rational map

Given an irreducible variety XX and a variety YY a rational map f:XYf: X\dashrightarrow Y (notice dashed arrow notation) is an equivalence class of partially defined maps, namely the pairs (U,f U)(U, f_U) where f Uf_U is a regular map f U:UYf_U: U\to Y defined on dense Zariski open subvarieties UXU\subset X and the equivalence is the agreement on the common intersection.

The notion of an image of a rational map is nontrivially defined, see that entry. A rational map f:XYf: X\dashrightarrow Y is dominant if its image as a rational map is the whole of YY.

The composition of rational maps gfg\circ f where f:XYf: X\dashrightarrow Y and g:YZg: Y\dashrightarrow Z is not always defined, namely it is even possible that the image of ff lies out of any dense open subset in YY, where gg is defined as a regular map. The composition is defined as the class of equivalence of pairs (g Vf U,f U 1(V))(g_V\circ f_U|, f_U^{-1}(V)) where UXU\subset X and VZV\subset Z are open dense subsets and f U 1(V)f_U^{-1}(V)\neq \emptyset if such exist and undefined otherwise.

If ff is dominant then in this situation is the composition gfg\circ f is always defined.

See also birational map, birational geometry, rational variety, unirational variety.

Created on May 19, 2010 18:34:10 by Zoran Škoda (