rational map

Given an irreducible variety $X$ and a variety $Y$ a rational map $f: X\dashrightarrow Y$ (notice dashed arrow notation) is an equivalence class of partially defined maps, namely the pairs $(U, f_U)$ where $f_U$ is a regular map $f_U: U\to Y$ defined on dense Zariski open subvarieties $U\subset X$ 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 $f: X\dashrightarrow Y$ is **dominant** if its image as a rational map is the whole of $Y$.

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

If $f$ is dominant then in this situation is the composition $g\circ f$ is always defined.

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

Created on May 19, 2010 at 18:34:10. See the history of this page for a list of all contributions to it.