ASL structures of some quadrics

Abstract: Let $K$ be a field and $X$, $Y$ denote matrices such that, the entries of $X$ are either indeterminates over $K$ or $0$ and the entries of $Y$ are indeterminates over $K$ which are different from those appearing in $X$. We consider ideals of the form $I_{1}(XY)$, which is the ideal generated by the $1\times 1$ minors of the matrix $XY$. We prove that the quotient ring $K[X, Y]/I_{1}(XY)$ admits an ASL structure for certain $X$ and $Y$.