Secant planes of projective curves embedded by nonspecial line bundles

ABSTRACT

For a nonspecial line bundle ℒ on a smooth curve X we consider a presentation ℒ≃𝒦_X−D+E which is minimal with respect to deg E. If ℒ is very ample, then this minimality means that any n-points of φ_ℒ(X) with n≤deg E−1 are in general position while φ_ℒ(E) spans a (deg E−2)-plane. In this work, we investigate conditions on D and E for ℒ≃𝒦_X−D+E to be minimal. We also observe s-secant (s−k−1)-planes which are minimal with respect to the secant degree s for a given k. We apply minimal presentations to problems about the exactness of Green-Lazarsfeld’s conjecture on property (N_p).

KEYWORDS:

• Algebraic curve
• linear system
• nonspecial line bundle
• secant plane

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 14C20
• 14H51
• 14E25
• 16E05

Acknowledgment

The author thanks KIAS for warm hospitality when she was an associate member in KIAS.