In algebraic geometry, a stable curve is an algebraic curve that is asymptotically stable in the sense of geometric invariant theory.
This is equivalent to the condition that it is a complete connected curve whose only singularities are ordinary double points and whose automorphism group is finite.
The condition that the automorphism group is finite can be replaced by the condition that it is not of arithmetic genus one and every non-singular rational component meets the other components in at least 3 points (Deligne & Mumford 1969).
A semi-stable curve is one satisfying similar conditions, except that the automorphism group is allowed to be reductive rather than finite (or equivalently its connected component may be a torus).
Alternatively the condition that non-singular rational components meet the other components in at least three points is replaced by the condition that they meet in at least two points.
Similarly a curve with a finite number of marked points is called stable if it is complete, connected, has only ordinary double points as singularities, and has finite automorphism group.
Over the complex numbers, a connected curve is stable if and only if, after removing all singular and marked points, the universal covers of all its components are isomorphic to the unit disk.
such that the geometric fibers are reduced, connected 1-dimensional schemes
such that These technical conditions are necessary because (1) reduces the technical complexity (also Picard-Lefschetz theory can be used here), (2) rigidifies the curves so that there are no infinitesimal automorphisms of the moduli stack constructed later on, and (3) guarantees that the arithmetic genus of every fiber is the same.
Note that for (1) the types of singularities found in Elliptic surfaces can be completely classified.
This example can be generalized to the case of a one-parameter family of smooth hyperelliptic curves degenerating at finitely many points.
In the general case of more than one parameter care has to be taken to remove curves which have worse than double-point singularities.
One of the most important properties of stable curves is the fact that they are local complete intersections.