On the symmetric algebra of certain first syzygy modules

Let ( R , m ) be a standard graded K -algebra over a field K . Then R can be written as S / I , where I ⊆ ( x 1 ,…, x n ) 2 is a graded ideal of a polynomial ring S = K [ x 1 ,…, x n ]. Assume that n ⩽ 3 and I is a strongly stable monomial ideal. We study the symmetric algebra Sym R (Syz 1 ( m )) of the first syzygy module Syz 1 ( m ) of m . When the minimal generators of I are all of degree 2, the dimension of Sym R (Syz 1 ( m )) is calculated and a lower bound for its depth is obtained. Under suitable conditions, this lower bound is reached.