2005/04/28 by Vincenzo Di Gennaro, Di Gennaro, Vincenzo
Computer Science · Engineering · Mathematics · #14H99 #14M05 #14N15 #14N30 #Advanced Numerical Analysis Techniques #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Polynomial and algebraic computation #math.AG #msc:14H99 #msc:14M05 #msc:14N15 #msc:14N30
paper · pdf · doi:10.48550/arxiv.math/0504576
10 pages, revised version, to appear in Proc. Amer. Math. Soc
openalex publication_date 2005/04/28 · arxiv created 2007/06/23 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Fix integers r,s1,...,sl such that 1≤ l≤ r-1 and sl≥ r-l+1, and let \Cal C(r;s1,...,sl) be the set of all integral, projective and nondegenerate curves C of degree s1 in the projective space \bold Pr, such that, for all i=2,...,l, C does not lie on any integral, projective and nondegenerate variety of dimension i and degree <si. We say that a curve C satisfies the \itflag condition (r;s1,...,sl) if C belongs to \Cal C(r;s1,...,sl). Define G(r;s1,...,sl)=max\pa(C): C∈ \Cal C(r;s1,...,sl) \, where pa(C) denotes the arithmetic genus of C. In the present paper, under the hypothesis s1>>...>>sl, we prove that a curve C satisfying the flag condition (r;s1,...,sl) and of maximal arithmetic genus pa(C)=G(r;s1,...,sl) must lie on a unique flag such as C=Vs11⊂ Vs22⊂ ... ⊂ Vsll⊂ \bold Pr, where, for any i=1,...,l, Vsii denotes an integral projective subvariety of \bold Pr of degree si and dimension i, such that its general linear curve section satisfies the flag condition (r-i+1;si,...,sl) and has maximal arithmetic genus G(r-i+1;si,...,sl). This proves the existence of a sort of a hierarchical structure of the family of curves with maximal genus verifying flag conditions.