This allows one to compose two rational maps between projective varieties.





Raising a map to the negative first power means computing the inverse birational map. Raising a map to the first power simply returns the map itself. In the next example we compute the blowup of a point on $P^2$ and its inverse.











Note that one can only raise maps to powers (with the exception of 1 and 1) if the source and target are the same. In that case, raising a map to a negative power means compose the inverse of a map with itself. We illustrate this with the quadratic transformation on $P^2$ that we started with (an transformation of order 2 in the Cremona group).


