Polarized K3 surfaces of genus thirteen and curves of genus three

2023/10/03 by Akihiro Kanemitsu, Kanemitsu, Akihiro, Shigeru Mukai +1
Mathematics · #14H60 #14J28 #14J60 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2310.02078

openalex publication_date 2023/10/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We describe a general (primitively) polarized K3 surface (S,h) with (h²)=24 as a complete intersection variety with respect to vector bundles on the 6-dimensional moduli space N^- of the stable vector bundles of rank two with fixed odd determinant on a curve C of genus 3. If the curve C is hyperelliptic, then N^- is a subvariety of the 12-dimensional Grassmann variety Gr(C⁸,2) defined by a pencil of quadric forms. In this case, our description implies that a general (S,h) is the intersection of two (7-dimensional) contact homogeneous varieties of Spin(7) in the Grassmann variety Gr(C⁸,2).

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