Algebraic geometry is a branch of Mathematics is commonly defined as the study of patterns of structure, change, and space; more informally, one might say it is the study of figures and numbers. Mathematical knowledge is constantly growing, through research and application, but mathematics itself is not usually considered a natural science. One reason is that... mathematics which, as the name suggests, combines Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. The term abstract algebra is used to distinguish the field from elementary algebra or high school algebra which teaches the correct rules for manipulating formulas and algebraic expressions involving real... abstract algebra, especially In abstract algebra, commutative algebra is the field of study of commutative rings, their ideals, modules and algebras. It is foundational both for algebraic geometry and for algebraic number theory. The most prominent example for commutative rings are polynomial rings. The subjects real founder, in the days when it... commutative algebra, with Geometry (from the Greek words Ge = earth and metro = measure) is the branch of mathematics first introduced by Theaetetus dealing with spatial relationships. From experience, or possibly intuitively, people characterize space by certain fundamental qualities, which are termed axioms in geometry. Such axioms are insusceptible to proof, but can be... geometry. It can be seen as the study of In mathematics, a solution set for a collection of polynomials over some ring is defined to be the set . Examples 1. The solution set of over the real numbers is the set {0}. 2. For any non-zero polynomial over the complex numbers in one variable, the solution set is... solution sets of systems of algebraic equations. When there is more than one variable, geometric considerations enter, and are important to understand the phenomenon. One can say that the subject starts where In mathematics, equation solving is the problem of finding what values (numbers, functions, sets...) fulfil a condition stated as an equality (an equation). Usually, this condition involves expressions with variables (or unknowns), which are to be substituted by values in order for the equality to hold. More precisely, an equation... 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.

Zeroes of simultaneous polynomials

In classical algebraic geometry, the main objects of interest are the vanishing sets of collections of In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. Here, simple means they are constructed using only multiplication and addition. Smooth means they are infinitely differentiable, i.e., they have derivatives of all finite orders. Because of their simple structure, polynomials are very easy... polynomials, meaning the set of all points that simultaneously satisfy one or more polynomial equations. For instance, the two-dimensional For other uses, see sphere (disambiguation). A sphere is, roughly speaking, a ball-shaped object. In non-mathematical usage, the term sphere is often used for something solid (which mathematicians call ball). But in mathematics, sphere refers to the boundary of a ball, which is hollow. This article deals with... sphere in three-dimensional In mathematics and astronomy, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. The generalization applies Euclids concept of distance, and the related concepts of length and angle, to a coordinate system in any number of dimensions. It is the standard example of... Euclidean space could be defined as the set of all points (x,y,z) with

x² + y² + z² − 1 = 0.

A "slanted" circle in can be defined as the set of all points (x,y,z) which satisfy the two polynomial equations

x² + y² + z² − 1 = 0,

x + y + z = 0.

Affine varieties

First we start with a 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 associative, commutative, and distributive rules hold, which are familiar from the arithmetic of ordinary numbers. Fields are important objects of study in... field k. In classical algebraic geometry, this field was always C, the complex numbers, but many of the same results are true if we assume only that k is In mathematics, a field F is said to be algebraically closed if every polynomial of degree at least 1, with coefficients in F, has a zero in F. In that case, every such polynomial splits into linear factors. It can be shown that a field is algebraically closed if and... algebraically closed. We define , called 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, is, for the moment, just a collection of points.

Henceforth we will drop the k in and instead write .

Define a function

to be regular if it can be written as a polynomial, that is, if there is a polynomial p in

k[x₁,...,x_n]

such that for each point

(t₁,...,t_n)

of ,

f(t₁,...,t_n) = p(t₁,...,t_n).

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 as .

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 . The vanishing set of S (or vanishing locus) is the set V(S) of all points in where every polynomial in S vanishes. In other words,

V(S)={(t₁,...,t_n) | for all p in S, p(t₁,...,t_n) = 0}.

A subset of 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 V of which is a variety, can one recover the set of polynomials which generate it? If V is any subset of , define I(V) to be the set of all polynomials whose vanishing set contains V. The I stands for In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. For instance, in rings one studies prime ideals instead of prime numbers, one defines coprime ideals as a generalization of coprime numbers, and one can prove a... ideal: if two polynomials f and g both vanish on V, then f+g vanishes on V, and if h is any polynomial, then hf vanishes on V, so I(V) is always an ideal of .

Two natural questions to ask are: given a subset V of , when is

V = V(I(V))?

Given a set S of polynomials, when is

S = I(V(S))?

The answer to the first question is provided by introducing the In mathematics, the Zariski topology is a structure basic to algebraic geometry, especially since 1950. It is named after its originator, Oscar Zariski. The Zariski topology is defined by defining the closed sets to be the sets consisting of the mutual zeroes of a set of polynomials. (See affine varieties... Zariski topology, a topology on which directly reflects the algebraic structure of . Then V = V(I(V)), if and only if V 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 In ring theory, a branch of mathematics, the radical of a ring isolates certain bad properties of the ring. There are several different kinds of radicals, such as the nilradical and the Jacobson radical, as well as a theory of general radical properties. Nilradicals Let R be a commutative ring... prime radical of the ideal generated by S. In more abstract language, there is a 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. They find applications in various mathematical theories as well as in the theory of programming. A Galois connection... Galois connection, giving rise to two In mathematics, given a partially ordered set (P, ≤), a closure operator on P is a function C : P → P with the following properties: if x ≤ y, then C(x) ≤ C(y), i.e. C is montonically increasing x ≤ C(x) for all x C... closure operators; they can be identified, and naturally play a basic role in the theory.

For various reasons we may not always want to work with the entire ideal corresponding to an algebraic set V. Hilbert's Basis Theorem implies that ideals in are always 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 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. Prime ideals have a simpler description for commutative rings, so we consider this case separately below. This article only covers ideals of ring theory. Prime... prime ideal of the polynomial ring.

Regular functions

Just as In mathematics, a continuous function is one in which arbitrarily small changes in the input produce arbitrarily small changes in the output. If small changes in the input can produce a broken jump in the changes of the output (or the value of the ouput is not defined), the function... continuous functions are the natural maps on Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion. The branch of mathematics that studies topological spaces in their own right is called topology. Definition A topological space... topological spaces and In mathematics, a smooth function is one that is infinitely differentiable, i.e., has derivatives of all finite orders. A function is called C1 if it has a derivative that is a continuous function; such functions are also called continuously differentiable. A function is called Cn for n ≥ 1... smooth functions are the natural maps on In mathematics, a manifold M is a type of space, characterised 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. Therefore, the Euclidean space itself gives... 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 is defined to be the restriction of a regular function on , in the sense we defined above.

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 In topology and related branches of mathematics, normal spaces, T4 spaces, and T5 spaces are particularly nice kinds of topological spaces. These conditions are examples of separation axioms. Definitions Suppose that X is a topological space. X is a normal space if, given any disjoint closed sets E and F... normal Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion. The branch of mathematics that studies topological spaces in their own right is called topology. Definition A topological space... topological space, where the The Tietze extension theorem in topology states that, if X is a normal topological space and f : A → 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 →... Tietze extension theorem guarantees that a continuous function on a closed subset always extends to the ambient topological space.

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 , there should be a relationship between their coordinate rings. Specifically, to get a function in k[V] we took a function in , 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 .

The category of affine varieties

Using regular functions from an affine variety to , 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 . Choose m regular functions on V, and call them f₁,...,f_m. We define a regular function f from V to by letting f(t₁,...,t_n)=(f₁,...,f_m). In other words, each f_i determines one coordinate of the In mathematics, the range of a function is the set of all values produced by a function. Sometimes called the image. Given a function , the set f(A) is called the range of f. The range is not to be confused with the codomain B. Generally the range is only... range of f.

If V' is a variety contained in , 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 theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. It is half-jokingly known as generalized abstract nonsense. The use of this phrase does not mean that mathematicians consider category theory to be fuzzy or non-rigorous, merely that a... category, where the objects are affine varieties and the In mathematics, a morphism is an abstraction of a function or mapping between two spaces. The word can mean different things depending on the type of space in question. In set theory, for example, morphisms are just functions, in group theory they are group homomorphisms, while in topology they are... morphisms are regular maps. The following theorem characterizes the category of affine varieties:

The category of affine varieties is the 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... opposite category to the category of finitely generated In ring theory, a ring R is said to be reduced if it has no non-zero nilpotent elements. Categories: Stub ... reduced k- 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. (Some authors use the term algebra synonymously with associative algebra... algebras and their homomorphisms.

Projective space

Consider the variety V(y=x²). If we draw it, we get a A parabola A parabola (from the Greek: παραβολή) is a conic section generated by the intersection of a cone, and a plane tangent to the cone or parallel to some plane tangent to the cone. If the plane is itself tangent to the... parabola. As x increases, the slope of the line from the origin to the point (x,x²) becomes larger and larger. As x decreases, the slope of the same line becomes smaller and smaller.

Compare this to the variety V(y=x³). This is a A cubic equation is a polynomial equation in which the highest occurring power of the unknown is the third power. An example is the equation 2x3 - 4x2 + 3x - 4 = 0 and the general form may be written as α3x3 + α2x2 + α1x + α0 = 0. Usually, the coefficients... cubic equation. As x increases, the slope of the line from the origin to the point (x,x³) 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.

The remedy to this is to work in In mathematics, a projective space is a fundamental construction from any vector space. It generalises the projective plane that may be constructed from a three-dimensional vector space, over any field. While the theory of projective planes has an aspect that belongs to combinatorics too, that is absent in the... projective space. Projective space has properties analogous to those of a In mathematics, a compact space is a space that resembles a closed and bounded subset of Euclidean space Rn in that it is small in a certain sense and contains all its limit points. The modern general definition calls a topological space compact if every open cover of it has... compact In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. The Hausdorff condition is one in a series of separation axioms that can be imposed on a topological space, however it is the one that is most frequently... 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 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. See singularity theory for general discussion of the geometric theory, which... singularity at one of those extra points, but V(y=x²) is smooth.

While In a historical perspective on mathematics, the field of geometry that developed in the first half of the nineteenth century under the name projective geometry was a stepping stone from analytic geometry to algebraic geometry. When treated in terms of homogeneous co-ordinates it looks like an extension or technical... projective geometry was originally established on a 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. The geometry of Euclid was indeed synthetic, though... synthetic foundation, the use of In mathematics, homogeneous co-ordinates, introduced by August Ferdinand Möbius, make calculations possible in projective space just as Cartesian co-ordinates do in Euclidean space. The homogeneous co-ordinates of a point of projective space of dimension n are usually written as (x:y:z: ... :w), a row... homogenous 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.

The modern viewpoint

The modern approach to algebraic geometry redefines the basic objects. Varieties are subsumed in 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. He was awarded the Fields Medal in 1966 and coawarded the Crafoord Prize with Pierre Deligne in 1988. He declined the latter... Alexander Grothendieck's concept of a In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern algebraic geometry. Technically, 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.

Notes and history

Algebraic geometry was developed largely by the In relation with the history of mathematics, the Italian school of algebraic geometry refers to the work over half a century or more (flourishing roughly 1885-1935) done internationally in birational geometry, particularly on algebraic surfaces. There were in the region of 30 to 40 leading mathematicians who made major... Italian geometers in the early part of the 20th century. Enriques classified In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the complex number field, an algebraic surface is therefore of complex dimension two (as a complex manifold) and so of dimension four as a smooth manifold. The theory of algebraic surfaces is... algebraic surfaces up to birational isomorphism. The style of the Italian school was very intuitive and does not meet the modern standards of For the medical term see rigor (medicine) Rigour (American English: rigor) has a number of meanings in relation to intellectual life and discourse. These are separate from judicial and political applications with their suggestion of laws enforced to the letter, or political absolutism. A religion, too, may be worn lightly... rigor.

By the Events and trends Technology Jet engine invented Science Nuclear fission discovered by Otto Hahn, Lise Meitner and Fritz Strassmann Pluto, the ninth planet from the Sun, is discovered by Clyde Tombaugh British biologist Arthur Tansley coins term ecosystem War, peace and politics Socialists proclaim The death of Capitalism Rise to... 1930s and Centuries: 19th century - 20th century - 21st century Decades: 1890s 1900s 1910s 1920s 1930s - 1940s - 1950s 1960s 1970s 1980s 1990s Years: 1940 1941 1942 1943 1944 1945 1946 1947 1948 1949 Events and trends Technology First nuclear bomb First cruise missile, the V1 flying bomb and the first ballistic missile, the... 1940s, Oscar Zariski, André Weil and others realized that algebraic geometry needed to be rebuilt on foundations of In abstract algebra, commutative algebra is the field of study of commutative rings, their ideals, modules and algebras. It is foundational both for algebraic geometry and for algebraic number theory. The most prominent example for commutative rings are polynomial rings. The subjects real founder, in the days when it... commutative algebra and 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. Informally, it is an assignment of particular values to the variables in a mathematical statement or equation. So for... valuation theory. Commutative algebra (earlier known as In algebraic geometry, elimination theory is the classical name for algorithmic approaches to eliminating between polynomials of several variables. The linear case would now routinely be handled by Gauss-Jordan elimination, rather than the theoretical solution provided by Cramers rule. In the same way, computational techniques for elimination can... elimination theory and then In mathematics, ideal theory is the theory of ideals in commutative rings; and is the precursor name for the contemporary subject of commutative algebra. The name grew out of the central considerations, such as the Lasker-Noether theorem in algebraic geometry, and the ideal class group in algebraic number theory... ideal theory, and refounded as the study of commutative rings and their In abstract algebra, a module is a generalization of a vector space. In a vector space the set of scalars forms a field whereas in a module the scalars just form a ring. Much of the theory of modules consists of recovering desirable properties of vector spaces in the realm... modules) had been and was being developed by David Hilbert David Hilbert (January 23, 1862 – February 14, 1943) was a German mathematician born in Wehlau, near Königsberg, Prussia (now Znamensk, near Kaliningrad, Russia) who is recognized as one of the most influential mathematicians of the 19th and early 20th centuries. His own discoveries alone would have... David Hilbert, Max Noether, Emanuel Lasker (December 24, 1868 – January 11, 1941) was a German chess player and mathematician, born at Berlinchen in Brandenburg (now Barlinek in Poland). Emanuel Lasker Chess champion In 1894 he became the second World Chess Champion by defeating Steinitz with 10 wins, 4 draws and 5 losses. He... Emanuel Lasker, Emmy Noether (March 23, 1882 – April 14, 1935) was one of the most talented mathematicians of the early 20th century, with penetrating insights that she used to develop elegant abstractions which she formalized beautifully. Emmy Noether She was born Amalie Noether in Erlangen, Bavaria, Germany. Her father, Max Noether... Emmy Noether, Wolfgang Krull (1899 - 1971) was a German mathematician, after whom Krull dimension, the Krull topology, and Krulls principal ideal theorem are named. Categories: People stubs | 1899 births | 1971 deaths | Mathematicians ... Wolfgang Krull, and others. For a while there was no standard foundation for algebraic geometry.

In the Events and trends Technology United States tests the first fusion bomb. See History of nuclear weapons Sputnik, the first man-made satellite, and thus the Sputnik crisis The De Havilland Comet enters service as the worlds first jet airliner Charles Townes builds a maser in 1953 at Columbia University... 1950s and Centuries: 19th century - 20th century - 21st century Decades: 1900s 1910s 1920s 1930s 1940s 1950s - 1960s - 1970s 1980s 1990s 2000s 2010s Years: 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 Events and trends The 1960s was a turbulent decade of change around the world. Many of the trends of... 1960s 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. He was educated at the Lycée de Nimes and then from 1945 to 1948 at the Ecole Normale Supérieure in Paris. Serre was... Jean-Pierre Serre and 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. He was awarded the Fields Medal in 1966 and coawarded the Crafoord Prize with Pierre Deligne in 1988. He declined the latter... Alexander Grothendieck recast the foundations making use of the theory of 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... sheaf theory. Later, from about 1960, the idea of In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern algebraic geometry. Technically, a... schemes was worked out, in conjunction with a very refined apparatus of Homological algebra is the branch of mathematics which studies the methods of homology and cohomology in a general setting. These concepts originated in algebraic topology. Cohomology theories have been defined for many different objects such as topological spaces, sheaves, groups, rings, Lie algebras, and C-star algebras. The study of... homological techniques. After a decade of rapid development the field stabilised in the 1970s, and new applications were made, both to Traditionally, number theory is that branch of pure mathematics concerned with the properties of integers and contains many open problems that are easily understood even by non-mathematicians. More generally, the field has come to be concerned with a wider class of problems that arose naturally from the study of... number theory and to more classical geometric questions on algebraic varieties, For non-mathematical singularity theories, see singularity. In mathematics, singularity theory is the study of the failure of manifold structure. A loop of string can serve as an example of a one-dimensional manifold, if one neglects its width. What is meant by a singularity can be seen by dropping... singularities and In algebraic geometry, the moduli problem is to describe the parameters on which algebraic varieties depend. The use of the term modulus here for such a parameter goes back to the same source as in modular form: a modular form in general is some kind of differential form (or tensor... moduli.

An important class of varieties, not easily understood directly from their defining equations, are the For the purposes of algebraic geometry over the complex numbers, an abelian variety is a complex torus (a torus of real dimension 2n that is a complex manifold) that is also a projective algebraic variety of dimension n, i.e. can be defined in projective space by algebraic equations. In... abelian varieties, which are the projective varieties whose points form an abelian In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. The branch of mathematics which studies groups is called group theory. The historical origin of group theory goes back to the works of Evariste Galois (1830), concerning the... group. The prototypical examples are the In mathematics, elliptic curves are defined by certain cubic (the superscript exponent is three, a.k.a. third degree) equations. They have been used in the proof of Fermats last theorem and they also find applications in cryptography (for details, see the article elliptic curve cryptography) and integer factorization... elliptic curves, which have a rich theory. They were instrumental in the proof of Pierre de Fermat Fermats last theorem (sometimes abbreviated as FLT and also called Fermats great theorem) is one of the most famous theorems in the history of mathematics. It states that: There are no positive integers x, y, and z such that in which n is a natural... Fermat's last theorem and are also used in Elliptic curve cryptography (ECC) is an approach to public-key cryptography based on the mathematics of elliptic curves. The use of elliptic curves in cryptography was suggested independently by Neal Koblitz and Victor Miller in 1985. The main benefit of ECC is that under certain situations it uses smaller keys... elliptic curve cryptography.

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 A computer algebra system (CAS) is a software program that facilitates symbolic mathematics. Typically, these systems include arbitrary precision (bignum) arithmetic, allowing for instance to evaluate pi to 10,000 digits. symbolic manipulation engine, to simplify algebraic expressions, differentiate and integrate functions and solve equations graphing facility, to produce graphs... computer algebra systems.

See also

• This is a list of important publications in mathematics, organized by field. Some reasons why a particular publication might be regarded as important: Topic creator – A publication that created a new topic Breakthrough – A publication that changed scientific knowledge significantly Introduction – A publication that is a good... Important publications in algebraic geometry

References

A classical textbook, predating schemes:

• This article is about a mathematician. For the companion of Samuel Johnson, see Hodge (cat). William Vallance Douglas Hodge (17 June 1903 - 7 July 1975) was a Scottish mathematician, specifically a geometer. His discovery of topological relations between algebraic geometry and differential geometry - now called Hodge theory and pertaining more... Hodge, W. V. D., and Pedoe, Daniel, Methods of Algebraic Geometry: Volume 1, The headquarters of the Cambridge University Press, in Trumpington Street, Cambridge. The Cambridge University Press is a publisher given a Royal Charter by Henry VIII in 1534, and one of the two privileged presses. It published its first book in 1584, making it the longest-established printing house in the... Cambridge University Press, 1994, ISBN 0521469007
• Hodge, W. V. D., and Pedoe, Daniel, Methods of Algebraic Geometry: Volume 2, Cambridge University Press, 1994, ISBN 0521469015
• Hodge, W. V. D., and Pedoe, Daniel, Methods of Algebraic Geometry: Volume 3, Cambridge University Press, 1994, ISBN 0521467756

Modern textbooks that do not use the language of schemes:

• Cox, Little, and O'Shea, Ideals, Varieties, and Algorithms (second edition), Springer, 1997, ISBN 0387946802
• Griffiths, Phillip, and Harris, Joe, Principles of Algebraic Geometry, Wiley-Interscience, 1994, ISBN 0471050598
• Harris, Joe, Algebraic Geometry: A First Course, Springer-Verlag, 1995, ISBN 0387977163
• David Bryant Mumford (born 11 June 1937) is an American mathematician known for distinguished work in algebraic geometry, and then for research into vision and pattern theory. He is currently a professor in the Division of Applied Mathematics at Brown University, having previously had a long academic career at Harvard... Mumford, David, Algebraic Geometry I: Complex Projective Varieties, 2nd ed., Springer-Verlag, 1995, ISBN 3540586571
• Reid, Miles, Undergraduate Algebraic Geometry, Cambridge University Press, 1988, ISBN 0521356628
• Shafarevich, Igor, Basic Algebraic Geometry I: Varieties in Projective Space, Springer-Verlag, 2nd ed., 1995, ISBN 0387548122

Textbooks and references for schemes:

• Eisenbud, David, and Harris, Joe, The Geometry of Schemes, Springer-Verlag, 1998, ISBN 0387986375
• 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. He was awarded the Fields Medal in 1966 and coawarded the Crafoord Prize with Pierre Deligne in 1988. He declined the latter... Grothendieck, Alexander, Éléments de géométrie algébrique, Publications mathématiques de l'IHÉS, vols. 4, 8, 11, 17, 20, 24, 28, 32, 1960, 1961, 1963, 1964, 1965, 1966, 1967
• Grothendieck, Alexander, Éléments de géométrie algébrique, vol. 1, 2nd ed., Springer-Verlag, 1971, ISBN 3540051139
• Hartshorne, Robin, Algebraic Geometry, Springer-Verlag, 1997, ISBN 0387902449
• Mumford, David, The Red Book of Varieties and Schemes: Includes the Michigan Lectures (1974) on Curves and Their Jacobians, 2nd ed., Springer-Verlag, 1999, ISBN 354063293X
• Shafarevich, Igor, Basic Algebraic Geometry II: Schemes and Complex Manifolds, Springer-Verlag, 2nd ed., 1995, ISBN 0387548122

On the internet:

• Kevin R. Coombes: Algebraic Geometry: A Total Hypertext Online System (http://odin.mdacc.tmc.edu/~krc/agathos/)
• Algebraic geometry (http://planetmath.org/encyclopedia/AlgebraicGeometry.html) entry on PlanetMath (http://planetmath.org/)