Date of Graduation
5-2024
Document Type
Dissertation
Degree Name
Doctor of Philosophy in Mathematics (PhD)
Degree Level
Graduate
Department
Mathematical Sciences
Advisor/Mentor
Lance E. Miller
Committee Member
Matthew B. Day
Second Committee Member
Wenbo Niu
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.
Citation
Walker, E. (2024). Derived Jet Schemes and Arc Spaces, and Arithmetic Arc Space Representability. Graduate Theses and Dissertations Retrieved from https://scholarworks.uark.edu/etd/5296