it is smoothable in any embedding dimension, provided dim K 𝔪2/𝔪3 ≤ 4. We also prove that the generic Artinian Gorenstein local K-algebra with 𝔪4 = 0 has rational Poincaré series.