On rank 3 quadratic equations of projective varieties. (English) Zbl 07876013
Summary: 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 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 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
\[\Sigma (X,L) \!=\! \{ (A,B) \mid A,B \!\in \! \text{Pic}(X),~L \!=\! A^2 \otimes B,~h^0 (X,A) \!\geq \! 2,~h^0 (X,B) \!\geq \! 1 \}.\]
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
\[\Phi_3 (X,L) ~=~ \bigcup_{(A,B) \in \Sigma (X,L)} W(A,B)\] holds when \(X\) is locally factorial. Thus this is an irreducible decomposition of \(\Phi_3 (X,L)\) when \(\text{Pic} (X)\) 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 \(\text{Pic}(X)\) is generated by a very ample line bundle.
MSC:
| 14E25 | Embeddings in algebraic geometry |
| 13C05 | Structure, classification theorems for modules and ideals in commutative rings |
| 14M15 | Grassmannians, Schubert varieties, flag manifolds |
| 14A10 | Varieties and morphisms |
Keywords:
rank of quadratic equation; low rank loci; minimal irreducible decomposition; Veronese embeddingReferences:
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