On rank 3 quadratic equations of projective varieties

Let $X \subset \P^r$ be a linearly normal variety defined by a very ample line bundle $L$ on a projective variety $X$. Recently it is shown in \cite{HLMP} that there are many cases where $(X,L)$ satisfies property $\textsf{QR} (3)$ in the sense that the homogeneous ideal $I(X,L)$ of $X$ is generated by quadratic polynomials of rank $3$. The locus $\Phi_3 (X,L)$ of rank $3$ quadratic equations of $X$ in $\P \left( I(X,L)_2 \right)$ is a projective algebraic set, and property $\textsf{QR} (3)$ of $(X,L)$ is equivalent to that $\Phi_3 (X)$ is nondegenerate in $\P \left( I(X)_2 \right)$. In this paper we study geometric structures of $\Phi_3 (X,L)$ such as its minimal irreducible decomposition. Let \begin{equation*} \Sigma (X,L) = \{ (A,B) ~|~ A,B \in {\rm Pic}(X),~L = A^2 \otimes B,~h^0 (X,A) \geq 2,~h^0 (X,B) \geq 1 \}. \end{equation*} We first construct a projective subvariety $W(A,B) \subset \Phi_3 (X,L)$ for each $(A,B)$ in $\Sigma (X,L)$. Then we prove that the equality \begin{equation*} \Phi_3 (X,L) ~=~ \bigcup_{(A,B) \in \Sigma (X,L)} W(A,B) \end{equation*} holds when $X$ is locally factorial. Thus this is an irreducible decomposition of $\Phi_3 (X,L)$ when ${\rm Pic} (X)$ is finitely generated and hence $\Sigma(X,L)$ is a finite set. Also we find a condition that the above irreducible decomposition is minimal. For example, it is a minimal irreducible decomposition of $\Phi_3 (X,L)$ if ${\rm Pic}(X)$ is generated by a very ample line bundle.