This course provides an introduction to the geometry of complex projective algebraic surfaces. In particular, the classical theory, dating back to Castelnuovo and Enriques, will be revisited in the framework of contemporary algebraic geometry.
A solid undergraduate-level knowledge of algebraic surfaces from the point of view of complex algebraic geometry.
1. Background material Complex manifolds. Holomorphic and meromorphic maps. Subvarieties. Divisors and holomorphic line bundles. The canonical bundle. Projective algebraic varieties. Degree and dimension. Examples. Linear systems. Some general results. Ample line bundles and their properties. Basics from the theory of curves. 2. Basics of the theory of projective algebraic surfaces Generalities on compact complex surfaces. Numerical characters. Algebraic surfaces: examples. Curves on a surface. Intersection theory on an algebraic surface. Topological interpretation. The Néron-Severi group and numerical equivalence. The Riemann-Roch theorem. Noether's formula. Genus formula. Virtual arithmetic genus of a divisor. 3. The cone of curves The Hodge index theorem. Interpretation in the real Néron-Severi space. The Nakai-Moishezon criterion. The ample cone. Nef divisors. The nef cone viewed as closure of the ample cone. The Mori cone and Kleiman's criterion. Remarkable examples illustrating the various cones. Nef treshold of an ample divisor and the Kawamata rationality theorem. 4. Birational maps and minimal models Rational maps and linear systems. Birational maps. Examples. Blowing-ups and their properties. Castelnuovo's contraction theorem. Examples. The structure of birational morphisms. Birational invariants. Minimal models. Ruled surfaces and fibrations in rational curves. The Noether-Enriques theorem. Enriques' theorem on minimal models. 5. Ruled and rational surfaces Numerical characters of ruled and of rational surfaces. Castelnuovo's rationality criterion. Minimal models of ruled and of rational surfaces. Numerical effectivity of the canonical bundle on a nonruled minimal surface (key lemma). Enriques' ruledness criterion. Characterizations of ruled surfaces via adjunction. The cubic surface; del Pezzo surfaces. 6. Nonruled surfaces Kodaira dimension. Behaviour of the plurigenera and other aspects of the tricothomy. Examples. Numerical characters of surfaces with Kodaira dimension zero. A glance at elliptic surfaces and surfaces of general type. Examples. The Godeaux surface.