Weak approximation for Fano complete intersections in positive characteristic

2018/11/06 by Starr, Jason Michael, Tian, Zhiyu, Zong, Runhong · 1 citation
#14G05 #14J45 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1811.02466

Abstract

For a smooth curve B over an algebraically closed field k, for every B-flat complete intersection X_B in B×_{Spec k} ℙⁿ_k of type (d₁,…,d_c), if the Fano index is ≥ 2 and if char(k)>max(d₁,…,d_c), we prove weak approximation of \widehatO_B,b-points of X_B by k(B)-points at all places of (strong) potentially good reduction, including all places of good reduction. The key step is the proof that such complete intersections are separably uniruled by lines, and even separably rationally connected, whenever smooth. We prove that the inequality is close to sharp. We prove a similar theorem for Fano manifolds of Picard number 1 and Fano index 1.

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• Separable rational connectedness and k-plane sections of hypersurfaces

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