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.

Qualification of what was actually proved is necessary because of the foundational difficulties. These included intensive use of birational models in dimension 3 of surfaces that can have non-singular models only when embedded in higher-dimensional projective space. That is, the theory wasn't posed in an intrinsic way. To get round that, a sophisticated theory of handling a linear system of divisors was developed (in effect, a line bundle theory for hyperplane sections of putative embeddings in projective space). Many of the modern techniques were found, in embryo form, and in some cases the articulation of those exceeded the available technical language.

The roll of honour of the school includes the following major Italians: Albanese, Bertini, Campedelli. Castelnuovo, Chisini, Enriques, De Franchis, Del Pezzo, B.Segre, C.Segre, Severi, Zappa (with contributions also from Cremona, Fano, Rosati, Torelli, Veronese). Elsewhere it involved H.F. Baker and P. Duval (UK), A.B. Coble and Oscar Zariski (USA), Emile Picard (France), L. Godeaux (Belgium), G. Humbert, Schubert and Max Noether, and later Erich Kähler(Germany), H.G. Zeuthen (Denmark). These figures were all involved in algebraic geometry, rather than the pursuit of projective geometry as synthetic geometry, which during the period under discussion was a huge (in volume terms) but secondary subject (when judged by its importance as research).

The new algebraic geometry that would succeed the Italian school was distinguished also by the intensive use of algebraic topology. The founder of that tendency was Henri Poincaré; during the 1930s it was developed by Lefschetz, Hodge and Todd. The modern synthesis brought together their work, that of the Cartan school, and of W.L. Chow and Kunihiko Kodaira, with the traditional material.