Italian School Of Algebraic Geometry
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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 rationa ...
, particularly on algebraic surfaces, 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
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 ...
of
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 sign ...
,
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 algebr ...
, who were involved in some of the deepest discoveries, as well as setting the style.


Algebraic surfaces

The emphasis on algebraic surfaces—
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 number ...
of
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...
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 ...
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 c ...
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 ...
in all its refinements (via the detailed geometry of the
theta-divisor 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. It is there ...
). The
classification of algebraic surfaces Classification is a process related to categorization, the process in which ideas and objects are recognized, differentiated and understood. Classification is the grouping of related facts into classes. It may also refer to: Business, organizat ...
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 living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial n ...
''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 ve ...
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, 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 ...
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 func ...
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 ...
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 Japane ...
in the 1950s, and refined to include mod ''p'' phenomena by Zariski, 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 geometr ...
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 Embedded or embedding (alternatively imbedded or imbedding) may refer to: Science * Embedding, in mathematics, one instance of some mathematical object contained within another instance ** Graph embedding * Embedded generation, a distributed ge ...
in higher-dimensional projective space. 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 is "considered the founder of the Italian school of algebraic geometry". Later they explain that in
Turin Turin ( , 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 from 1861 to 1865. Th ...
the collaboration of Enrico D'Ovidio and Corrado Segre "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
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 sign ...
and
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 ...
. 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 Giacomo Albanese (11 July 1890 – 8 June 1947) was an Italian mathematician known for his work in algebraic geometry. He took a permanent position in the University of São Paulo, Brazil, in 1936. Biography Albanese attended the school in Palerm ...
, Eugenio Bertini, Luigi Campedelli, Oscar Chisini, Michele De Franchis, Pasquale del Pezzo, Beniamino Segre,
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 algebr ...
,
Guido Zappa Guido Zappa (7 December 1915 – 17 March 2015) was an Italian mathematician and a noted group theorist: his other main research interests were geometry and also the history of mathematics. Zappa was particularly known for some examples of alge ...
(with contributions also from Gino Fano, Carlo Rosati, Giuseppe Torelli, Giuseppe Veronese). Elsewhere it involved H. F. Baker and Patrick du Val (UK), Arthur Byron Coble (USA), Georges Humbert and Charles Émile Picard (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 n ...
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 (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, pr ...
as
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, compas ...
, 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 mentioned the Italian school in connection with
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 ...
s. Henry Forder (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 ve ...
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 functi ...
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” of higher dimensions. Herein a combination of the theory of
integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
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 invariants that classify topological spaces up to homeomorphism, though usually most classif ...
. 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, 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 and
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 Japane ...
, 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 __NOTOC__ In algebraic geometry, a Barth surface is one of the complex nodal surfaces in 3 dimensions with large numbers of double points found by . Two examples are the Barth sextic of degree 6 with 65 double points, and the Barth decic of degre ...
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 algebraic ...
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 theorem o ...
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, ...
. * * *. * * *{{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-p ...
, 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 Buzzardbr>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