$d$-sequence and Regular sequence of Quadrics

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 homogeneous polynomials of degree $2$ given by the $1\times 1$ minors of the matrix $XY$. We prove that $d$-sequences and regular sequences arise naturally as part of generators of $I_{1}(XY)$ for some special cases. We use this information to calculate the equations defining the Rees algebra of $I_{1}(XY)$.

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