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image of a rational map

As rational map f:XYf: X \dashrightarrow Y among projective algebraic varieties is not defined everywhere there is also a nontrivial definition of an image of a rational map: it is defined by help of a graph of a rational map.

For this embed YY in a projective space P n=P k nP^n = P^n_k of dimension nn; now the image of ff is the image of the composition still denoted f:XP nf: X\dashrightarrow P^n. It is defined as a regular map on some open dense subset f U:UP nf_U: U\to P^n. Let the set theoretical graph be setgraphf UU×P nsetgraph f_U \subset U\times P^n; then define the graph of the rational map ff as the closure of the setgraphf Usetgraph f_U within X×P nX\times P^n. This does not depend on the choice of the dense open subset UU and agrees with the usual graph when ff is regular.

Notice that not every kk-point in the image of a rational map is actually set theoretically in image of the underlying map of sets of kk-points.

A rational map f:XYf: X\dashrightarrow Y is dominant if its image as a rational map is the whole of YY.

Revised on March 13, 2013 18:25:06 by Anonymous Coward (134.157.251.69)