2022/08/26 by Park, Euisung · 2 citations
#Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2208.12481
Let X ⊂ ¶r be a linearly normal variety defined by a very ample line bundle L on a projective variety X. Recently it is shown in \citeHLMP that there are many cases where (X,L) satisfies property \textsfQR (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 ¶( I(X,L)2 ) is a projective algebraic set, and property \textsfQR (3) of (X,L) is equivalent to that Φ3 (X) is nondegenerate in ¶( 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 ∈ \rm 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) ~=~ \bigcup(A,B) ∈ Σ(X,L) W(A,B) holds when X is locally factorial. Thus this is an irreducible decomposition of Φ3 (X,L) when \rm 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 \rm Pic(X) is generated by a very ample line bundle.