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
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 ...
, and other contributions in
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
.
Biography
Enriques was born in
Livorno
Livorno () is a port city on the Ligurian Sea on the western coast of Tuscany, Italy. It is the capital of the Province of Livorno, having a population of 158,493 residents in December 2017. It is traditionally known in English as Leghorn (pronou ...
, and brought up in
Pisa
Pisa ( , or ) is a city and ''comune'' in Tuscany, central Italy, straddling the Arno just before it empties into the Ligurian Sea. It is the capital city of the Province of Pisa. Although Pisa is known worldwide for its leaning tower, the cit ...
, in a
Sephardi Jew
Sephardic (or Sephardi) Jews (, ; lad, Djudíos Sefardíes), also ''Sepharadim'' , Modern Hebrew: ''Sfaradim'', Tiberian: Səp̄āraddîm, also , ''Ye'hude Sepharad'', lit. "The Jews of Spain", es, Judíos sefardíes (or ), pt, Judeus sefar ...
ish family of
Portuguese
Portuguese may refer to:
* anything of, from, or related to the country and nation of Portugal
** Portuguese cuisine, traditional foods
** Portuguese language, a Romance language
*** Portuguese dialects, variants of the Portuguese language
** Portu ...
descent. His younger brother was zoologist
Paolo Enriques
Paolo Enriques (17 August 1878 in Livorno – 26 December 1932 in Rome) was an Italian zoologist of Portuguese-Jewish descent.
He was the brother of mathematician Federigo Enriques and the brother-in-law of another mathematician Guido Castelnuov ...
who was also the father of Enzo Enriques Agnoletti and
Anna Maria Enriques Agnoletti. He became a student 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 signi ...
(who later became his brother-in-law after marrying his sister Elbina), and became an important member of the
Italian school of algebraic geometry
In relation to the history of mathematics, the Italian school of algebraic geometry refers to mathematicians and their work in birational geometry, particularly on algebraic surfaces, centered around Rome roughly from 1885 to 1935. There were 30 ...
. He also worked on
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
. He collaborated with Castelnuovo,
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 ...
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 ...
. He had positions at the
University of Bologna
The University of Bologna ( it, Alma Mater Studiorum – Università di Bologna, UNIBO) is a public research university in Bologna, Italy. Founded in 1088 by an organised guild of students (''studiorum''), it is the oldest university in continuo ...
, and then the
University of Rome La Sapienza
The Sapienza University of Rome ( it, Sapienza – Università di Roma), also called simply Sapienza or the University of Rome, and formally the Università degli Studi di Roma "La Sapienza", is a public research university located in Rome, Ita ...
. He lost his position in 1938, when the
Fascist
Fascism is a far-right, Authoritarianism, authoritarian, ultranationalism, ultra-nationalist political Political ideology, ideology and Political movement, movement,: "extreme militaristic nationalism, contempt for electoral democracy and pol ...
government enacted the "leggi razziali" (racial laws), which in particular banned Jews from holding professorships in Universities.
The Enriques classification, of complex
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 up to birational equivalence, was into five main classes, and was background to further work until
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 ...
reconsidered the matter in the 1950s. The largest class, in some sense, was that of
surfaces of general type In algebraic geometry, a surface of general type is an algebraic surface with Kodaira dimension 2. Because of Algebraic geometry and analytic geometry#Chow.27s theorem, Chow's theorem any compact complex manifold of dimension 2 and with Kodaira ...
: those for which the consideration of
differential form
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
s provides
linear system
In systems theory, a linear system is a mathematical model of a system based on the use of a linear operator.
Linear systems typically exhibit features and properties that are much simpler than the nonlinear case.
As a mathematical abstraction o ...
s that are large enough to make all the geometry visible. The work of the Italian school had provided enough insight to recognise the other main birational classes.
Rational surface
In algebraic geometry, a branch of mathematics, a rational surface is a surface birationally equivalent to the projective plane, or in other words a rational variety of dimension two. Rational surfaces are the simplest of the 10 or so classes of su ...
s and more generally
ruled surface
In geometry, a surface is ruled (also called a scroll) if through every point of there is a straight line that lies on . Examples include the plane, the lateral surface of a cylinder or cone, a conical surface with elliptical directrix, the ...
s (these include
quadric
In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and hyperbolas). It is a hypersurface (of dimension ''D'') in a -dimensional space, and it is de ...
s and
cubic surface
In mathematics, a cubic surface is a surface in 3-dimensional space defined by one polynomial equation of degree 3. Cubic surfaces are fundamental examples in algebraic geometry. The theory is simplified by working in projective space rather than a ...
s in projective 3-space) have the simplest geometry.
Quartic surface
In mathematics, especially in algebraic geometry, a quartic surface is a surface defined by an equation of degree 4.
More specifically there are two closely related types of quartic surface: affine and projective. An ''affine'' quartic surfac ...
s in 3-spaces are now classified (when
non-singular
In the mathematical field of algebraic geometry, a singular point of an algebraic variety is a point that is 'special' (so, singular), in the geometric sense that at this point the tangent space at the variety may not be regularly defined. In cas ...
) as cases of
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; the classical approach was to look at the
Kummer surface
In algebraic geometry, a Kummer quartic surface, first studied by , is an irreducible nodal surface of degree 4 in \mathbb^3 with the maximal possible number of 16 double points. Any such surface is the Kummer variety of the Jacobian variety o ...
s, which are singular at 16 points.
Abelian surface In mathematics, an abelian surface is a 2-dimensional abelian variety.
One-dimensional complex tori are just elliptic curves and are all algebraic, but Riemann discovered that most complex tori of dimension 2 are not algebraic via the Riemann bi ...
s give rise to Kummer surfaces as quotients. There remains the class of
elliptic surface 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, which are
fiber bundle
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...
s over a curve with
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 ...
s as fiber, having a finite number of modifications (so there is a bundle that is
locally trivial
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...
actually over a curve less some points). The question of classification is to show that any surface, lying in
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 ...
of any dimension, is in the birational sense (after
blowing up
In mathematics, blowing up or blowup is a type of geometric transformation which replaces a subspace of a given space with all the directions pointing out of that subspace. For example, the blowup of a point in a plane replaces the point with th ...
and
blowing down of some curves, that is) accounted for by the models already mentioned.
No more than other work in the Italian school would the proofs by Enriques now be counted as complete and
rigorous
Rigour (British English) or rigor (American English; American and British English spelling differences#-our, -or, see spelling differences) describes a condition of stiffness or strictness. These constraints may be environmentally imposed, su ...
. Not enough was known about some of the technical issues: the geometers worked by a mixture of inspired guesswork and close familiarity with examples.
Oscar Zariski
, birth_date =
, birth_place = Kobrin, Russian Empire
, death_date =
, death_place = Brookline, Massachusetts, United States
, nationality = American
, field = Mathematics
, work_institutions = ...
started to work in the 1930s on a more refined theory of birational mappings, incorporating
commutative algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent ...
methods. He also began work on the question of the classification for
characteristic p
In mathematics, the characteristic of a ring , often denoted , is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0). If this sum never reaches the additive ide ...
, where new phenomena arise. The schools of Kunihiko Kodaira and
Igor 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. ...
had put Enriques' work on a sound footing by about 1960.
Works
* Enriques F.
Lezioni di geometria descrittiva'. Bologna, 1920.
* Enriques F. ''Lezioni di geometria proiettiva''
Italian ed. 1898an
German ed. 1903
* Enriques F. & Chisini, O. ''Lezioni sulla teoria geometrica delle equazioni e delle funzioni algebriche''. Bologna, 1915-1934
Volume 1Volume 2 Vol. 3, 1924; Vol. 4, 1934.
* Severi F. ''Lezioni di geometria algebrica : geometria sopra una curva, superficie di Riemann-integrali abeliani''.
Italian ed. 1908
* Enriques F. ''Problems of Science'' (trans. ''Problemi di Scienza''). Chicago, 1914.
* Enriques F. ''Zur Geschichte der Logik''. Leipzig, 1927.
* Castelnouvo G., Enriques F. ''Die algebraischen Flaechen''/
* Enriques F.
Le superficie algebriche'. Bologna, 1949.
Articles
On ''
Scientia
Scientia is the Latin word for knowledge. It may refer to:
* 7756 Scientia
*'' The Triumph of Science over Death'', a sculpture of Filipino hero José Rizal
* ''Scientia'' (UTFSM journal), a scientific journal published by Universidad Técnica F ...
''.
*
ed evoluzione*
numeri e l'infinito*
pragmatismo*
principio di ragion sufficiente nel pensiero greco*
problema della realtà*
significato della critica dei principii nello sviluppo delle matematiche*
della storia del pensiero scientifico nella cultura nazionale*
dans la pensee des grecs*
nella storia del pensiero*
mathematique de Klein*
connaissance historique et la connaissance scientifique dans la critique de Enrico De Michelis*
filosofia positiva e la classificazione delle scienze*
motivi della filosofia di Eugenio Rignano
References
External links
*
*
Official home page of center for Enriques studies (Italian language)*
{{DEFAULTSORT:Enriques, Federigo
1871 births
1946 deaths
Livornese Jews
Sapienza University of Rome faculty
20th-century Italian mathematicians
Italian people of Portuguese descent
20th-century Italian philosophers
Algebraic geometers
Italian algebraic geometers
Italian historians of mathematics
Members of the Lincean Academy
19th-century Italian Jews
20th-century Italian Jews