Browsing by Author "Spaulding, Zac James"

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    From Infinite to Finite: Rational Reductions of del Pezzo Surfaces
    (2024-08-08) Spaulding, Zac James; Varilly-Alvarado, Anthony
    It is well-known that all del Pezzo surfaces of degree at least 5 over a finite field are rational, i.e., birational to the projective plane, but this is generally not true for those of lower degree. If we fix a del Pezzo surface X of degree d < 5, defined over a number field k, and consider the primes p of k of good reduction for X, then we may ask: how often do we expect X_p, the reduction of X modulo p, to be rational? To answer this question, we combine a result of Colliot-Thélène from 2019 with the Chebotarev Density Theorem to determine the natural density of the set \pi_{rat}(X,k) --- the set of primes of k for which the reduction X_p is F_p-rational --- in the set of all primes of k. We present an algorithm to determine this natural density with input data being the action of the absolute Galois group of k on the geometric Picard group. We implement this algorithm in magma, exhibiting the nonzero uniform lower bound 1/1920 for this natural density, independent of starting data.