Varilly-Alvarado, Anthony2024-08-302024-08-302024-082024-08-08August 202Spaulding, Zac James. From Infinite to Finite: Rational Reductions of del Pezzo Surfaces. (2024). PhD diss., Rice University. https://hdl.handle.net/1911/117770https://hdl.handle.net/1911/117770It 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.application/pdfengCopyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder.Algebraic geometrydel Pezzo surfacesNumber theoryRationalityThesis2024-08-30