## Abstract

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).

Original language | English |
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Pages (from-to) | 175-184 |

Number of pages | 10 |

Journal | Journal of Algebra |

Volume | 351 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2012 Feb 1 |

### Bibliographical note

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.

## Keywords

- Green-Lazarsfeld index
- Minimal free resolution
- Primary

## ASJC Scopus subject areas

- Algebra and Number Theory