Non-trivial Linear Systems on Smooth Plane Curves

1993/01/06 by Marc Coppens, Coppens, Marc, Takao Kato +1 · 1 citation
Computer Science · Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Polynomial and algebraic computation #alg-geom #math.AG

paper · pdf · doi:10.48550/arxiv.alg-geom/9301003

15 pages, LaTeX 2.09

arxiv created 1993/01/06 · openalex publication_date 1993/01/06 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let C be a smooth plane curve of degree d defined over an algebraically closed field k. A base point free complete very special linear system g^r_n on C is trivial if there exists an integer m≥ 0 and an effective divisor E on C of degree md-n such that g^r_n=|mg²_d-E| and r=(m²+3m)/2-(md-n). In this paper, we prove the following: Theorem Let g^r_n be a base point free very special non-trivial complete linear system on C. Write r=(x+1)(x+2)/2-b with x, b integers satisfying x≥ 1, 0≤ b ≤ x. Then n≥ n(r):=(d-3)(x+3)-b. Moreover, this inequality is best possible.

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