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.
In: Journal of Algebra, Vol. 351, No. 1, 01.02.2012, p. 175-184.Research output: Contribution to journal › Article
}
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
AN - SCOPUS:84455205531
VL - 351
SP - 175
EP - 184
JO - Journal of Algebra
JF - Journal of Algebra
SN - 0021-8693
IS - 1
ER -