On Syzygies of Projected Algebraic Curves

Wanseok Lee, Euisung Park

Research output: Contribution to journalArticle

Abstract

Let C ⊂ ℙr be a linearly normal projective integral curve of arithmetic genus g ≥ 1 and degree d = 2g + 1 + p for some p ≥ 1. It is well known that C is cut out by quadric and satisfies Green-Lazarsfeld's property Np. Recently it is known that for any q ∈ ℙr\C such that the linear projection πq: C → ℙr-1 of C from q is an embedding, the projected image Cq:= πq(C) ⊂ ℙr-1 is 3-regular, and hence its homogeneous ideal is generated by quadratic and cubic equations. In this article we study the problem when Cq is still cut out by quadrics. Our main result in this article shows that if the relative location of q with respect to C is general then the homogeneous ideal of C q is still generated by quadrics and the syzygies among them are generated by linear syzygies for the first a few steps.

Original languageEnglish
Pages (from-to)2092-2099
Number of pages8
JournalCommunications in Algebra
Volume41
Issue number6
DOIs
Publication statusPublished - 2013 May 1

Fingerprint

Syzygies
Algebraic curve
Quadric
Linear Projection
Cubic equation
Quadratic equation
Genus
Linearly
Curve

Keywords

  • Linear projection
  • Minimal free resolution
  • Projective curve

ASJC Scopus subject areas

  • Algebra and Number Theory

Cite this

On Syzygies of Projected Algebraic Curves. / Lee, Wanseok; Park, Euisung.

In: Communications in Algebra, Vol. 41, No. 6, 01.05.2013, p. 2092-2099.

Research output: Contribution to journalArticle

Lee, Wanseok ; Park, Euisung. / On Syzygies of Projected Algebraic Curves. In: Communications in Algebra. 2013 ; Vol. 41, No. 6. pp. 2092-2099.
@article{aee1d50c1f25407baaa6bfae07a7be57,
title = "On Syzygies of Projected Algebraic Curves",
abstract = "Let C ⊂ ℙr be a linearly normal projective integral curve of arithmetic genus g ≥ 1 and degree d = 2g + 1 + p for some p ≥ 1. It is well known that C is cut out by quadric and satisfies Green-Lazarsfeld's property Np. Recently it is known that for any q ∈ ℙr\C such that the linear projection πq: C → ℙr-1 of C from q is an embedding, the projected image Cq:= πq(C) ⊂ ℙr-1 is 3-regular, and hence its homogeneous ideal is generated by quadratic and cubic equations. In this article we study the problem when Cq is still cut out by quadrics. Our main result in this article shows that if the relative location of q with respect to C is general then the homogeneous ideal of C q is still generated by quadrics and the syzygies among them are generated by linear syzygies for the first a few steps.",
keywords = "Linear projection, Minimal free resolution, Projective curve",
author = "Wanseok Lee and Euisung Park",
year = "2013",
month = "5",
day = "1",
doi = "10.1080/00927872.2011.653464",
language = "English",
volume = "41",
pages = "2092--2099",
journal = "Communications in Algebra",
issn = "0092-7872",
publisher = "Taylor and Francis Ltd.",
number = "6",

}

TY - JOUR

T1 - On Syzygies of Projected Algebraic Curves

AU - Lee, Wanseok

AU - Park, Euisung

PY - 2013/5/1

Y1 - 2013/5/1

N2 - Let C ⊂ ℙr be a linearly normal projective integral curve of arithmetic genus g ≥ 1 and degree d = 2g + 1 + p for some p ≥ 1. It is well known that C is cut out by quadric and satisfies Green-Lazarsfeld's property Np. Recently it is known that for any q ∈ ℙr\C such that the linear projection πq: C → ℙr-1 of C from q is an embedding, the projected image Cq:= πq(C) ⊂ ℙr-1 is 3-regular, and hence its homogeneous ideal is generated by quadratic and cubic equations. In this article we study the problem when Cq is still cut out by quadrics. Our main result in this article shows that if the relative location of q with respect to C is general then the homogeneous ideal of C q is still generated by quadrics and the syzygies among them are generated by linear syzygies for the first a few steps.

AB - Let C ⊂ ℙr be a linearly normal projective integral curve of arithmetic genus g ≥ 1 and degree d = 2g + 1 + p for some p ≥ 1. It is well known that C is cut out by quadric and satisfies Green-Lazarsfeld's property Np. Recently it is known that for any q ∈ ℙr\C such that the linear projection πq: C → ℙr-1 of C from q is an embedding, the projected image Cq:= πq(C) ⊂ ℙr-1 is 3-regular, and hence its homogeneous ideal is generated by quadratic and cubic equations. In this article we study the problem when Cq is still cut out by quadrics. Our main result in this article shows that if the relative location of q with respect to C is general then the homogeneous ideal of C q is still generated by quadrics and the syzygies among them are generated by linear syzygies for the first a few steps.

KW - Linear projection

KW - Minimal free resolution

KW - Projective curve

UR - http://www.scopus.com/inward/record.url?scp=84878562417&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84878562417&partnerID=8YFLogxK

U2 - 10.1080/00927872.2011.653464

DO - 10.1080/00927872.2011.653464

M3 - Article

AN - SCOPUS:84878562417

VL - 41

SP - 2092

EP - 2099

JO - Communications in Algebra

JF - Communications in Algebra

SN - 0092-7872

IS - 6

ER -