In relation to the

history of mathematics The history of mathematics deals with the origin of discoveries in mathematics and the mathematical methods and notation of the past. Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments ...

, the Italian school of algebraic geometry refers to mathematicians and their work in

birational geometry In mathematics, birational geometry is a field of algebraic geometry in which the goal is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets. This amounts to studying mappings that are given by rational fu ...

, particularly on

algebraic surface In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two (as a complex manifold, when it is non-singular) and so of di ...

s, centered around

Rome , established_title = Founded , established_date = 753 BC , founder = King Romulus (legendary) , image_map = Map of comune of Rome (metropolitan city of Capital Rome, region Lazio, Italy).svg , map_caption ...

roughly from 1885 to 1935. There were 30 to 40 leading mathematicians who made major contributions, about half of those being Italian. The leadership fell to the group in

Guido Castelnuovo Guido Castelnuovo (14 August 1865 – 27 April 1952) was an Italian mathematician. He is best known for his contributions to the field of algebraic geometry, though his contributions to the study of statistics and probability theory are also signi ...

Federigo Enriques Abramo Giulio Umberto 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 algebrai ...

and

Francesco Severi Francesco Severi (13 April 1879 – 8 December 1961) was an Italian mathematician. He was the chair of the committee on Fields Medal on 1936, at the first delivery. Severi was born in Arezzo, Italy. He is famous for his contributions to algeb ...

, who were involved in some of the deepest discoveries, as well as setting the style.

Algebraic surfaces

The emphasis on

s—

algebraic varieties Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Mo ...

dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...

two—followed on from an essentially complete geometric theory of

algebraic curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane c ...

s (dimension 1). The position in around 1870 was that the curve theory had incorporated with

Brill–Noether theory In algebraic geometry, Brill–Noether theory, introduced by , is the study of special divisors, certain divisors on a curve that determine more compatible functions than would be predicted. In classical language, special divisors move on the cur ...

the

Riemann–Roch theorem The Riemann–Roch theorem is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeros and allowed poles. It rel ...

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 algebraic curves by their

genus Genus ( plural genera ) is a taxonomic rank used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In the hierarchy of biological classification, genus com ...

''g''. The division of curves corresponds to the rough classification into the three types: ''g'' = 0 (

projective line In mathematics, a projective line is, roughly speaking, the extension of a usual line by a point called a ''point at infinity''. The statement and the proof of many theorems of geometry are simplified by the resultant elimination of special cases; ...

); ''g'' = 1 (

elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...

); and ''g'' > 1 (

Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...

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 fibration In mathematics, an elliptic surface is a surface that has an elliptic fibration, in other words a proper morphism with connected fibers to an algebraic curve such that almost all fibers are smooth curves of genus 1. (Over an algebraically closed fi ...

s, and

K3 surface In mathematics, a complex analytic K3 surface is a compact connected complex manifold of dimension 2 with trivial canonical bundle and irregularity zero. An (algebraic) K3 surface over any field means a smooth proper geometrically connected alg ...

s, as they would now be called) being with the case of two-dimension

abelian varieties In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular function ...

in the 'middle' territory. This was an essentially sound, breakthrough set of insights, recovered in modern

complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic. The term complex manifold is variously used to mean a com ...

language by

Kunihiko Kodaira 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. He was awarded a Fields Medal in 1954, being the first Japanese ...

in the 1950s, and refined to include mod ''p'' phenomena by

Zariski , birth_date = , birth_place = Kobrin, Russian Empire , death_date = , death_place = Brookline, Massachusetts, United States , nationality = American , field = Mathematics , work_institutions = ...

, the

Shafarevich Igor Rostislavovich Shafarevich (russian: И́горь Ростисла́вович Шафаре́вич; 3 June 1923 – 19 February 2017) was a Soviet and Russian mathematician who contributed to algebraic number theory and algebraic geometry ...

school and others by around 1960. The form of the Riemann–Roch theorem on a surface was also worked out.

Foundational issues

Some proofs produced by the school are not considered satisfactory because of foundational difficulties. These included frequent use of birational models in dimension three of surfaces that can have non-singular models only when embedded in higher-dimensional

projective space In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...

. In order to avoid these issues, a sophisticated theory of handling a

linear system of divisors In algebraic geometry, a linear system of divisors is an algebraic generalization of the geometric notion of a family of curves; the dimension of the linear system corresponds to the number of parameters of the family. These arose first in the fo ...

was developed (in effect, a

line bundle In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the ''tangent bundle'' is a way of organisin ...

theory for hyperplane sections of putative embeddings in projective space). Many modern techniques were found, in embryonic form, and in some cases the articulation of these ideas exceeded the available technical language.

The geometers

According to Guerraggio & Nastasi (page 9, 2005),

Luigi Cremona Antonio Luigi Gaudenzio Giuseppe Cremona (7 December 1830 – 10 June 1903) was an Italian mathematician. His life was devoted to the study of geometry and reforming advanced mathematical teaching in Italy. He worked on algebraic curves and alge ...

is "considered the founder of the Italian school of algebraic geometry". Later they explain that in

Turin Turin ( , Piedmontese language, Piedmontese: ; it, Torino ) is a city and an important business and cultural centre in Northern Italy. It is the capital city of Piedmont and of the Metropolitan City of Turin, and was the first Italian capital ...

the collaboration of

Enrico D'Ovidio Enrico D'Ovidio (1842-1933) was an Italian mathematician who is known by his works on geometry. Life and work D'Ovidio, son of a liberal parents involved in the Italian independence movement, studied at the university of Naples under his uncle ...

and

Corrado Segre Corrado Segre (20 August 1863 – 18 May 1924) was an Italian mathematician who is remembered today as a major contributor to the early development of algebraic geometry. Early life Corrado's parents were Abramo Segre and Estella De Ben ...

"would bring, either by their own efforts or those of their students, Italian algebraic geometry to full maturity". A one-time student of Segre, H.F. Baker wrotethat Corrado Segre "may probably be said to be the father of that wonderful Italian school which has achieved so much in the birational theory of algebraical loci." On this topic, Brigaglia & Ciliberto (2004) say "Segre had headed and maintained the school of geometry that Luigi Cremona had established in 1860." Reference to the

Mathematics Genealogy Project The Mathematics Genealogy Project (MGP) is a web-based database for the academic genealogy of mathematicians.. By 31 December 2021, it contained information on 274,575 mathematical scientists who contributed to research-level mathematics. For a ty ...

shows that, in terms of ''Italian doctorates'', the real productivity of the school began with

and

. In the USA

Oscar Zariski , birth_date = , birth_place = Kobrin, Russian Empire , death_date = , death_place = Brookline, Massachusetts, United States , nationality = American , field = Mathematics , work_institutions = ...

inspired many Ph.D.s. The roll of honour of the school includes the following other Italians: Giacomo Albanese,

Eugenio Bertini Eugenio Bertini (8 November 1846 – 24 February 1933) was an Italian mathematician who introduced Bertini's theorem. He was born at Forlì Forlì ( , ; rgn, Furlè ; la, Forum Livii) is a ''comune'' (municipality) and city in Emilia-Romagna, ...

, Luigi Campedelli,

Oscar Chisini Oscar Chisini (14 March 1889Biography and Work of Chisini
at the

Michele De Franchis,

Pasquale del Pezzo Pasquale del Pezzo, Duke of Caianello and Marquis of Campodisola (2 May 1859 – 20 June 1936), was an Italian mathematician. He was born in Berlin (where his father was a representative of the Neapolitan king) on 2 May 1859. He died in Naples ...

Beniamino Segre 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 finite geometry. Life and career He was born and studied in Turin. ...

, Guido Zappa (with contributions also from

Gino Fano Gino Fano (5 January 18718 November 1952) was an Italian mathematician, best known as the founder of finite geometry. He was born to a wealthy Jewish family in Mantua, in Italy and died in Verona, also in Italy. Fano made various contributions ...

, Carlo Rosati, Giuseppe Torelli,

Giuseppe Veronese Giuseppe Veronese (7 May 1854 – 17 July 1917) was an Italian mathematician. He was born in Chioggia, near Venice. Education Veronese earned his laurea in mathematics from the Istituto Tecnico di Venezia in 1872. Work Although Veronese's work w ...

). Elsewhere it involved

H. F. Baker Henry Frederick Baker FRS FRSE (3 July 1866 – 17 March 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 ...

and

Patrick du Val Patrick du Val (March 26, 1903 – January 22, 1987) was a British mathematician, known for his work on algebraic geometry, differential geometry, and general relativity. The concept of Du Val singularity of an algebraic surface is named aft ...

(UK),

Arthur Byron Coble Arthur Byron Coble (November 3, 1878 – December 8, 1966) was an American mathematician. He did research on finite geometries and the group theory related to them, Cremona transformations associated with the Galois theory of equations, and ...

(USA),

Georges Humbert Marie Georges Humbert (7 January 1859 Paris, France – 22 January 1921 Paris, France) was a French mathematician who worked on Kummer surfaces and the Appell–Humbert theorem and introduced Humbert surfaces. His son was the mathematician Pierr ...

and

Charles Émile Picard Charles is a masculine given name predominantly found in English and French speaking countries. It is from the French form ''Charles'' of the Proto-Germanic name (in runic alphabet) or ''*karilaz'' (in Latin alphabet), whose meaning was ...

(France), Lucien Godeaux (Belgium),

Hermann Schubert __NOTOC__ Hermann Cäsar Hannibal Schubert (22 May 1848 – 20 July 1911) was a German mathematician. Schubert was one of the leading developers of enumerative geometry, which considers those parts of algebraic geometry that involve a finite nu ...

and

Max Noether Max Noether (24 September 1844 – 13 December 1921) was a German mathematician who worked on algebraic geometry and the theory of algebraic functions. He has been called "one of the finest mathematicians of the nineteenth century". He was the ...

, and later

Erich Kähler Erich Kähler (; 16 January 1906 – 31 May 2000) was a German mathematician with wide-ranging interests in geometry and mathematical physics, who laid important mathematical groundwork for algebraic geometry and for string theory. Education an ...

(Germany), H. G. Zeuthen (Denmark). These figures were all involved in algebraic geometry, rather than the pursuit of

projective geometry In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, pro ...

synthetic geometry Synthetic geometry (sometimes referred to as axiomatic geometry or even pure geometry) is the study of geometry without the use of coordinates or formulae. It relies on the axiomatic method and the tools directly related to them, that is, compa ...

, 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

In 1950

Henry Forder Henry George Forder (27 September 1889 – 21 September 1981) was a New Zealand mathematician. Academic career Born in Shotesham All Saints, near Norwich, he won a scholarships first to a Grammar school and then to University of Cambridge. A ...

mentioned the Italian school in connection with

(1950) ''Geometry'', page 166

Further development of the theory of plane curves is only fruitful when it is connected with the theory of
Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
s and
Abelian function In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular func ...
s. This has been a favorite study during the last fifty years, of the Italian geometers, and they have also made contributions of great beauty to a similar theory of surfaces and of “
Varieties Variety may refer to: Arts and entertainment Entertainment formats * Variety (radio) * Variety show, in theater and television Films * ''Variety'' (1925 film), a German silent film directed by Ewald Andre Dupont * ''Variety'' (1935 film), ...
” of higher dimensions. Herein a combination of the theory of
integral In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
s on the varieties, and of their topology, yields decisive results. The theory of curves and surfaces is thus connected with modern algebra and topology...

The new algebraic geometry that would succeed the Italian school was distinguished by the intensive use of

algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...

. The founder of that tendency was

Henri Poincaré Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The ...

; during the 1930s it was developed by

Lefschetz Solomon Lefschetz (russian: Соломо́н Ле́фшец; 3 September 1884 – 5 October 1972) was an American mathematician who did fundamental work on algebraic topology, its applications to algebraic geometry, and the theory of non-linear o ...

, Hodge and

Todd Todd or Todds may refer to: Places ;Australia: * Todd River, an ephemeral river ;United States: * Todd Valley, California, also known as Todd, an unincorporated community * Todd, Missouri, a ghost town * Todd, North Carolina, an unincorporated ...

. The modern synthesis brought together their work, that of the Cartan school, and of

W.L. Chow Chow Wei-Liang (; October 1, 1911, Shanghai – August 10, 1995, Baltimore) was a Chinese mathematician and stamp collector born in Shanghai, known for his work in algebraic geometry. Biography Chow was a student in the US, graduating from the ...

and

, with the traditional material.

Collapse of the school

In the earlier years of the Italian school under Castelnuovo, the standards of

rigor Rigour (British English) or rigor (American English; see spelling differences) describes a condition of stiffness or strictness. These constraints may be environmentally imposed, such as "the rigours of famine"; logically imposed, such as ma ...

were as high as most areas of mathematics. Under Enriques it gradually became acceptable to use somewhat more informal arguments instead of complete rigorous proofs, such as the "principle of continuity" saying that what is true up to the

limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...

is true at the limit, a claim that had neither a rigorous proof nor even a precise statement. At first this did not matter too much, as Enriques's intuition was so good that essentially all the results he claimed were in fact correct, and using this more informal style of argument allowed him to produce spectacular results about algebraic surfaces. Unfortunately, from about 1930 onwards under Severi's leadership the standards of accuracy declined further, to the point where some of the claimed results were not just inadequately proved, but were incorrect. For example, in 1934 Severi claimed that the space of rational equivalence classes of cycles on an algebraic surface is finite-dimensional, but showed that this is false for surfaces of positive geometric genus, and in 1946 Severi published a paper claiming to prove that a degree-6 surface in 3-dimensional projective space has at most 52 nodes, but the Barth sextic has 65 nodes. Severi did not accept that his arguments were inadequate, leading to some acrimonious disputes as to the status of some results. From about 1950 to 1980 there was considerable effort to salvage as much as possible, and convert it into the rigorous algebraic style of algebraic geometry set up by Weil and Zariski. In particular in the 1960s Kodaira and Shafarevich and his students rewrote the

Enriques classification Abramo Giulio Umberto 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 algebrai ...

of algebraic surfaces in a more rigorous style, and also extended it to all compact complex surfaces, while in the 1970s Fulton and MacPherson put the classical calculations of

intersection theory In mathematics, intersection theory is one of the main branches of algebraic geometry, where it gives information about the intersection of two subvarieties of a given variety. The theory for varieties is older, with roots in Bézout's theore ...

on rigorous foundations.

References

*. * Aldo Brigaglia (2001) "The creation and the persistence of national schools: The case of Italian algebraic geometry", Chapter 9 (pages 187–206) of ''Changing Images in Mathematics'', Umberto Bottazzini and Amy Delmedico editors,

Routledge Routledge () is a British multinational publisher. It was founded in 1836 by George Routledge, and specialises in providing academic books, journals and online resources in the fields of the humanities, behavioural science, education, law, and ...

. * * *. * * *{{Citation , last = Vesentini , first = Edoardo , author-link = Edoardo Vesentini , title = Beniamino Segre and Italian geometry , journal =

Rendiconti di Matematica e delle sue Applicazioni , abbreviation = Rend. Mat. Appl. , publisher = "Guido Castelnuovo" Department of Mathematics, Sapienza University of Rome and Istituto Nazionale di Alta Matematica Francesco Severi , country = Italy , frequency = Biannual , history = 1913-pr ...

, issue = 2 , volume = 25 , pages = 185–193 , year = 2005 , url = http://www.mat.uniroma1.it/ricerca/rendiconti/2005%282%29/185-193.pdf , mr = 2197882 , zbl = 1093.01009 .

External links

*David Mumfor
email about the errors of the Italian algebraic geometry school under Severi
*

Kevin Buzzard Kevin Mark Buzzard (born 21 September 1968) is a British mathematician and currently a professor of pure mathematics at Imperial College London. He specialises in arithmetic geometry and the Langlands program. Biography While attending the Roy ...

br>what mistakes did the Italian algebraic geometers actually make?
*A. Brigaglia, C. Ciliberto, & E. Sernes

at

University of Palermo The University of Palermo ( it, Università degli Studi di Palermo) is a university located in Palermo, Italy, and founded in 1806. It is organized in 12 Faculties. History The University of Palermo was officially founded in 1806, although its ...

. History of mathematics Algebraic geometry History of geometry