Birational map and birational morphism in algebraic geometry

Question

In algebraic geometry do the two terms "birational map" and "birational morphism" indicate the same object?

By reading wikipedia the answer seems to be NO:

• A birational map from $X$ to $Y$ is a rational map $f: X \rightarrow Y$ such that there is a rational map $Y\rightarrow X$ inverse to $f$.
• A birational morphism $f: X → Y$, is a morphism which is birational. That is, $f$ is defined everywhere, but its inverse may not be.

The above distinction is also true for example in the book "Beaville - Complex algebraic surfaces"? The author doesn't give the basic definitions.

Answer 1

You are correct: they are not the same. In general, when people say "morphism", they mean an honest, defined-everywhere morphism. A "map" usually just means a rational map, defined on an open subset.

For a basic surface example, let $X$ be $\mathbb P^2$ blown up at the point $[1,0,0]$. There is a blow-down $\pi : X \to \mathbb P^2$: this is a birational morphism (defined everywhere). There is also an inverse map $\mathbb P^2 \dashrightarrow X$, defined everywhere except $[1,0,0]$. This is a birational map, but isn't a morphism.

Answer 2

Consider the hyperbola $H=\{(a,b)\in K^2 : ab=1\}$ and the projection $p:H\rightarrow K$ to the first coordinate. Then $p$ is an injective morphism with the rational map $q:K\rightarrow H$, $a\mapsto (a,a^{-1})$ as an inverse. However $q$ is only defined on the open set $K\setminus 0$. ($K$ is an (algebraically closed) field.)