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Results tagged with moduli-spaces
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user 3927
Given a concrete category C, with objects denoted Obj(C), and an equivalence relation ~ on Obj(C) given by morphisms in C. The moduli set for Obj(C) is the set of equivalence classes with respect to ~; denoted Iso(C). When Iso(C) is an object in the category Top, then the moduli set is called a moduli space.
14
votes
When is a coarse moduli space also a fine moduli space?
Since nobody gave a reference yet, in my paper "Artithmetic moduli of generalized elliptic curves" I included a proof that an Artin stack whose geometric points have trivial automorphism schemes is ne …
- 7,108
21
votes
What is a good introductory text for moduli theory?
Read Katz-Mazur, "Arithmetic moduli of elliptic curves" (and for your purposes you can ignore the last chapter, even though it was their motivation for writing the book).
18
votes
Accepted
Non-representable functor, representable on locally Noetherian schemes?
Define $F(X) = {\rm{Hom}}_{\mathbf{C}}(X,{\rm{Spec}}(R/tR))$ where $R$ is the valuation ring of an algebraic closure of $\mathbf{C}((t))$. Note that every element of the maximal ideal of $R/tR$ is nil …
15
votes
Existence of fine moduli space for curves and elliptic curves
If you want to work over a base ring such as $\mathbf{Z}[1/n]$ rather than over $\mathbf{Q}$ or $\mathbf{C}$ then the relevant numerical condition is that the part of $N$ coprime to $n$ not be "too sm …
- 7,108
25
votes
Are Jacobians principally polarized over non-algebraically closed fields?
There's a more down to earth way to deal with this, which is already explained in Mumford's GIT: make an fppf (or even etale) surjective base change to acquire a section, use that to define the princ …
- 7,108