Algebraic Surfaces
The emphasis on algebraic surfaces — algebraic varieties of dimension two — followed on from an essentially complete geometric theory of algebraic curves (dimension 1). The position in around 1870 was that the curve theory had incorporated with Brill–Noether theory the Riemann–Roch theorem in all its refinements (via the detailed geometry of the theta-divisor).
The classification of algebraic surfaces was a bold and successful attempt to repeat the division of curves by their genus g. It corresponds to the rough classification into the three types: g= 0 (projective line); g = 1 (elliptic curve); and g > 1 (Riemann surfaces with independent holomorphic differentials). In the case of surfaces, the Enriques classification was into five similar big classes, with three of those being analogues of the curve cases, and two more (elliptic fibrations, and K3 surfaces, as they would now be called) being with the case of two-dimension abelian varieties in the 'middle' territory. This was an essentially sound, breakthrough set of insights, recovered in modern complex manifold language by Kunihiko Kodaira in the 1950s, and refined to include mod p phenomena by Zariski, the Shafarevich school and others by around 1960. The form of the Riemann–Roch theorem on a surface was also worked out.
Read more about this topic: Italian School Of Algebraic Geometry
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