Stable curve
Encyclopedia
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 .
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 3 points is replaced by the condition that they meet in at least 2 points.
Similarly a curved with a finite number of marked points is called stable if is complete, connected, has only ordinary double points as singularities, and has finite automorphism group. For example an elliptic curve (a non-singular genus 1 curve with 1 marked point) is stable.
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 compoinents are isomorphic to the unit disk.
Algebraic curve
In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.- Plane algebraic curves...
that is asymptotically stable in the sense of geometric invariant theory
Geometric invariant theory
In mathematics Geometric invariant theory is a method for constructing quotients by group actions in algebraic geometry, used to construct moduli spaces...
.
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 .
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 3 points is replaced by the condition that they meet in at least 2 points.
Similarly a curved with a finite number of marked points is called stable if is complete, connected, has only ordinary double points as singularities, and has finite automorphism group. For example an elliptic curve (a non-singular genus 1 curve with 1 marked point) is stable.
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 compoinents are isomorphic to the unit disk.