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Encyclopedia > Algebraic geometry

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. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex analysis, topology and number theory. Initially a study of polynomial equations in many variables, the subject of algebraic geometry starts where equation solving leaves off, and it becomes at least as important to understand the totality of solutions of a system of equations as to find some solution; this does lead into some of the deepest waters in the whole of mathematics, both conceptually and in terms of technique. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ... In abstract algebra, commutative algebra studies commutative rings, their ideals, and modules over such rings. ... Calabi-Yau manifold Geometry (Greek Î³ÎµÏ‰Î¼ÎµÏ„ÏÎ¯Î±; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. ... Complex analysis is the branch of mathematics investigating functions of complex numbers, and is of enormous practical use in many branches of mathematics, including applied mathematics. ... A MÃ¶bius strip, an object with only one surface and one edge; such shapes are an object of study in topology. ... Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ... In mathematics, equation solving is the problem of finding what values (numbers, functions, sets, etc. ...

The fundamental objects of study in algebraic geometry are algebraic varieties, geometric manifestations of solutions of systems of polynomial equations. Plane algebraic curves, which include lines, circles, parabolas, lemniscates, and Cassini ovals, form one of the best studied classes of algebraic varieties. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve relative position of different curves and relations between the curves given by different equations. In mathematics, an algebraic variety is essentially a set of common zeroes of a set of polynomials. ... In mathematics, a solution set for a collection of polynomials over some ring is defined to be the set . ... In mathematics, a polynomial is an expression that is constructed from one or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ... In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. ... A representation of one line Three lines â€” the red and blue lines have same slope, while the red and green ones have same y-intercept. ... Circle illustration This article is about the shape and mathematical concept of circle. ... A parabola A graph showing the reflective property, the directrix (light blue), and the lines connecting the focus and directrix to the parabola (blue) In mathematics, the parabola (from the Greek: Ï€Î±ÏÎ±Î²Î¿Î»Î®) (IPA pronunciation: ) is a conic section generated by the intersection of a right circular conical surface and a plane... A lemniscate In mathematics, a lemniscate is a type of curve described by a Cartesian equation of the form: Graphing this equation produces a curve similar to . ... In mathematics, a Cassini oval is a set of points in the plane such that each point p on the oval bears a special relation to two other, fixed points q1 and q2, namely that the product is constant. ...

Descartes's idea of coordinates is central to algebraic geometry, but it has undergone a series of remarkable transformations beginning in the early 19th century. Before then, the coordinates were assumed to be tuples of real numbers, but first complex numbers, and then elements of an arbitrary field became acceptable. Homogeneous coordinates of projective geometry offered an extension of the notion of coordinate system in different direction, and enriched the scope of algebraic geometry. Much of the development of algebraic geometry in the 20th century occurred within abstract algebraic framework, with increasing emphasis being placed on 'intrinsic' properties of algebraic varieties not dependent on any particular way of embedding the variety in an ambient coordinate space; this parallels developments in topology and complex geometry. René Descartes René Descartes (IPA: , March 31, 1596 &#8211; February 11, 1650), also known as Cartesius, worked as a philosopher and mathematician. ... See Cartesian coordinate system or Coordinates (elementary mathematics) for a more elementary introduction to this topic. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = âˆ’1. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, homogeneous coordinates, introduced by August Ferdinand MÃ¶bius, allow affine transformations to be easily represented by a matrix. ... Projective geometry is a non-metrical form of geometry. ... A MÃ¶bius strip, an object with only one surface and one edge; such shapes are an object of study in topology. ... In mathematics, complex geometry is the application of complex numbers to plane geometry. ...

One key distinction between classical projective geometry of 19th century and modern algebraic geometry, in the form given to it by Grothendieck and Serre, is that the former is concerned with the more geometric notion of a point, while the latter emphasizes the more analytic concepts of a regular function and a regular map and extensively draws on sheaf theory. Another important difference lies in the scope of the subject. Grothendieck's idea of scheme provides the language and the tools for geometric treatment of arbitrary commutative rings and, in particular, bridges algebraic geometry with algebraic number theory. Andrew Wiles's celebrated proof of Fermat's last theorem is a vivid testament to the power of this approach. André Weil, Grothendieck, and Deligne also demonstrated that the fundamental ideas of topology of manifolds have deep analogues in algebraic geometry over finite fields. Alexander Grothendieck (born March 28, 1928, Berlin) is one of the greatest mathematicians of the 20th century, with major contributions to algebraic geometry, homological algebra, and functional analysis. ... Jean-Pierre Serre (born September 15, 1926) is one of the leading mathematicians of the twentieth century, active in algebraic geometry, number theory and topology. ... In mathematics, a regular function in the sense of algebraic geometry is an everywhere-defined, polynomial function on an algebraic variety V with values in the field K over which V is defined. ... In mathematics, a sheaf F on a given topological space X gives a set or richer structure F(U) for each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and gluing smaller open sets to obtain... In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. ... In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ... This article or section does not cite its references or sources. ... For the French mathematician with work in the area of elliptic curves, see AndrÃ© Weil. ... The Taniyamaâ€“Shimura theorem (also called the modularity theorem) establishes an important connection between elliptic curves, which are objects from algebraic geometry, and modular forms, which are certain periodic holomorphic functions investigated in number theory. ... Pierre de Fermats conjecture written in the margin of his copy of Arithmetica proved to be one of the most intriguing and enigmatic mathematical problems ever devised. ... AndrÃ© Weil (May 6, 1906 - August 6, 1998) was one of the great mathematicians of the 20th century. ... Pierre Deligne, March 2005 Pierre Deligne (born 3 October 1944) is a Belgian mathematician. ... In algebraic geometry, a motive (or sometimes motif) refers to some essential part of an algebraic variety. Mathematically, the theory of motives is then the conjectural universal cohomology theory for such objects. ... In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... In abstract algebra, a finite field or Galois field (so named in honor of Ã‰variste Galois) is a field that contains only finitely many elements. ...

## Zeros of simultaneous polynomials GA_googleFillSlot("encyclopedia_square");  Sphere and slanted circle

In classical algebraic geometry, the main objects of interest are the vanishing sets of collections of polynomials, meaning the set of all points that simultaneously satisfy one or more polynomial equations. For instance, the two-dimensional sphere in three-dimensional Euclidean space R3 could be defined as the set of all points (x,y,z) with Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... In mathematics, a polynomial is an expression that is constructed from one or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ... A sphere is a symmetrical geometrical object. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... $x^2+y^2+z^2-1=0.,$

A "slanted" circle in R3 can be defined as the set of all points (x,y,z) which satisfy the two polynomial equations $x^2+y^2+z^2-1=0,,$ $x+y+z=0.,$

## Affine varieties

First we start with a field k. In classical algebraic geometry, this field was always the complex numbers C, but many of the same results are true if we assume only that k is algebraically closed. We define An(k) (or more simply An, when k is clear from the context), called the affine n-space over k, to be kn. The purpose of this apparently superfluous notation is to emphasize that one `forgets' the vector space structure that kn carries. Abstractly speaking, An is, for the moment, just a collection of points. In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, a field is said to be algebraically closed if every polynomial in one variable of degree at least , with coefficients in , has a zero (root) in . ...

A function f : AnA1 is said to be regular if it can be written as a polynomial, that is, if there is a polynomial p in k[x1,...,xn] such that f(t1,...,tn) = p(t1,...,tn) for every point (t1,...,tn) of An.

Regular functions on affine n-space are thus exactly the same as polynomials over k in n variables. We will write the regular functions on An as k[An].

We say that a polynomial vanishes at a point if evaluating it at that point gives zero. Let S be a set of polynomials in k[An]. The vanishing set of S (or vanishing locus) is the set V(S) of all points in An where every polynomial in S vanishes. In other words, $V(S) = {(t_1,dots,t_n) | mbox{for all }p mbox{ in }S, p(t_1,dots,t_n) = 0}.,$

A subset of An which is V(S), for some S, is called an algebraic set. The V stands for variety (a specific type of algebraic set to be defined below).

Given a subset U of An, can one recover the set of polynomials which generate it? If U is any subset of An, define I(U) to be the set of all polynomials whose vanishing set contains U. The I stands for ideal: if two polynomials f and g both vanish on U, then f+g vanishes on U, and if h is any polynomial, then hf vanishes on U, so I(U) is always an ideal of k[An]. In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. ...

Two natural questions to ask are:

• Given a subset U of An, when is U = V(I(U))?
• Given a set S of polynomials, when is S = I(V(S))?

The answer to the first question is provided by introducing the Zariski topology, a topology on An which directly reflects the algebraic structure of k[An]. Then U = V(I(U)) if and only if U is a Zariski-closed set. The answer to the second question is given by Hilbert's Nullstellensatz. In one of its forms, it says that I(V(S)) is the prime radical of the ideal generated by S. In more abstract language, there is a Galois connection, giving rise to two closure operators; they can be identified, and naturally play a basic role in the theory; the example is elaborated at Galois connection. In mathematics, namely algebraic geometry, the Zariski topology is a particular topology chosen for algebraic varieties that reflects the algebraic nature of their definition but is only weakly related to their geometric properties; it is due to Oscar Zariski and took a place of particular importance in the field around... Hilberts Nullstellensatz (German: theorem of zeros) is a theorem in algebraic geometry that relates varieties and ideals in polynomial rings over algebraically closed fields. ... In ring theory, a branch of mathematics, the radical of a ring isolates certain bad properties of the ring. ... In mathematics, especially in order theory, a Galois connection is a particular correspondence between two partially ordered sets (posets). Galois connections generalize the correspondence between subgroups and subfields investigated in Galois theory. ... In mathematics, given a partially ordered set (P, â‰¤), a closure operator on P is a function C : P â†’ P with the following properties: x â‰¤ C(x) for all x, i. ... In mathematics, especially in order theory, a Galois connection is a particular correspondence between two partially ordered sets (posets). Galois connections generalize the correspondence between subgroups and subfields investigated in Galois theory. ...

For various reasons we may not always want to work with the entire ideal corresponding to an algebraic set U. Hilbert's basis theorem implies that ideals in k[An] are always finitely generated. In mathematics, Hilberts basis theorem, first proved by David Hilbert in 1888, states that, if k is a field, then every ideal in the ring of multivariate polynomials k[x1, x2, ..., xn] is finitely generated. ...

An algebraic set is called irreducible if it cannot be written as the union of two smaller algebraic sets. An irreducible algebraic set is also called a variety. It turns out that an algebraic set is a variety if and only if the polynomials defining it generate a prime ideal of the polynomial ring. In mathematics, the concept of irreducible component is used to make formal the idea that a set such as defined by the equation XY = 0 is the union of the two lines X = 0 and Y = 0. ... In mathematics, an algebraic variety is essentially a set of common zeroes of a set of polynomials. ... In mathematics, a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers. ...

## Regular functions

Just as continuous functions are the natural maps on topological spaces and smooth functions are the natural maps on differentiable manifolds, there is a natural class of functions on an algebraic set, called regular functions. A regular function on an algebraic set V contained in An is defined to be the restriction of a regular function on An, in the sense we defined above. In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. ... This article or section is in need of attention from an expert on the subject. ...

It may seem unnaturally restrictive to require that a regular function always extend to the ambient space, but it is very similar to the situation in a normal topological space, where the Tietze extension theorem guarantees that a continuous function on a closed subset always extends to the ambient topological space. In topology and related branches of mathematics, normal spaces, T4 spaces, and T5 spaces are particularly nice kinds of topological spaces. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... The Tietze extension theorem in topology states that, if X is a normal topological space and f : A &#8594; R is a continuous map from a closed subset A of X into the real numbers carrying the standard topology, then there exists a continuous map F : X &#8594; R with...

Just as with the regular functions on affine space, the regular functions on V form a ring, which we denote by k[V]. This ring is called the coordinate ring of V.

Since regular functions on V come from regular functions on An, there should be a relationship between their coordinate rings. Specifically, to get a function in k[V] we took a function in k[An], and we said that it was the same as another function if they gave the same values when evaluated on V. This is the same as saying that their difference is zero on V. From this we can see that k[V] is the quotient k[An]/I(V).

## The category of affine varieties

Using regular functions from an affine variety to A1, we can define regular functions from one affine variety to another. First we will define a regular function from a variety into affine space: Let V be a variety contained in An. Choose m regular functions on V, and call them f1, ..., fm. We define a regular function f from V to Am by letting f(t1, ..., tn) = (f1, ..., fm). In other words, each fi determines one coordinate of the range of f. In mathematics, the range of a function is the set of all output values produced by that function. ...

If V' is a variety contained in Am, we say that f is a regular function from V to V' if the range of f is contained in V'.

This makes the collection of all affine varieties into a category, where the objects are affine varieties and the morphisms are regular maps. The following theorem characterizes the category of affine varieties: In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ... In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ...

The category of affine varieties is the opposite category to the category of finitely generated integral k-algebras and their homomorphisms.

In category theory, an abstract branch of mathematics, the dual of a category C is the category formed by reversing all the morphisms of C. That is, we take Cop to be the category with objects that are those of C, but with the morphisms from X to Y in... In abstract algebra, an integral domain is a commutative ring with an additive identity 0 and a multiplicative identity 1 such that 0 â‰  1, in which the product of any two non-zero elements is always non-zero; that is, there are no zero divisors. ... In mathematics, an algebra over a field K, or a K-algebra, is a vector space A over K equipped with a compatible notion of multiplication of elements of A. A straightforward generalisation allows K to be any commutative ring. ...

## Projective space  parabola (y=x2, blue) and cubic (y=x3, red) in projective space

Consider the variety V(y - x2). If we draw it, we get a parabola. As x increases, the slope of the line from the origin to the point (xx2) becomes larger and larger. As x decreases, the slope of the same line becomes smaller and smaller. Image File history File links Size of this preview: 800 Ã— 474 pixelsFull resolution (903 Ã— 535 pixel, file size: 9 KB, MIME type: image/png) In:= n=250; s=1. ... Image File history File links Size of this preview: 800 Ã— 474 pixelsFull resolution (903 Ã— 535 pixel, file size: 9 KB, MIME type: image/png) In:= n=250; s=1. ... A parabola A graph showing the reflective property, the directrix (light blue), and the lines connecting the focus and directrix to the parabola (blue) In mathematics, the parabola (from the Greek: Ï€Î±ÏÎ±Î²Î¿Î»Î®) (IPA pronunciation: ) is a conic section generated by the intersection of a right circular conical surface and a plane...

Compare this to the variety V(y - x3). This is a cubic equation. As x increases, the slope of the line from the origin to the point (xx3) becomes larger and larger just as before. But unlike before, as x decreases, the slope of the same line again becomes larger and larger. So the behavior "at infinity" of V(y-x3) is different from the behavior "at infinity" of V(y - x2). It is, however, difficult to make the concept of "at infinity" meaningful, if we restrict to working in affine space. Graph of a cubic polynomial: y = x3/4 + 3x2/4 âˆ’ 3x/2 âˆ’ 2 = (1/4)(x + 4)(x + 1)(x âˆ’ 2) In mathematics, a cubic equation is a polynomial equation in which the highest occurring power of the unknown is the third power. ...

The remedy to this is to work in projective space. Projective space has properties analogous to those of a compact Hausdorff space. Among other things, it lets us make precise the notion of "at infinity" by including extra points. The behavior of a variety at those extra points then gives us more information about it. As it turns out, V(y - x3) has a singularity at one of those extra points, but V(y - x2) is smooth. This article does not cite its references or sources. ... In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ... In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ... In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability. ...

While projective geometry was originally established on a synthetic foundation, the use of homogeneous coordinates allowed the introduction of algebraic techniques. Furthermore, the introduction of projective techniques made many theorems in algebraic geometry simpler and sharper: For example, Bézout's theorem on the number of intersection points between two varieties can be stated in its sharpest form only in projective space. For this reason, projective space plays a fundamental role in algebraic geometry. 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. ... In mathematics, homogeneous coordinates, introduced by August Ferdinand MÃ¶bius, allow affine transformations to be easily represented by a matrix. ... This article refers to BÃ©zouts theorem in algebraic geometry. ...

## The modern viewpoint

The modern approach to algebraic geometry redefines the basic objects. Varieties are subsumed in Alexander Grothendieck's concept of a scheme. Schemes start with the observation that if finitely generated reduced k-algebras are geometrical objects, then perhaps arbitrary commutative rings should also be geometrical objects. As such, schemes become both a more general algebro-geometric object, and a convenient language to describe those objects. This language of schemes has proved to be a valuable way of dealing with geometric concepts and has become a cornerstone of modern algebraic geometry. Alexander Grothendieck (born March 28, 1928 in Berlin, Germany) is one of the most important mathematicians of the 20th century. ... In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. ...

## History

Image File history File links Circle-question-red. ...

### Prehistory: Before the 19th century

Some of the roots of algebraic geometry date back to the later work of the Hellenistic Greeks such as Archimedes and Apollonius on conic sections. The geometrical methods of the Greeks were first applied in a recognizably algebraic setting by the Persian mathematician Omar Khayyám (born 1048 A.D.), who discovered the general method of solving cubic equations by intersecting a parabola with a circle. Each of these primordial developments in algebraic geometry dealt with questions of finding and describing the intersections of algebraic curves. The Hellenistic period of Greek history was the period between the death of Alexander the Great in 323 BC and the annexation of the Greek peninsula and islands by Rome in 146 BC. Although the establishment of Roman rule did not break the continuity of Hellenistic society and culture, which... Archimedes of Syracuse (Greek: c. ... Apollonius of Perga [Pergaeus] (ca. ... In mathematics, a conic section (or just conic) is a curved locus of points, formed by intersecting a cone with a plane. ... For information about all peoples of Iran, see Demographics of Iran. ... GhiyÄs od-DÄ«n Abul-Fatah OmÄr ibn IbrÄhÄ«m KhayyÄm NishÄbÅ«rÄ« (Persian: ØºÛŒØ§Ø« Ø§Ù„Ø¯ÛŒÙ† Ø§Ø¨Ùˆ Ø§Ù„ÙØªØ­ Ø¹Ù…Ø± Ø¨Ù† Ø§Ø¨Ø±Ø§Ù‡ÛŒÙ… Ø®ÛŒØ§Ù… Ù†ÛŒØ´Ø§Ø¨ÙˆØ±ÛŒ) or Omar Khayyam (Nishapur, Persia, May 18, 1048 â€“ December 4, 1131) was a Persian poet, mathematician, philosopher and astronomer who lived in Persia. ... Graph of a cubic polynomial: y = x3/4 + 3x2/4 âˆ’ 3x/2 âˆ’ 2 = (1/4)(x + 4)(x + 1)(x âˆ’ 2) In mathematics, a cubic equation is a polynomial equation in which the highest occurring power of the unknown is the third power. ...

Such techniques of applying geometrical constructions to algebraic problems were also adopted by a number of Renaissance mathematicians such as Gerolamo Cardano and Niccolò Fontana Tartaglia on their studies of the cubic equation. The geometrical approach to construction problems, rather than the algebraic one, was favored by most 16th and 17th century mathematicians, notably Blaise Pascal who argued against the use of algebraic and analytical methods in geometry. The French mathematicians Franciscus Vieta and later René Descartes and Pierre de Fermat revolutionized the conventional way of thinking about construction problems through the introduction of coordinate geometry. They were interested primarily in the properties of algebraic curves, such as those defined by Diophantine equations (in the case of Fermat), and the algebraic reformulation of the classical Greek works on conics and cubics (in the case of Descartes). The Renaissance (French for rebirth, or Rinascimento in Italian), was a cultural movement in Italy (and in Europe in general) that began in the late Middle Ages, and spanned roughly the 14th through the 17th century. ... Gerolamo Cardano. ... NiccolÃ² Fontana Tartaglia. ... Blaise Pascal (pronounced ), (June 19, 1623â€“August 19, 1662) was a French mathematician, physicist, and religious philosopher. ... François Viète. ... RenÃ© Descartes (French IPA: ) (March 31, 1596 â€“ February 11, 1650), also known as Renatus Cartesius (latinized form), was a highly influential French philosopher, mathematician, scientist, and writer. ... Pierre de Fermat Pierre de Fermat IPA: (August 17, 1601â€“January 12, 1665) was a French lawyer at the Parlement of Toulouse, France, and a mathematician who is given credit for early developments that led to modern calculus. ... Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry, is the study of geometry using the principles of algebra. ... In mathematics, a Diophantine equation is an equation between two polynomials with integer coefficients with any number of unknowns. ...

During the same period, Blaise Pascal and Gérard Desargues approached geometry from a different perspective, developing the synthetic notions of projective geometry. Pascal and Desargues also studied curves, but from the purely geometrical point of view: the analog of the Greek ruler and compass construction. Ultimately, the analytic geometry of Descartes and Fermat won out, for it supplied the 18th century mathematicians with concrete quantitative tools needed to study physical problems using the new calculus of Isaac Newton and Gottfried Wilhelm Leibniz. However, by the end of the 18th century, most of the algebraic character of coordinate geometry was subsumed by the calculus of infinitesimals of Joseph Louis Lagrange and Leonard Euler. GÃ©rard Desargues (February 21 or March 2, 1591-October 1661) was a French mathematician and engineer, who is considered one of the founders of projective 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. ... Projective geometry is a non-metrical form of geometry. ... Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of geometry using the principles of algebra. ... Sir Isaac Newton FRS (4 January 1643 â€“ 31 March 1727) [ OS: 25 December 1642 â€“ 20 March 1727] was an English physicist, mathematician, astronomer, natural philosopher, and alchemist. ... Gottfried Leibniz Gottfried Wilhelm von Leibniz (July 1, 1646 in Leipzig - November 14, 1716 in Hannover) was a German philosopher, scientist, mathematician, diplomat, librarian, and lawyer of Sorb descent. ... Joseph-Louis Lagrange, comte de lEmpire (January 25, 1736 â€“ April 10, 1813; b. ... Leonhard Euler aged 49 (oil painting by Emanuel Handmann, 1756) Leonhard Euler (April 15, 1707 - September 18, 1783) (pronounced oiler) was a Swiss mathematician and physicist. ...

See e.g. Dieudonné, Jean: "The historical development of algebraic geometry", Amer. Math. Monthly 79 (1972), 827--866. (MR46#7232) or his more complete "History of algebraic geometry. An outline of the history and development of algebraic geometry", Wadsworth Mathematics Series. Wadsworth International Group, Belmont, Calif., 1985. 186 pp. ISBN: 0-534-03723-2 (MR86h:01004)

### Nineteenth and early 20th century

It took the simultaneous 19th century developments of non-Euclidean geometry and Abelian integrals in order to bring the old algebraic ideas back into the geometrical fold. The first of these new developments was seized up by Edmond Laguerre and Arthur Cayley, who attempted to ascertain the generalized metric properties of projective space. Cayley introduced the idea of homogeneous polynomial forms, and more specifically quadratic forms, on projective space. Subsequently, Felix Klein studied projective geometry (along with other sorts of geometry) from the viewpoint that the geometry on a space is encoded in a certain class of transformations on the space. By the end of the 19th century, projective geometers were studying more general kinds of transformations on figures in projective space. Rather than the projective linear transformations which were normally regarded as giving the fundamental Kleinian geometry on projective space, they concerned themselves also with the higher degree birational transformations. This weaker notion of congruence would later lead members of the 20th century Italian school of algebraic geometry to classify algebraic surfaces up to birational isomorphism. Behavior of lines with a common perpendicular in each of the three types of geometry The term non-Euclidean geometry describes hyperbolic, elliptic and absolute geometry, which are contrasted with Euclidean geometry. ... In mathematics, an abelian integral in Riemann surface theory is a function related to the indefinite integral of a differential of the first kind. ... Edmond Nicolas Laguerre (April 9, 1834 - August 14, 1886) was French mathematician who was born in Bar-le-Duc France and died in Bar-le-Duc France. ... Arthur Cayley (August 16, 1821 - January 26, 1895) was a British mathematician. ... In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. ... Felix Christian Klein (April 25, 1849, DÃ¼sseldorf, Germany â€“ June 22, 1925, GÃ¶ttingen) was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory. ... The symmetry group of a geometric figure is the group of congruencies under which it is invariant, with composition as the operation. ... In mathematics, a Klein geometry is a type of geometry motivated by Felix Klein in his influential Erlangen program. ... 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 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. ... In mathematics, an algebraic surface is an algebraic variety of dimension two. ... Definition In mathematics, two varieties are birationally isomorphic if there is a bijective birational map from one variety to the other, defined over . ...

The second early 19th century development, that of Abelian integrals, would lead Bernhard Riemann to the development of Riemann surfaces. Bernhard Riemann. ...

### Twentieth century

van der Waerden, Oscar Zariski, André Weil and others attempted to develop a rigorous foundation for algebraic geometry based on contemporary commutative algebra, including valuation theory and the theory of ideals. Bartel Leendert van der Waerden (February 2, 1903-January 12, 1996) was a Dutch mathematician who born in Amsterdam, Netherlands and died in Zürich, Switzerland. ... Oscar Zariski was one of the most influential mathematicians working in the field of algebraic geometry in the twentieth century. ... AndrÃ© Weil (May 6, 1906 - August 6, 1998) was one of the great mathematicians of the 20th century. ... In abstract algebra, commutative algebra studies commutative rings, their ideals, and modules over such rings. ... Model Theory In logic and model theory, a valuation is a map from the set of variables of a first-order language to the universe of some interpretation of that language. ... iDEAL is an Internet payment method in The Netherlands, based on online banking. ...

In the 1950s and 1960s Jean-Pierre Serre and Alexander Grothendieck recast the foundations making use of sheaf theory. Later, from about 1960, the idea of schemes was worked out, in conjunction with a very refined apparatus of homological techniques. After a decade of rapid development the field stabilised in the 1970s, and new applications were made, both to number theory and to more classical geometric questions on algebraic varieties, singularities and moduli. Jean-Pierre Serre (born September 15, 1926) is one of the leading mathematicians of the twentieth century, active in algebraic geometry, number theory and topology. ... Alexander Grothendieck (born March 28, 1928 in Berlin, Germany) is one of the most important mathematicians of the 20th century. ... In mathematics, a sheaf F on a given topological space X gives a set or richer structure F(U) for each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and gluing smaller open sets to obtain... In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. ... Homological algebra is the branch of mathematics which studies the methods of homology and cohomology in a general setting. ... Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ... For non-mathematical singularity theories, see singularity. ... In algebraic geometry, a moduli space is a parameter space for families of algebraic objects (such as algebraic varieties, morphisms, vector bundles). ...

An important class of varieties, not easily understood directly from their defining equations, are the abelian varieties, which are the projective varieties whose points form an abelian group. The prototypical examples are the elliptic curves, which have a rich theory. They were instrumental in the proof of Fermat's last theorem and are also used in elliptic curve cryptography. 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. ... This picture illustrates how the hours on a clock form a group under modular addition. ... In mathematics, an elliptic curve is a plane curve defined by an equation of the form y2 = x3 + a x + b, which is non-singular; that is, its graph has no cusps or self-intersections. ... Pierre de Fermats conjecture written in the margin of his copy of Arithmetica proved to be one of the most intriguing and enigmatic mathematical problems ever devised. ... Elliptic curve cryptography (ECC) is an approach to public-key cryptography based on the algebraic structure of elliptic curves over finite fields. ...

While much of algebraic geometry is concerned with abstract and general statements about varieties, methods for effective computation with concretely-given polynomials have also been developed. The most important is the technique of Gröbner bases which is employed in all computer algebra systems. In computer algebra, computational algebraic geometry, and computational commutative algebra, a GrÃ¶bner basis is a particular kind of generating subset of an ideal I in a polynomial ring R. One can view it as a multivariate, non-linear generalization of: the Euclidean algorithm for computation of univariate greatest common... A computer algebra system (CAS) is a software program that facilitates symbolic mathematics. ...

This is a list of important publications in mathematics, organized by field. ... This is a list of named (classes of) algebraic surfaces. ... A root-finding algorithm is a numerical method, or algorithm, for finding a value x such that f(x) = 0, for a given function f. ... Results from FactBites:

 PlanetMath: algebraic geometry (2500 words) Algebraic geometry is the study of algebraic objects using geometrical tools. Algebraic groups are essentially matrix group schemes, and as such allow the tools of algebraic geometry to be applied to their study. This is version 11 of algebraic geometry, born on 2004-03-19, modified 2005-02-26.
 Encyclopedia: Algebraic geometry (417 words) Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. In algebraic geometry, the geometric objects studied are defined as the set of zeroes of a number of polynomials: meaning the set of common zeroes, or equally the set defined by one or several simultaneous polynomial equations. Algebraic geometry was developed largely by the Italian geometers in the early part of the 20-th century.
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