On Syzygies of Projected Algebraic Curves

Abstract

Let C ⊂ ℙ ^r 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 N _p . Recently it is known that for any q ∈ ℙ ^r ∖C such that the linear projection π _q : C → ℙ ^r−1 of C from q is an embedding, the projected image C _q : = π _q (C) ⊂ ℙ ^r−1 is 3-regular, and hence its homogeneous ideal is generated by quadratic and cubic equations. In this article we study the problem when C _q 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.

Key Words:

• Linear projection
• Minimal free resolution
• Projective curve

2010 Mathematics Subject Classification:

• Primary 14N15, 13D02
• Secondary 51N35

ACKNOWLEDGMENT

Wanseok Lee was supported by the second stage of BK21 Project. Euisung Park was supported by the Korea Reach Foundation Grant funded by the Korea government (KRF-2008-331-C00013).

Notes

Communicated by L. Ein.