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. 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 components are isomorphic to the unit disk.
In algebraic geometry, a stablecurve is an algebraic curve that is asymptotically stable in the sense of geometric invariant theory. This is equivalent...
"boundary" points which correspond to stable nodal curves (together with their isomorphisms). A curve is stable if it is complete, connected, has no singularities...
This leads to the notion of a stablecurve of genus g ≥ 2 {\displaystyle g\geq 2} , a not-necessarily-smooth complete curve with no terribly bad singularities...
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...
closures intersect. Example: (Deligne & Mumford 1969) A stablecurve is a reduced connected curve of genus ≥2 such that its only singularities are ordinary...
The Phillips curve is an economic model, named after Bill Phillips, that correlates reduced unemployment with increasing wages in an economy. While Phillips...
Laffer curve illustrates a theoretical relationship between rates of taxation and the resulting levels of the government's tax revenue. The Laffer curve assumes...
Bremmer's J Curve describes the relationship between a country's openness and its stability; focusing on the notion that while many countries are stable because...
domain of a stable map need not be a stablecurve. However, one can contract its unstable components (iteratively) to produce a stablecurve, called the...
In mathematics, a nilcurve is a pointed stablecurve over a finite field with an indigenous bundle whose p-curvature is square nilpotent. Nilcurves were...
curve is any of a variety of J-shaped diagrams where a curve initially falls, then steeply rises above the starting point. In economics, the "J curve"...
The Bell Curve: Intelligence and Class Structure in American Life is a 1994 book by psychologist Richard J. Herrnstein and political scientist Charles...
stable bundles of given rank and degree over a nonsingular curve is a quasiprojective algebraic variety. The cohomology of the moduli space of stable...
The Elephant Curve, also known as the Lakner-Milanovic graph or the global growth incidence curve, is a graph that illustrates the unequal distribution...
and lowest vapor pressure in a distillation region. All residue curves end at stable nodes. Unstable node: This is the pure component or the azeotropic...
in particular the study of dynamical systems, the idea of stable and unstable sets or stable and unstable manifolds give a formal mathematical definition...
elliptic curve may be described more concretely as an elliptic curve that has bad reduction only of multiplicative type. Suppose E is an elliptic curve defined...
integral over the moduli space of stablecurves. Several fundamental results in the intersection theory of moduli spaces of curves can be deduced from the ELSV...
distributed. A normal distribution is sometimes informally called a bell curve. However, many other distributions are bell-shaped (such as the Cauchy,...
curves Moduli stack of elliptic curves Moduli spaces of K-stable Fano varieties Modular curve Picard functor Moduli of semistable sheaves on a curve Kontsevich...
By the Jordan curve theorem, every closed trajectory divides the plane into two regions, the interior and the exterior of the curve. Given a limit cycle...
groups. stable 1. A stablecurve is a curve with some "mild" singularity, used to construct a good-behaving moduli space of curves. 2. A stable vector...
In mathematics, a Frey curve or Frey–Hellegouarch curve is the elliptic curve y 2 = x ( x − α ) ( x + β ) {\displaystyle y^{2}=x(x-\alpha )(x+\beta )}...
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