On rank $3$ quadratic equations of projective varieties

Abstract:

Let $X \subset \mathbb {P}^r$ be a linearly normal variety defined by a very ample line bundle $L$ on a projective variety $X$. Recently it is shown by Kangjin Han, Wanseok Lee, Hyunsuk Moon, and Euisung Park [Compos. Math. 157 (2021), pp. 2001–2025] that there are many cases where $(X,L)$ satisfies property $\mathsf {QR} (3)$ 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 $\mathbb {P}\left ( I(X,L)_2 \right )$ is a projective algebraic set, and property $\mathsf {QR} (3)$ of $(X,L)$ is equivalent to that $\Phi _3 (X)$ is nondegenerate in $\mathbb {P}\left ( I(X)_2 \right )$.

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) \mid A,B \!\in \! {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 ${Pic} (X)$ 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 ${Pic}(X)$ is generated by a very ample line bundle.