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• Tags
  • #AG #higher-algebra/stacks #AG/moduli-spaces
• Refs:
  • #todo/add-references #todo/too-long
• Links:
  • stacks MOC
  • Unsorted/stacks MOC
  • moduli space
  • representable
  • elliptic curve
  • moduli stack of abelian varieties
  • derived moduli stack
  • Chow ring
  • level structure
  • cohomological field theory
  • Gromov-Witten

moduli stack of elliptic curves

Moduli of curves

Compatifications

Notes

The moduli space \({\mathcal{M}}_g\) is not representable in \({\mathsf{Sch}}\): if it were, then every isotrivial family of curves would be equivalent to a trivial family. Counterexample to this: take the family \begin{align*} X_t: \quad y^{2}=x^{3}+t, t \neq 0 \end{align*} All fibers are abstractly isomorphic by computing the monodromy action \(\pi_1({\mathbf{C}}^{\times}; 1)\curvearrowright H^1(X_1, {\mathbf{C}})\)

It does admit a coarse moduli space.

Teichmüller space is a cover of \(M_g\)?

Lattices

G structures

## Stable Curves

Stable Reduction

Homology and Smoothness

# Coarse vs fine

Elliptic curves

Stable curves and compactification

Contraction morphism

Gluing morphism

Graphs

Non-representability: nontrivial families

Deformations and the tangent space