Topic: Derived Geometry and Stacks

# Motivation for Stacks

- What we would really like is a quasi-projective variety \({\mathcal{M}_g}\) with a universal family \(U_{g} \rightarrow {\mathcal{M}_g}\) such that the fiber of a point \([C] \in {\mathcal{M}_g}\) is precisely that curve. This is where the difficulty lies-automorphisms of curves obstruct the existence of such a family-and this is the main reason we want to expand our notion of a geometric space from schemes to algebraic stacks. Algebraic stacks provide a nice approach ensuring the existence of a universal family but it is by no means the only approach.

# Problems with representability

# Enlarging the category of schemes

# Misc

# Prestacks

# Misc