2024/09/30 by Ito, Tetsushi, Kanemitsu, Akihiro, Takamatsu, Teppei +1 · 1 citation
#11G35 #14C34 #14J45 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2409.20046
We study arithmetic finiteness of prime Fano threefolds of genus 7 and their higher dimensional generalization, called Mukai varieties of genus 7. For prime Fano threefolds of genus 7, we provide an arithmetic refinement of the Torelli theorem, obtain Shafarevich-type finiteness results, and show the failure of the Néron--Ogg--Shafarevich criterion of good reduction. For Mukai varieties of genus 7, we prove that Shafarevich-type finiteness results hold in dimensions 9 and 10, but fail in dimension 6. In addition, we show that Mukai n-folds of genus 7 over ℤ do not exist for n ≤ 4, whereas they exist for 5 ≤ n ≤ 10.