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In
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, a stable curve is an
algebraic curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane c ...
that is asymptotically stable in the sense of
geometric invariant theory In mathematics, geometric invariant theory (or GIT) is a method for constructing quotients by Group action (mathematics), group actions in algebraic geometry, used to construct moduli spaces. It was developed by David Mumford in 1965, using ideas ...
. This is equivalent to the condition that it is a complete connected curve whose only singularities are ordinary
double point In geometry, a singular point on a curve is one where the curve is not given by a smooth embedding of a parameter. The precise definition of a singular point depends on the type of curve being studied. Algebraic curves in the plane Algebraic curv ...
s and whose
automorphism group In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...
is finite. The condition that the automorphism group is finite can be replaced by the condition that it is not of
arithmetic genus In mathematics, the arithmetic genus of an algebraic variety is one of a few possible generalizations of the genus of an algebraic curve or Riemann surface. Projective varieties Let ''X'' be a projective scheme of dimension ''r'' over a field '' ...
one and every non-singular
rational Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abili ...
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 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. For example, an
elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...
(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 cover A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties. Definition Let X be a topological space. A covering of X is a continuous map : \pi : E \rightarrow X such that there exists a discrete spa ...
s of all its components are isomorphic to the unit disk.


Definition

Given an arbitrary scheme S and setting g \geq 2 a ''stable'' genus g curve over S is defined as a proper flat morphism \pi: C \to S such that the geometric fibers are reduced, connected 1-dimensional schemes C_s such that # C_s has only ordinary double-point singularities # Every rational component E meets other components at more than 2 points # \dim H^1(\mathcal_) = g 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 In mathematics, an elliptic surface is a surface that has an elliptic fibration, in other words a proper morphism with connected fibers to an algebraic curve such that almost all fibers are smooth curves of genus 1. (Over an algebraically closed fie ...
can be completely classified.


Examples

One classical example of a family of stable curves is given by the Weierstrass family of curves : \begin \operatorname\left( \frac \right) \\ \downarrow \\ \operatorname(\mathbb \end where the fibers over every point \neq 0,1 are smooth and the degenerate points only have one double-point singularity. This example can be generalized to the case of a one-parameter family of smooth hyperelliptic curves degenerating at finitely many points.


Non-examples

In the general case of more than one parameter care has to be taken to remove curves which have worse than double-point singularities. For example, consider the family over \mathbb^2_ constructed from the polynomials : y^2 = x(x-s)(x-t)(x-1)(x-2) since along the diagonal s = t there are non-double-point singularities. Another non-example is the family over \mathbb^1_t given by the polynomials : x^3 -y^2 + t which are a family of elliptic curves degenerating to a rational curve with a cusp.


Properties

One of the most important properties of stable curves is the fact that they are local complete intersections. This implies that standard Serre-duality theory can be used. In particular, it can be shown that for every stable curve \omega_^ is a relatively very-ample sheaf; it can be used to embed the curve into \mathbb^_S. Using the standard Hilbert Scheme theory we can construct a moduli scheme of curves of genus g embedded in some projective space. The Hilbert polynomial is given by : P_g(n) = (6n-1)(g-1) There is a sublocus of stable curves contained in the Hilbert scheme : H_g \subset \textbf^_ This represents the functor : \mathcal_g(S) \cong \left. \left\\Bigg/ \right. \cong \operatorname(S,H_g) where \sim are isomorphisms of stable curves. In order to make this the moduli space of curves without regard to the embedding (which is encoded by the isomorphism of projective spaces) we have to mod out by PGL(5g - 6). This gives us the moduli stack : \mathcal_g := underline_g / \underline(5g-6)


See also

*
Moduli of algebraic curves In algebraic geometry, a moduli space of (algebraic) curves is a geometric space (typically a scheme or an algebraic stack) whose points represent isomorphism classes of algebraic curves. It is thus a special case of a moduli space. Depending on ...
* Stable map of curves


References

* Artin, M.; Winters, G. (1971-11-01).
Degenerate fibres and stable reduction of curves
. ''Topology''. 10 (4): 373–383. doi:10.1016/0040-9383(71)90028-0.
ISSN An International Standard Serial Number (ISSN) is an eight-digit serial number used to uniquely identify a serial publication, such as a magazine. The ISSN is especially helpful in distinguishing between serials with the same title. ISSNs ...
 0040-9383. * * * {{Algebraic curves navbox Algebraic curves Moduli theory