What are the odds that a conic bundle over a finite field is rational?

2026/07/17 by Amanda Hernandez · 1 citation
#math.AG

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Abstract

We study rationality of conic bundles over finite fields defined over the line. In particular, for a fixed number n of degenerate fibers, what is the proportion of good conic bundles that are rational over a finite field of order q? The Galois action on their Picard groups factors through the Weyl group W(D_n), reducing rationality to a condition on signed permutation types. Using work of Colliot-Thélène and Yang together with Chebotarev density, we compute the asymptotic proportion of such conic bundles that are rational as q goes to infinity.

Citations

• Moduli of conic surfaces over the line

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• Moduli of conic surfaces over the line

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