can be classified as follows. If q > 0 then the map to the Albanese variety has fibers that are projective lines (if the surface is minimal) so the surface is a ruled surface. If q = 0 this argument does not work as the Albanese variety is a point, but in this case Castelnuovo's theorem implies that the surface is rational. For non-algebraic surfaces Kodaira found an extra class of surfaces, called type VII, which are still not well understood. Rational surfaces Rational surface means surface birational to the complex projective plane P2. These are all algebraic. The minimal rational surfaces are P2 itself and the Hirzebruch surfaces Σn for n = 0 or n ≥ 2. (The Hirzebruch surface Σn is the P1 bundle over P1 associated to the sheaf O(0) + O(n). The surface Σ0 is isomorphic to P1 × P1, and Σ1 is isomorphic to P2 blown up at a point so is not minimal.) Invariants: The plurigenera are all 0 and the fundamental group is trivial. Hodge diamond: Examples: P2, P1 × P1 = Σ0, Hirzebruch surfaces Σn, quadrics, cubic surfaces, del Pezzo surfaces, Veronese surface. Many of these examples are non-minimal. Ruled surfaces of genus > 0 Ruled surfaces of genus g have a smooth morphism to a curve of genus g whose fibers are lines P1. They are all algebraic. (The ones of genus 0 are the Hirzebruch surfaces and are rational.) Any ruled surface is birationally equivalent to P1 × C for a unique curve C, so the classification of ruled surfaces up to birational equivalence is essentially the same as the classification of curves. A ruled surface not isomorphic to P1 × P1 has a unique ruling (P1 × P1 has two). Invariants: The plurigenera are all 0. Hodge diamond: Examples: The product of any curve of genus > 0 with P1. Surfaces of class VII These surfaces are never algebraic or Kähler. The minimal ones with b2 = 0 have been classified by Bogomolov, and are either Hopf surfaces or Inoue surfaces. Examples with positive second Betti number include Inoue-Hirzebruch surfaces, Enoki surfaces, and more generally Kato surfaces. The global spherical shell conjecture implies that all minimal class VII surfaces with positive second Betti number are Kato surfaces, which would more or less complete the classification of the type VII surfaces. Invariants: q = 1, h1,0 = 0. All plurigenera