Pointed algebraic curve

Definition

A pointed algebraic curve over a scheme $S$ is a family of algebraic curves $\pi:X\to S$ together with specified sections $x_1,\ldots,x_n:S\to X$ .

Over a field, pairwise distinct rational marked points determine the reduced effective divisor

D=x_1+\cdots+x_n

and the complement punctured curve $U=X\setminus D$ .

The sections may be ordered, labeled, or merely collected into a divisor; these are different moduli problems. Stability is not part of the bare definition.

Additional hypotheses

Common additional hypotheses require the marked sections to be pairwise disjoint or to pass through the smooth locus.

Langlands role

Ramified geometric Langlands studies local systems on $U$ with specified behavior around $D$ , and $G$ -bundles on $X$ with compatible level or parabolic structure at the marked points.

References

Pierre Deligne and David Mumford, “The irreducibility of the space of curves of given genus,” Publications Mathématiques de l’IHÉS 36 (1969), 75–109. DOI.