TY - JOUR

T1 - Projective subvarieties having large Green-Lazarsfeld index

AU - Park, Euisung

N1 - Funding Information:
This work was supported by the Korea Science and Engineering Foundation (KOSEF) grant funded by the Korea government (No. 20090073305). The author thanks the referee for his/her encouragement to study the case index(X)= c − 3 in Section 4.

PY - 2012/2/1

Y1 - 2012/2/1

N2 - Let X⊂Pn+c be a nondegenerate projective irreducible subvariety of degree d and codimension c≥ 1. The Green-Lazarsfeld index of X, denoted by index(X), is defined as the largest p such that the homogeneous ideal of X is generated by quadrics and the syzygies among them are generated by linear syzygies until the (p-1)-th stage. Thus index(X) is an important invariant in order to describe the minimal free resolution of X. Recently it is shown that d= c+1 if and only if index(X) ≥ c, and X is a del Pezzo variety if and only if index( X) = c-1. In this paper, we prove that index(X) = c-2 (c≥3) if and only if X is either a complete intersection of three quadrics or else an arithmetically Cohen-Macaulay variety with d=c+3 (Theorem 1.1). Also we classify X with index(X) = c-3 (c≥ 4) for the cases when d= c+2 (Theorem 4.1) and when X is a smooth surface (Theorem 4.3).

AB - Let X⊂Pn+c be a nondegenerate projective irreducible subvariety of degree d and codimension c≥ 1. The Green-Lazarsfeld index of X, denoted by index(X), is defined as the largest p such that the homogeneous ideal of X is generated by quadrics and the syzygies among them are generated by linear syzygies until the (p-1)-th stage. Thus index(X) is an important invariant in order to describe the minimal free resolution of X. Recently it is shown that d= c+1 if and only if index(X) ≥ c, and X is a del Pezzo variety if and only if index( X) = c-1. In this paper, we prove that index(X) = c-2 (c≥3) if and only if X is either a complete intersection of three quadrics or else an arithmetically Cohen-Macaulay variety with d=c+3 (Theorem 1.1). Also we classify X with index(X) = c-3 (c≥ 4) for the cases when d= c+2 (Theorem 4.1) and when X is a smooth surface (Theorem 4.3).

KW - Green-Lazarsfeld index

KW - Minimal free resolution

KW - Primary

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U2 - 10.1016/j.jalgebra.2011.10.041

DO - 10.1016/j.jalgebra.2011.10.041

M3 - Article

AN - SCOPUS:84455205531

VL - 351

SP - 175

EP - 184

JO - Journal of Algebra

JF - Journal of Algebra

SN - 0021-8693

IS - 1

ER -