Logarithmic good reduction and the index

2017/11/30 by Mitsui, Kentaro, Smeets, Arne
#14D06 #14G20 #14H10 #14H25 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1711.11547

Abstract

Let K be the fraction field of a complete discrete valuation ring, with algebraically closed residue field of characteristic p > 0. This paper studies the index of a smooth, proper K-variety X with logarithmic good reduction. We prove that it is prime to p in `most' cases, for example if the Euler number of X does not vanish, but (perhaps surprisingly) not always. We also fully characterise curves of genus 1 with logarithmic good reduction, thereby completing classical results of T. Saito and Stix valid for curves of genus at least 2.

Related

• Logarithmic good reduction of abelian varieties
• The specialization index of a variety over a discretely valued field
• On the indices of curves over local fields
• Log smoothness and polystability over valuation rings
• Reduction and isogenies of elliptic curves