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A variety X over an uncountable algebraically closed field k is uniruled if and only if there is a rational curve passing through every k-point of X. By contrast, there are varieties over the algebraic closure k of a finite field which are not uniruled but have a rational curve through every k -point.
Every irreducible complex algebraic curve is birational to a unique smooth projective curve, so the theory for curves is trivial. The case of surfaces was first investigated by the geometers of the Italian school around 1900; the contraction theorem of Guido Castelnuovo essentially describes the process of constructing a minimal model of any smooth projective surface.
A 3-dimensional foliated chart with n = 3 and q = 1. The plaques are 2-dimensional and the transversals are 1-dimensional. A rectangular neighborhood in R n is an open subset of the form B = J 1 × ⋅⋅⋅ × J n, where J i is a (possibly unbounded) relatively open interval in the ith coordinate axis. If J 1 is of the form (a,0], it is said ...
Chern numbers of minimal complex surfaces. The Enriques–Kodaira classification of compact complex surfaces states that every nonsingular minimal compact complex surface is of exactly one of the 10 types listed on this page; in other words, it is one of the rational, ruled (genus > 0), type VII, K3, Enriques, Kodaira, toric, hyperelliptic, properly quasi-elliptic, or general type surfaces.
It is sometimes called a normal form of f by G. In general this form is not uniquely defined because there are, in general, several elements of G that can be used for reducing f; this non-uniqueness is the starting point of Gröbner basis theory. The definition of the reduction shows immediately that, if h is a normal form of f by G, one has
Equivalently, a variety is rationally connected if every two points are connected by a rational curve contained in the variety. [3] This definition differs from that of path connectedness only by the nature of the path, but is very different, as the only algebraic curves which are rationally connected are the rational ones.
A variety is called convex if the pullback of the tangent bundle to a stable rational curve: has globally generated sections. [2] Geometrically this implies the curve is free to move around infinitesimally without any obstruction. Convexity is generally phrased as the technical condition
From a foundational perspective, the above theorem is usually used as the definition of (,,). That is, the Chow variety is usually defined as a subvariety of P V k , d , n {\displaystyle \mathbb {P} V_{k,d,n}} , and only then shown to be a fine moduli space for the moduli problem in question.