Punctured algebraic curve

Definition

Let $X$ be an algebraic curve over a field $k$ , and let $D\subset X$ be a finite reduced closed subscheme. The open subscheme

U=X\setminus D

is the punctured algebraic curve obtained by removing $D$ .

If $X$ is smooth and $D=\{x_1,\ldots,x_n\}$ consists of distinct marked points, then the pointed curve $(X;x_1,\ldots,x_n)$ determines $U$ . The pointed curve retains the marked sections and their labels, while $U$ generally does not.

Langlands role

Ramified local systems live on $U$ together with conditions describing their behavior around the missing divisor $D$ . Those boundary conditions are extra data, not part of the definition of the punctured curve.

References

Pierre Deligne, Équations différentielles à points singuliers réguliers, Lecture Notes in Mathematics 163, Springer, 1970. DOI.