- Italian school of algebraic geometry
In relation with the history of

mathematics , the**Italian school of**refers to the work over half a century or more (flourishing roughly 1885-1935) done internationally inalgebraic geometry birational geometry , particularly onalgebraic surface s. There were in the region of 30 to 40 leading mathematicians who made major contributions; about half of those being in fact Italian. There is no question that the leadership fell to the group inRome ofGuido Castelnuovo ,Federigo Enriques andFrancesco Severi ; who were involved in some of the deepest discoveries, as well as setting the style.**Algebraic surfaces**The emphasis on

algebraic surface s — algebraic varieties of dimension two — followed on from an essentially complete geometric theory ofalgebraic curve s (dimension 1). The position in around 1870 was that the curve theory had incorporated withBrill-Noether theory theRiemann-Roch theorem in all its refinements (via the detailed geometry of thetheta-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 surface s 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, andK3 surface s, 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 moderncomplex manifold language byKunihiko Kodaira in the 1950s, and refined to include mod p phenomena by Zariski, theShafarevich school and others by around 1960. The form of theRiemann-Roch theorem on a surface was also worked out.**Foundational issues**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 alinear system of divisors was developed (in effect, aline 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 geometers**The roll of honour of the school includes the following major Italians:

Giacomo Albanese , Bertini, Campedelli,Guido Castelnuovo ,Oscar Chisini ,Federigo Enriques ,Michele De Franchis ,Pasquale del Pezzo ,Beniamino Segre ,Corrado Segre ,Francesco Severi ,Guido Zappa (with contributions also fromLuigi Cremona ,Gino Fano , Rosati, Torelli,Giuseppe Veronese ).Elsewhere it involved

H. F. Baker andPatrick du Val (UK),A. B. Coble andOscar Zariski (USA),Charles Émile Picard (France),Lucien Godeaux (Belgium), G. Humbert,Hermann Schubert andMax Noether , and laterErich Kähler (Germany),H. G. Zeuthen (Denmark).These figures were all involved in algebraic geometry, rather than the pursuit of

projective geometry assynthetic geometry , which during the period under discussion was a huge (in volume terms) but secondary subject (when judged by its importance as research).**Advent of topology**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 wasHenri Poincaré ; during the 1930s it was developed byLefschetz , Hodge and Todd. The modern synthesis brought together their work, that of the Cartan school, and ofW.L. Chow andKunihiko Kodaira , with the traditional material.**From the 1950s**The fashion and foundational attitude changed in algebraic geometry from 1950 onwards, leading to an axiomatisation and some acrimony as to the status of some results. For a while it may have seemed that the tradition of the Italian school would possibly be lost, in the sense that the old papers had become hard to read for the new generation of geometers.

The essentials were in fact transmitted, in particular through

Zariski 's students. Some of the areas opened up, such asmoduli space s for curves, have been at the centre of recent work related tophysics . Very many of the fundamental concepts in algebraic geometry still bear the names of those of the Italian school.**References*** [

*http://www.mat.uniroma1.it/~rendicon/2005(2)/185-193.pdf Beniamino Segre and Italian geometry (PDF)*] , article byEdoardo Vesentini

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