On the symmetric algebra of certain first syzygy modules

Let $(R,rak{m})$ be a standard graded $K$-algebra over a field $K$. Then $R$ can be written as $S/I$, where $Isubseteq(x_1,ldots,x_n)^2$ is a graded ideal of a polynomial ring $S=K[x_1,ldots,x_n]$. Assume that $ngeq3$ and $I$ is a strongly stable monomial ideal. We study the symmetric algebra ${ m Sym}_R({ m Syz}_1(rak{m}))$ of the first syzygy module ${ m Syz}_1(rak{m})$ of $rak{m}$. When the minimal generators of $I$ are all of degree 2, the dimension of ${ m Sym}_R({ m Syz}_1(rak{m}))$ is calculated and a lower bound for its depth is obtained. Under suitable conditions, this lower bound is reached.