unstable, semi-stable, or stable if it is the image of a point in with the same property. "Unstable" is the opposite of "semistable" (not "stable"). The unstable points form a Zariski closed set of projective space, while the semistable and stable points both form Zariski open sets (possibly empty). These definitions are from and are not equivalent to the ones in the first edition of Mumford's book. Many moduli spaces can be constructed as the quotients of the space of stable points of some subset of projective space by some group action. These spaces can often be compactified by adding certain equivalence classes of semistable points. Different stable orbits correspond to different points in the quotient, but two different semistable orbits may correspond to the same point in the quotient if their closures intersect. Example: A stable curve is a reduced connected curve of genus ≥2 such that its only singularities are ordinary double points and every non-singular rational component meets the other components in at least 3 points. The moduli space of stable curves of genus is the quotient of a subset of the Hilbert scheme of curves in with Hilbert polynomial by the group . Example: A vector bundle over an algebraic curve (or over a Riemann surface) is a stable vector bundle if and only if for all proper non-zero subbundles of and is semistable if this condition holds with < replaced by ≤. See also GIT quotient Geometric complexity theory Geometric quotient Categorical quotient Quantization commutes with reduction K-stability K-stability of Fano varieties Bridgeland stability condition Stability (algebraic geometry) References Kirwan, Frances, Cohomology of quotients in symplectic and algebraic geometry. Mathematical Notes, 31. Princeton University Press, Princeton, NJ, 1984. i+211 pp. Kraft, Hanspeter, Geometrische Methoden in der Invariantentheorie. (German) (Geometrical methods in invariant theory) Aspects of Mathematics, D1. Friedr. Vieweg & Sohn, Braunschweig, 1984. x+308 pp. ; (1st ed 1965); (2nd ed) V. L. Popov, E. B. Vinberg, Invariant theory, in Algebraic geometry. IV. Encyclopaedia of Mathematical Sciences, 55 (translated from 1989 Russian edition) Springer-Verlag, Berlin, 1994. vi+284 pp. Moduli theory Scheme theory Algebraic groups Invariant theory