General existence theorems in moduli theory
Abstract/Contents
- Abstract
- In this thesis, we prove that there is an algebraic stack parameterizing all curves. The curves that appear in this algebraic stack are allowed to be arbitrarily singular, non-reduced, disconnected, and reducible. We also prove the boundedness of the open substack parameterizing reduced and connected curves with fixed arithmetic genus g and at most e irreducible components. We also show that for essentially any algebraic stack, there is an algebraic stack, the Hilbert stack, parameterizing quasi-finite maps to the stack. The technical heart of this result is a generalization of formal GAGA to a non-separated morphism of algebraic stacks, something that was previously unknown for a morphism of schemes. We also employ derived algebraic geometry, in an essential way, to prove the algebraicity of the Hilbert stack. The Hilbert stack, for algebraic spaces, was claimed to exist by M. Artin (1974), but was left unproved due to a lack of foundational results for non-separated algebraic spaces. Finally, we generalize the fundamental GAGA results of J. P. Serre (1956) in three ways---to the non-separated setting, to stacks, and to families. As an application of these results, we show that analytic compactifications of the moduli stack of smooth curves possessing modular interpretations are algebraizable.
Description
| Type of resource | text |
|---|---|
| Form | electronic; electronic resource; remote |
| Extent | 1 online resource. |
| Publication date | 2011 |
| Issuance | monographic |
| Language | English |
Creators/Contributors
| Associated with | Hall, Jack Kingsbury | |
|---|---|---|
| Associated with | Stanford University, Department of Mathematics | |
| Primary advisor | Vakil, Ravi | |
| Thesis advisor | Vakil, Ravi | |
| Thesis advisor | Conrad, Brian, 1970- | |
| Thesis advisor | Li, Jun, (Mathematician) | |
| Advisor | Conrad, Brian, 1970- | |
| Advisor | Li, Jun, (Mathematician) |
Subjects
| Genre | Theses |
|---|
Bibliographic information
| Statement of responsibility | Jack Hall. |
|---|---|
| Note | Submitted to the Department of Mathematics. |
| Thesis | Thesis (Ph.D.)--Stanford University, 2011. |
| Location | electronic resource |
Access conditions
- Copyright
- © 2011 by Jack Kingsbury Hall
- License
- This work is licensed under a Creative Commons Attribution Non Commercial 3.0 Unported license (CC BY-NC).
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