From Infinite to Finite: Rational Reductions of del Pezzo Surfaces

Simple item page
dc.contributor.advisorVarilly-Alvarado, Anthonyen_US
dc.creatorSpaulding, Zac Jamesen_US
dc.date.accessioned2024-08-30T15:55:08Zen_US
dc.date.available2024-08-30T15:55:08Zen_US
dc.date.created2024-08en_US
dc.date.issued2024-08-08en_US
dc.date.submittedAugust 2024en_US
dc.date.updated2024-08-30T15:55:09Zen_US
dc.description.abstractIt 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.en_US
dc.format.mimetypeapplication/pdfen_US
dc.identifier.citationSpaulding, Zac James. From Infinite to Finite: Rational Reductions of del Pezzo Surfaces. (2024). PhD diss., Rice University. https://hdl.handle.net/1911/117770en_US
dc.identifier.urihttps://hdl.handle.net/1911/117770en_US
dc.language.isoengen_US
dc.rightsCopyright 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.en_US
dc.subjectAlgebraic geometryen_US
dc.subjectdel Pezzo surfacesen_US
dc.subjectNumber theoryen_US
dc.subjectRationalityen_US
dc.titleFrom Infinite to Finite: Rational Reductions of del Pezzo Surfacesen_US
dc.typeThesisen_US
dc.type.materialTexten_US
thesis.degree.departmentMathematicsen_US
thesis.degree.disciplineNatural Sciencesen_US
thesis.degree.grantorRice Universityen_US
thesis.degree.levelDoctoralen_US
thesis.degree.nameDoctor of Philosophyen_US
Files
Original bundle
Now showing 1 - 1 of 1
Thumbnail Image
Name:
SPAULDING-DOCUMENT-2024.pdf
Size:
819.63 KB
Format:
Adobe Portable Document Format
Download
License bundle
Now showing 1 - 2 of 2
No Thumbnail Available
Name:
PROQUEST_LICENSE.txt
Size:
5.84 KB
Format:
Plain Text
Description:
Download
No Thumbnail Available
Name:
LICENSE.txt
Size:
2.98 KB
Format:
Plain Text
Description:
Download
Collections
Rice University Theses and Dissertations