In algebraic geometry, varieties are classified up to isomorphism. More coarsely, it can be classified into birational equivalence. The complexity builds with higher dimensions. Given a surface S and a point p ∈ S: ∃ surface S' with morphism ε: S' → S unique up to isomorphism s.t.
- the restriction of ε to ε-1(S - {p}) is isomorphic to S - {p}
- ε-1(p) = E is isomorphic to ℙ1
ε is the blow-up of S at p and E its exceptional curve (II.1). The fundamental results in the birational geometry of surfaces uses additionally some facts about rational maps and linear systems. Varieties X, Y with X irreducible and a rational map Φ: X → Y a morphism from an open subset U of X to Y inextendable to any larger open subset. Φ is defined at x if x ∈ U. If X is a smooth surface, X - U = F is a finite set. If C is an irreducible curve on S, Φ is defined on C - F. Restruction induces an isomorphism between divisor groups of S and of S - F. This defines the inverse image under Φ of a divisor D on Y, written Φ*D (II.4). Divisors D on surface S form the set of all effective divisors |D| on S linearly equivalent to D. Non-vanishing sections of OS(D) define the elements of |D|, defined up to scalar multiplication. |D| is identified with the projective space associated to the vector space H0(OS(D)), a linear subspace P of |D| is a linear system on S and can be defined by a vector subspace of H0(OS(D)). P is complete if P = |D|. A 1D linear system is a pencil (II.5). A curve C is a fixed component of P if every divisor of P contains it, and the fixed part of P is the biggest divisor F that is contained in every element of P. ∀ D ∈ P: |D - F| has no fixed part. There is a bijection between { rational maps φ: P → ℙn s.t. φ(S) is contained in no hyperplane} and {linear systems on S w/o fixed part and with dim n} (II.6). The rational map φ: S → X from a surface to a projective variety implies a surface S' and morphism η: S' → S which is the composite of a finite number of blow-ups and a morphism f: S' → X. That defines a commutative diagram (II.7). If f: X→S is a birational morphism of surfaces and f-1 is undefined at a point p ∈ S, then f: X →g S' →ε S where g is a birational morphism and ε in the blow-up at p (II.8). An irreducible (possibly singular) surface S and smooth surface S' with birational morphism f: S → S' where f-1 is rational and undefined at p ∈ S', then f-1(p) is a curve on S (II.9). Rational φ S → S' of surface with φ-1 is undefined at p ∈ S' leads to a curve C on S with φ(C) = {p} (II.10). A birational morphism f:S → S0 implies a sequence of blow-ups εk: Sk → Sk-1 and an isomorphism between S and Sn so that f = ε1 ○ ... ○ ε1 ○ u (II.11). A surface S is minimal if B(S) is minimal, s.t. all birational morphisms S → S' are isomorphisms (II. 15). All surfaces dominate some minimal surface (II.16). The elements of B(S) are obtained by successive blow-ups of minimal surfaces. Every surface has a unique minimal model, so its classification decomposes into ruled surfaces (birationally known, unknown minimal models) and non-ruled surfaces (biregular classification is the same as birational one). In both case, it suffices to classify minimal surfaces (p. 19). A curve E ⊂ S is exceptional if ∃ ε: S → S', which makes it isomorphic to ℙ1, satisfying E2 = -1 (p. 20). This is Castelnuovo's contractibility criterion (II.17).