Stacks and Moduli

• Jarod Alper’s course: https://sites.math.washington.edu/~jarod/math582C.html
• Martin Olsson’s, Algebraic spaces and stacks.
• What is a stack?, by Dan Edidin.
• Equivariant geometry and the cohomology of the moduli space of curves, by Dan Edidin.
• Notes on the construction of the moduli space of curves, by Dan Edidin.
• Stacks for Everybody, by Barbara Fantechi.
• Picard groups of moduli problems, by David Mumford, and Daniel Litt’s exposition thereof (parts one and two).
• Notes on Grothendieck topologies, fibered categories and descent theory, by Angelo Vistoli (lovely comprehensive reference).
• More notes

Definitions

• What is the blowup of a scheme? The blowdown of a scheme?
• What is a site?
• What is a topos?
• What is the Hilbert scheme?
• Topologies:
  • What is the etale topology?
  • What is the fppf topology?
• What is the etale site?
• What is an algebraic space?
• What is Faithfully flat descent?
• What is a torsor?
• What is the Amtisur complex?
• What does it mean to have a closed subcategory?
• What does it mean for a subcategory to be local on the base? local on the domain?
• What does it mean to have a stable subcategory?
• What is an etale equivalence relation?
• What is a locally closed substack?
• What is an orbifold?
• What is a smooth stack?

Results

• What is the local criterion for flatness?
• What is the infinitesimal criterion for flatness?

Problems

• Representable functors:
  • What is ${\mathbb{A}}^n_{/ {k}} $ as a functor? What is it represented by?
    • For schemes with structure sheaves taking values in \(R{\hbox{-}}\)algebras, \begin{align*} {\mathbb{A}}^n_{/ {{-}}} \coloneqq{{\Gamma}\qty{({-}); {\mathcal{O}}_{{-}}} }{ {}^{ \scriptscriptstyle\times^{n} } }: {\mathsf{Sch}}^{\operatorname{op}}\to {\mathsf{Alg}}_{/ {R}} \\ X &\mapsto {{\Gamma}\qty{X; {\mathcal{O}}_X} }\carptpower{n} \\ f &\mapsto f^* ,\end{align*} which is represented by \(\operatorname{Spec}{\mathbb{Z}}[x_0, \cdots, x_n]\).
  • What is ${\mathbb{P}}^n_{/ {k}} $ as a functor? What is it represented by?
  • What is \({\mathbb{A}}^n_{/ {k}} \setminus\left\{{0}\right\}\) as a functor?
    • \(x\mapsto \mathbf{x} = {\left[ {x_1, \cdots, x_n} \right]}\) where not all of the \(x_i\) are zero.
• Show that if $X \subseteq {\mathbb{P}}^n_{/ {k}} $ is a closed subscheme and \({\mathcal{F}}\) is a coherent sheaf of \({\mathcal{O}}_X{\hbox{-}}\)modules, then \(\Gamma(X, {\mathcal{F}})\) is a finite dimensional \(k{\hbox{-}}\)vector space.
• Construct a ring \(R\) such that \(\operatorname{Spec}R\) is ${\mathbb{A}}^1_{/ {k}} $ punctured at \(0, 1\).
  • \(R = k[x] \left[ { \scriptstyle { { \qty{ x(x-1)} }^{-1}} } \right]\)
• Show that the cohomology of a quasicoherent sheaf on an affine scheme vanishes.