### 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 |
---|---|

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 |

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### Keywords

- Green-Lazarsfeld index
- Minimal free resolution
- Primary

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

**Projective subvarieties having large Green-Lazarsfeld index.** / Park, Euisung.

Research output: Contribution to journal › Article

*Journal of Algebra*, vol. 351, no. 1, pp. 175-184. https://doi.org/10.1016/j.jalgebra.2011.10.041

}

TY - JOUR

T1 - Projective subvarieties having large Green-Lazarsfeld index

AU - Park, Euisung

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

UR - http://www.scopus.com/inward/record.url?scp=84455205531&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84455205531&partnerID=8YFLogxK

U2 - 10.1016/j.jalgebra.2011.10.041

DO - 10.1016/j.jalgebra.2011.10.041

M3 - Article

VL - 351

SP - 175

EP - 184

JO - Journal of Algebra

JF - Journal of Algebra

SN - 0021-8693

IS - 1

ER -