Stable curve

Stable curve

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 elliptic and every non-singular rational component meets the other components in at least 3 points harv|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 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.

ee also

*Moduli of algebraic curves

References

*Citation | last1=Deligne | first1=Pierre | author1-link=Pierre Deligne | last2=Mumford | first2=David | author2-link=David Mumford | title=The irreducibility of the space of curves of given genus | url=http://www.numdam.org/item?id=PMIHES_1969__36__75_0 | id=MathSciNet | id = 0262240 | year=1969 | journal=Publications Mathématiques de l'IHÉS | issn=1618-1913 | issue=36 | pages=75–109
*Citation | last1=Gieseker | first1=D. | title=Lectures on moduli of curves | publisher=Published for the Tata Institute of Fundamental Research, Bombay | series=Tata Institute of Fundamental Research Lectures on Mathematics and Physics | isbn=978-3-540-11953-1 | id=MathSciNet | id = 691308 | year=1982 | volume=69
*Citation | last1=Harris | first1=Joe | author1-link=Joe Harris | last2=Morrison | first2=Ian | title=Moduli of curves | publisher=Springer-Verlag | location=Berlin, New York | series=Graduate Texts in Mathematics | isbn=978-0-387-98438-4; 978-0-387-98429-2 | id=MathSciNet | id = 1631825 | year=1998 | volume=187


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