(Zariski's main theorem)
Let \(f \colon X \to Y\) be a birational projective morphism of Noetherian integral schemes with \(Y\) normal. Then every fibre \(f^{-1}(y)\) is connected.
(Elimination of indeterminacy (Hironaka))
Let \(X\) and \(Y\) be smooth projective varieties over an algebraically closed field \(k\), and \(F \colon X \dashrightarrow Y\) a birational map. Assume that \(k\) has characteristic \(0\), or that \(\dim X = 2\). Then there exist a smooth projective variety \(W\) and birational morphisms \(g \colon W \to X\) and \(f \colon W \to Y\) fitting into
with \(F \circ g = f\) on the open subset of \(W\) where \(F \circ g\) is defined.
(Factorization)
Let \(f \colon X' \to X\) be a birational morphism of smooth projective surfaces and let \(p\) be a fundamental point of \(f^{-1}\). Then \(f\) factors through the blowup of \(X\) at \(p\). Consequently every birational map of smooth projective surfaces is a finite sequence of blowups followed by a finite sequence of blowdowns.
The connectedness theorem supplies the fibre structure used in the surface factorization statement. A birational morphism of smooth surfaces is an isomorphism away from finitely many points, and at each such point the fibre is connected of dimension one — so it is a curve that gets contracted, and contraction of a curve to a smooth point is a blowup.
Fundamental points are the points where the inverse map fails to be defined: the map is a morphism on a largest open \(U\), and \(X \setminus U\) is that finite set. The total transform of such a point is what Zariski’s theorem says is connected and positive-dimensional.
The factorization statement is the reason surfaces are classified up to birational equivalence rather than isomorphism, and why the invariants that matter — \(p_g\), \(q\), the plurigenera, \(\kappa\) — are exactly those unchanged by blowup.
Dependencies
T-SRFCASTCastelnuovo's contractibility criterionuses