Many results were obtained, but, in the Italian school of algebraic geometry , and are up to 100 years old.
The birational geometry of algebraic surfaces is rich, because of blowing up (also known as a monoidal transformation), under which a point is replaced by the curve of all limiting tangent directions coming into it (a projective line).
This states that any birational map between algebraic surfaces is given by a finite sequence of blowups and blowdowns.
be the abelian group consisting of all the divisors on S. Then due to the intersection theorem is viewed as a quadratic form.
The Hodge index theorem is used in Deligne's proof of the Weil conjecture.
Basic results on algebraic surfaces include the Hodge index theorem, and the division into five groups of birational equivalence classes called the classification of algebraic surfaces.
The general type class, of Kodaira dimension 2, is very large (degree 5 or larger for a non-singular surface in P3 lies in it, for example).
The third, h1,1, is not a birational invariant, because blowing up can add whole curves, with classes in H1,1.
The arithmetic genus pa is the difference This explains why the irregularity got its name, as a kind of 'error term'.
The Riemann-Roch theorem for surfaces was first formulated by Max Noether.
The families of curves on surfaces can be classified, in a sense, and give rise to much of their interesting geometry.