Rational Variety - Rationally Connected Variety

A rationally connected variety V is a projective algebraic variety over an algebraically closed field such that through every two points there passes the image of a regular map from the projective line into V. Equivalently, a variety is rationally connected if every two points are connected by a rational curve contained in the variety.

This definition differs form 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.

Every rational variety, including the projective spaces, is rationally connected, but the converse is false. The class of the rationally connected varieties is thus a generalization of the class of the rational varieties. Unirational varieties are rationally connected, but it is not known if the converse holds.

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