General Position - General Type

General Type

General position is a property of configurations of points, or more generally other subvarieties (lines in general position, so no three concurrent, and the like) – it is an extrinsic notion, which depends on an embedding as a subvariety. Informally, subvarieties are in general position if they cannot be described more simply than others. An intrinsic analog of general position is general type, and corresponds to a variety which cannot be described by simpler polynomial equations than others. This is formalized by the notion of Kodaira dimension of a variety, and by this measure projective spaces are the most special varieties, though there are other equally special ones, meaning having negative Kodaira dimension. For algebraic curves, the resulting classification is: projective line, torus, higher genus surfaces, and similar classifications occur in higher dimensions, notably the Enriques–Kodaira classification of algebraic surfaces.