Poincaré series and deformations of Gorenstein local algebras with low socle degree

Abstract

Let $K$ be an algebraically closed field of characteristic $0$, and let $A$ be an Artinian Gorenstein local commutative and Noetherian $K$--algebra, with maximal ideal $M$. In the present paper we prove a structure theorem describing such kind of $K$--algebras satisfying $M^4=0$. We use this result in order to prove that such a $K$--algebra $A$ has rational Poincaré series and it is always smoothable in any embedding dimension, if ${\dim }_K{M}^2/M^3\le 4$. We also prove that the generic Artinian Gorenstein local $K$--algebra with socle degree three has rational Poincaré series, in spite of the fact that such algebras are not necessarily smoothable.