ON RANK 3 QUADRATIC EQUATIONS OF PROJECTIVE VARIETIES

초록

Let X ⊂ Pr 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 Φ3(X, L) of rank 3 quadratic equations of X in P (I(X, L)2) is a projective algebraic set, and property QR(3) of (X, L) is equivalent to that Φ3(X) is nondegenerate in P (I(X)2). In this paper we study geometric structures of Φ3(X, L) such as its minimal irreducible decomposition. Let Σ(X, L) = {(A, B) | A, B ∈Pic(X), L= A2 × B, h0(X, A) ≥2, h0(X, B) ≥1}. We first construct a projective subvariety W(A, B) ⊂ Φ3(X, L) for each (A, B) in Σ(X, L). Then we prove that the equality ∪ Φ3(X, L) = W(A, B) (A,B)∈Σ(X,L) holds when X is locally factorial. Thus this is an irreducible decomposition of Φ3(X, L) when Pic(X) is finitely generated and hence Σ(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 Φ3(X, L) if Pic(X) is generated by a very ample line bundle. © 2023 American Mathematical Society.

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