Unirationality and R-equivalence for conic bundles over quasi-finite fields

2024/10/25 by Boughattas, Elyes
#14D06 #14E08 (Primary) 14D10 #14G05 #14G12 #14H30 (Secondary) #14M20 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2410.19686

Abstract

Yanchevskiĭ had asked whether conic bundle surfaces over P¹_k are unirational when k is a finite field. We give a partial answer to his question by showing that for quasi-finite fields k (e.g. finite fields) a regular conic bundle X over P¹_k is unirational if all non-split fibres lie over rational points. For large finite fields k, this beats a previous result of Mestre. Under the same assumption, we also prove that all rational points of X are R-equivalent.

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