# The classification of complex algebraic surfaces

## TL;DR

This paper reviews the classification of complex algebraic surfaces using Mori's theory, covering key theorems and programs that structure the understanding of these surfaces.

## Contribution

It provides a comprehensive overview of the classification of complex algebraic surfaces based on Mori's theory, including classical and modern results.

## Key findings

- Includes the P12-Theorem and Sarkisov's programme for surfaces
- Discusses Noether–Castelnuovo's classical theorem
- Synthesizes modern classification approaches

## Abstract

This text grew up from the notes of a graduate course I gave at the University of Roma ``Tor Vergata'' in the academic year 2018--19. The subject is the classification of complex algebraic surfaces following Mori's theory. It includes the $P_{12}$--Theorem, Sarkisov's programme in the surface case and Noether--Castelnuovo's Theorem in its classical version.