Date of Graduation

5-2024

Document Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematics (PhD)

Degree Level

Graduate

Department

Mathematical Sciences

Advisor/Mentor

Miller, Lance E.

Committee Member

Day, Matthew B.

Second Committee Member

Niu, Wenbo

Keywords

Jet schemes; Arc spaces; derivations

Abstract

Associated to a given scheme X one can define geometric and arithmetic notions of jet schemes and arc spaces. We develop a construction of the geometric jets and arcs in the setting of derived schemes and explore consequences thereof. In particular, we prove an analogous theorem to that of one by Tommaso de Fernex and Roi Docampo concerning the cotangent sheaves of geometric jets and arcs. Our version then produces many subsequent results which allow us to prove stronger versions of results concerning geometric jets and arcs by removing unnecessary hypotheses. Separately, we explore evidence as to why, in contrast to the geometric setting, the arithmetic arc space is not in general a scheme. The arc space is a natural limit of the jet schemes, and we show that this limit need not be a scheme in the arithmetic setting. The arc space also has a highly nontrivial description as a certain representing object of a representable functor, and we show that this functor need not be representable in the arithmetic setting.

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