In relation with the history of mathematics, the Italian school of algebraic geometry refers to the work over half a century or more (flourishing roughly 1885-1935) done internationally in birational geometry, particularly on algebraic surfaces. 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 in Rome of Guido Castelnuovo, Federigo Enriques and Francesco Severi; who were involved in some of the deepest discoveries, as well as setting the style. For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problematics of geometry. ... In mathematics, birational geometry is a part of the subject of algebraic geometry, that deals with the geometry of an algebraic variety that is dependent only on its function field. ... In mathematics, an algebraic surface is an algebraic variety of dimension two. ... For other uses, see Rome (disambiguation). ... Guido Castelnuovo (14 August 1865, Venice – 27 April 1952, Rome) was an Italian Jewish mathematician. ... Federigo Enriques (5 January 1871 –14 June 1946) was an Italian mathematician, now known principally as the first to give a classification of algebraic surfaces in birational geometry, and other contributions in algebraic geometry. ... Franceso Severi (13 April 1879, Arezzo, Italy - 8 December 1961, Rome) was an Italian mathematician. ...

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). In mathematics, an algebraic surface is an algebraic variety of dimension two. ... In mathematics, an algebraic variety is essentially a set of common zeroes of a set of polynomials. ... In algebraic geometry, the dimension of an algebraic variety V is defined, informally speaking, as the number of independent rational functions that exist on V. So, for example, an algebraic curve has by definition dimension 1. ... In algebraic geometry, an algebraic curve is an algebraic variety of dimension equal to 1. ... In mathematics, the Brill-Noether theory in algebraic geometry is the theory of special divisors on generic algebraic curves. ... In mathematics, specifically in complex analysis and algebraic geometry, the Riemann-Roch theorem is an important tool in the computation of the dimension of the space of meromorphic functions with prescribed zeroes and allowed poles. ... In mathematics, the theta-divisor Θ is the divisor in the sense of algebraic geometry defined on an abelian variety A over the complex numbers (and principally polarized) by the zero locus of the associated Riemann theta-function. ...

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. In mathematics, the Enriques-Kodaira classification is a classification of compact complex surfaces. ... In mathematics, the genus has few different meanings Topology The genus of a connected, oriented surface is an integer representing the maximum number of cuttings along closed simple curves without rendering the resultant manifold disconnected. ... A catalog of elliptic curves. ... Riemann surface for the function f(z) = sqrt(z) In mathematics, particularly in complex analysis, a Riemann surface, named after Bernhard Riemann, is a one-dimensional complex manifold. ... A K3 manifold is a hyperkähler manifold of real dimension 4, i. ... In mathematics, particularly in algebraic geometry, complex analysis and number theory, abelian variety is a term used to denote a complex torus that can be embedded into projective space as a projective variety. ... In differential geometry, a complex manifold is a manifold such that every neighborhood looks like the complex n-space in a coherent way. ... Kunihiko Kodaira (小平 邦彦 Kodaira Kunihiko, 16 March 1915 – 26 July 1997) was a Japanese mathematician known for distinguished work in algebraic geometry and the theory of complex manifolds; and as the founder of the Japanese school of algebraic geometers. ... Igor Rostislavovich Shafarevich (born 3 June 1923) is a Russian mathematician, founder of the major school of algebraic number theory and algebraic geometry in the USSR. He was also an important dissident figure under the Soviet regime, a public supporter of Andrei Sakharovs Human Rights Committee from 1970. ...

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 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. This article does not cite its references or sources. ... In mathematics, the concept of a linear system of divisors arose first in the form of a linear system of algebraic curves in the projective plane. ... In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. ...

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 from Luigi Cremona, Gino Fano, Rosati, Torelli, Giuseppe Veronese).-1... Guido Castelnuovo (14 August 1865, Venice – 27 April 1952, Rome) was an Italian Jewish mathematician. ... Federigo Enriques (5 January 1871 –14 June 1946) was an Italian mathematician, now known principally as the first to give a classification of algebraic surfaces in birational geometry, and other contributions in algebraic geometry. ... Pasquale Del Pezzo, Duke of Cajanello, (1859–1936), was the most Neapolitan of Neapolitan Mathematicians. He was born in Berlin (where his father was a representative of the Neapolitan king) on 2 May 1859. ... Beniamino Segre (16 February 1903-2 October 1977) was an Italian mathematician who is remembered today as a major contributor to algebraic geometry and one of the founders of combinatorial geometry. ... Corrado Segre (20 Aug 1863-18 May 1924) was an Italian mathematician who is remembered today as a major contributor to the early development of algebraic geometry. ... Franceso Severi (13 April 1879, Arezzo, Italy - 8 December 1961, Rome) was an Italian mathematician. ... Luigi Cremona (7 December 1830, Pavia - 10 June 1903) was an Italian mathematician. ... Gino Fano (5 January 1871 - 8 November 1952) was an Italian mathematician. ... Giuseppe Veronese (May 7, 1854 - July 17, 1917) was an Italian mathematician. ...

Elsewhere it involved H. F. Baker and P. Duval (UK), A. B. Coble and Oscar Zariski (USA), Charles Émile Picard (France), Lucien Godeaux (Belgium), G. Humbert, Hermann Schubert and Max Noether, and later Erich Kähler (Germany), H. G. Zeuthen (Denmark). Henry Frederick Baker (July 3, 1866 - March 17, 1956) was a British mathematician, working mainly in algebraic geometry, but also remembered for contributions to partial differential equations (related to what would become known as solitons), and Lie groups. ... Oscar Zariski was one of the most influential mathematicians working in the field of algebraic geometry in the twentieth century. ... Charles Émile Picard (July 24, 1856 - December 11, 1941) was a leading French mathematician. ... Max Noether (September 24, 1844 - December 13, 1921) was a German mathematician. ... Erich Kähler (16 January 1906 - 31 May 2000) was a German mathematician with wide-ranging geometrical interests. ... Hieronymus Georg Zeuthen (15 February 1839 – 6 January 1920) was a Danish mathematician. ...

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). Projective geometry is a non-metrical form of geometry. ... Synthetic geometry is a descriptive term that identifies a methodology of geometry which makes use of theorems and synthetic observations to create theorems or solve problems, as opposed to analytic geometry which uses algebra, numbers, computations to draw theorems or solve problems. ...

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 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. Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ... Jules Henri Poincaré (April 29, 1854 – July 17, 1912) (IPA: [1]) was one of Frances greatest mathematicians and theoretical physicists, and a philosopher of science. ... Solomon Lefschetz (3 September 1884-5 October 1972) was a US mathematician who did fundamental work on algebraic topology, its applications to algebraic geometry, and the theory of non-linear ordinary differential equations. ... This article is about a mathematician. ... John Arthur Todd (23 August 1908 - 22 December 1994) was a British geometer. ... Henri Cartan (born July 8, 1904) is a son of Élie Cartan, and is, as his father was, a distinguished and influential French mathematician. ... Kunihiko Kodaira (小平 邦彦 Kodaira Kunihiko, 16 March 1915 – 26 July 1997) was a Japanese mathematician known for distinguished work in algebraic geometry and the theory of complex manifolds; and as the founder of the Japanese school of algebraic geometers. ...

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 as moduli spaces for curves, have been at the centre of recent work related to physics. Very many of the fundamental concepts in algebraic geometry still bear the names of those of the Italian school. Oscar Zariski was one of the most influential mathematicians working in the field of algebraic geometry in the twentieth century. ... In algebraic geometry, a moduli space is a parameter space for families of algebraic objects (such as algebraic varieties, morphisms, vector bundles). ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ...

References

• Beniamino Segre and Italian geometry (PDF), article by Edoardo Vesentini