A speciality theorem for curves in \bold P⁵

2005/07/07 by Vincenzo Di Gennaro, Di Gennaro, Vincenzo, Davide Franco +1
Mathematics · #14H99 #14M10 #14N15 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Analysis and Curvature Flows #Meromorphic and Entire Functions #math.AG #msc:14H99 #msc:14M10 #msc:14N15

paper · pdf · doi:10.48550/arxiv.math/0507162

10 pages

arxiv created 2005/07/07 · openalex publication_date 2005/07/07 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

Let C⊂ \bold P^r be an integral projective curve. One defines the speciality index e(C) of C as the maximal integer t such that h⁰(C,ω_C(-t))>0, where ω_C denotes the dualizing sheaf of C. Extending a classical result of Halphen concerning the speciality of a space curve, in the present paper we prove that if C⊂ \bold P⁵ is an integral degree d curve not contained in any surface of degree < s, in any threefold of degree >s>>t>>u≥ 1, then e(C)≤ (d)/(s)+(s)/(t)+(t)/(u)+u-6. Moreover equality holds if and only if C is a complete intersection of hypersurfaces of degrees u, (t)/(u), (s)/(t) and (d)/(s). We give also some partial results in the general case C⊂ \bold P^r, r≥ 3.

Related

• A speciality Theorem for curves in ℙ⁵ contained in Noether-Lefschetz general fourfolds
• A remark on the intersection of plane curves
• Refining Castelnuovo-Halphen bounds
• On the genus of projective curves not contained in hypersurfaces of given degree
• On surfaces of maximal sectional regularity