The Stacks project

111.9 Examples of schemes, algebraic spaces, algebraic stacks

The Stacks project currently contains two chapters discussing moduli stacks and their properties, see Moduli Stacks, Section 106.1 and Moduli of Curves, Section 107.1. Over time we intend to add more, for example:

  1. $\mathcal{A}_ g$, i.e., principally polarized abelian schemes of genus $g$,

  2. $\mathcal{A}_1 = \mathcal{M}_{1, 1}$, i.e., $1$-pointed smooth projective genus $1$ curves,

  3. $\mathcal{M}_{g, n}$, i.e., smooth projective genus $g$-curves with $n$ pairwise distinct labeled points,

  4. $\overline{\mathcal{M}}_{g, n}$, i.e., stable $n$-pointed nodal projective genus $g$-curves,

  5. $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _ S(\mathcal{X}, \mathcal{Y})$, moduli of morphisms (with suitable conditions on the stacks $\mathcal{X}$, $\mathcal{Y}$ and the base scheme $S$),

  6. $\textit{Bun}_ G(X) = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _ S(X, BG)$, the stack of $G$-bundles of the geometric Langlands programme (with suitable conditions on the scheme $X$, the group scheme $G$, and the base scheme $S$),

  7. $\mathcal{P}\! \mathit{ic}_{\mathcal{X}/S}$, i.e., the Picard stack associated to an algebraic stack over a base scheme (or space).

More generally, the Stacks project is somewhat lacking in geometrically meaningful examples.


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