On the rationality of quadric surface bundles

Paulsen, Matthias

doi:10.5802/aif.3399

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Cited by 2 publications

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“…Rationality of conic bundles has been extensively studied and we have a precise conjectural rationality statement, we refer to [Pro18] for a comprehensive survey. Moreover, quadric bundles have recently received great attention especially concerning stable rationality [HKT16], [AO18], [Sch18], [BvB18], [HPT18], [HT19], [Sch19], [Pau21], [NO22]. We recall that a variety X is stably rational if X × P m k is rational for some m ≥ 0.…”

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confidence: 99%

On the unirationality of quadric bundles

Massarenti¹

2022

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We prove that a general n-fold quadric bundle Q n−1 → P 1 , over a number field, with anti-canonical divisor of positive volume and discriminant of odd degree δ Q n−1 is unirational. Furthermore, the same holds for quadric bundles over an arbitrary infinite field provided that Q n−1 has a point, is otherwise general and n ≤ 5. We also give similar results for quadric bundles over higher dimensional projective spaces defined over an arbitrary field. For instance, we prove the unirationality of a general n-fold quadric bundle Q h → P n−h with discriminant of odd degree δ Q h ≤ 3h + 4, and of any smooth 4-fold quadric bundle Q 2 → P 2 , over an algebraically closed field, with δ Q 2 ≤ 12.

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