On rank 3 quadratic equations of projective varieties

Abstract

Let X ⊂ P r X \subset \mathbb {P}^r be a linearly normal variety defined by a very ample line bundle L L on a projective variety X 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 ) (X,L) satisfies property Q R ( 3 ) \mathsf {QR} (3) in the sense that the homogeneous ideal I ( X , L ) I(X,L) of X X is generated by quadratic polynomials of rank 3 3 . The locus Φ 3 ( X , L ) \Phi _3 (X,L) of rank 3 3 quadratic equations of X X in P ( I ( X , L ) 2 ) \mathbb {P}\left ( I(X,L)_2 \right ) is a projective algebraic set, and property Q R ( 3 ) \mathsf {QR} (3) of ( X , L ) (X,L) is equivalent to that Φ 3 ( X ) \Phi _3 (X) is nondegenerate in P ( I ( X ) 2 ) \mathbb {P}\left ( I(X)_2 \right ) . In this paper we study geometric structures of Φ 3 ( X , L ) \Phi _3 (X,L) such as its minimal irreducible decomposition. Let Σ ( X , L ) = { ( A , B ) ∣ A , B ∈ P i c ( X ) , L = A 2 ⊗ B , h 0 ( X , A ) ≥ 2 , h 0 ( X , B ) ≥ 1 } . \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 ) ⊂ Φ 3 ( X , L ) W(A,B) \subset \Phi _3 (X,L) for each ( A , B ) (A,B) in Σ ( X , L ) \Sigma (X,L) . Then we prove that the equality Φ 3 ( X , L ) = ⋃ ( A , B ) ∈ Σ ( X , L ) W ( A , B ) \begin{equation*} \Phi _3 (X,L) ~=~ \bigcup _{(A,B) \in \Sigma (X,L)} W(A,B) \end{equation*} holds when X X is locally factorial. Thus this is an irreducible decomposition of Φ 3 ( X , L ) \Phi _3 (X,L) when P i c ( X ) {Pic} (X) is finitely generated and hence Σ ( X , L ) \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 Φ 3 ( X , L ) \Phi _3 (X,L) if P i c ( X ) {Pic}(X) is generated by a very ample line bundle.