2024/02/01 by Francesco Bastianelli, Bastianelli, Francesco, Nicola Picoco +1 · 1 citation
Mathematics · Social Sciences · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Analysis and Curvature Flows #Vietnamese History and Culture Studies
paper · pdf · doi:10.48550/arxiv.2402.00753
openalex publication_date 2024/02/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper is a sequel of arXiv:2208.00990. Let C be a smooth complex projective curve of genus g and let C(k) be its k-fold symmetric product. The covering gonality of C(k) is the least gonality of an irreducible curve E⊂ C(k) passing through a general point of C(k). It follows from previous works of the authors that if 2≤ k≤ 4 and g≥ k+4, the covering gonality of C(k) equals the gonality of C. In this paper, we prove that under mild assumptions of generality on C, the only curves E⊂ C(k) computing the covering gonality of C(k) are copies of C of the form C+p, for some point p∈ C(k-1). As a byproduct, we deduce that the connecting gonality of C(k) (i.e. the least gonality of an irreducible curve E⊂ C(k) connecting two general points of C(k)) is strictly larger than the covering gonality.