Algebraic geometry occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields ascomplex analysis, topology andnumber theory. As a study of systems of polynomial equations in several variables, the subject of algebraic geometry begins with finding specific solutions viaequation solving, and then proceeds to understand the intrinsic properties of the totality of solutions of a system of equations. This understanding requires both conceptual theory and computational technique.
In the 20th century, algebraic geometry split into several subareas.
The mainstream of algebraic geometry is devoted to the study of the complex points of the algebraic varieties and more generally to the points with coordinates in analgebraically closed field.
A large part ofsingularity theory is devoted to the singularities of algebraic varieties.
Computational algebraic geometry is an area that has emerged at the intersection of algebraic geometry andcomputer algebra, with the rise of computers. It consists mainly ofalgorithm design andsoftware development for the study of properties of explicitly given algebraic varieties.
Much of the development of the mainstream of algebraic geometry in the 20th century occurred within an 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,differential andcomplex geometry. One key achievement of this abstract algebraic geometry isGrothendieck'sscheme theory which allows one to usesheaf theory to study algebraic varieties in a way which is very similar to its use in the study ofdifferential andanalytic manifolds. This is obtained by extending the notion of point: In classical algebraic geometry, a point of an affine variety may be identified, throughHilbert's Nullstellensatz, with amaximal ideal of thecoordinate ring, while the points of the corresponding affine scheme are all prime ideals of this ring. This means that a point of such a scheme may be either a usual point or a subvariety. This approach also enables a unification of the language and the tools of classical algebraic geometry, mainly concerned with complex points, and of algebraic number theory.Wiles' proof of the longstanding conjecture calledFermat's Last Theorem is an example of the power of this approach.
In classical algebraic geometry, the main objects of interest are the vanishing sets of collections ofpolynomials, meaning the set of all points that simultaneously satisfy one or morepolynomial equations. For instance, thetwo-dimensionalsphere of radius 1 in three-dimensionalEuclidean spaceR3 could be defined as the set of all points with
A "slanted" circle inR3 can be defined as the set of all points which satisfy the two polynomial equations
First we start with afieldk. In classical algebraic geometry, this field was always the complex numbersC, but many of the same results are true if we assume only thatk isalgebraically closed. We consider theaffine space of dimensionn overk, denotedAn(k) (or more simplyAn, whenk is clear from the context). When one fixes a coordinate system, one may identifyAn(k) withkn. The purpose of not working withkn is to emphasize that one "forgets" the vector space structure thatkn carries.
A functionf :An →A1 is said to bepolynomial (orregular) if it can be written as a polynomial, that is, if there is a polynomialp ink[x1,...,xn] such thatf(M) =p(t1,...,tn) for every pointM with coordinates (t1,...,tn) inAn. The property of a function to be polynomial (or regular) does not depend on the choice of a coordinate system inAn.
When a coordinate system is chosen, the regular functions on the affinen-space may be identified with the ring ofpolynomial functions inn variables overk. Therefore, the set of the regular functions onAn is a ring, which is denotedk[An].
We say that a polynomialvanishes at a point if evaluating it at that point gives zero. LetS be a set of polynomials ink[An]. Thevanishing set of S (orvanishing locus orzero set) is the setV(S) of all points inAn where every polynomial inS vanishes. Symbolically,
A subset ofAn which isV(S), for someS, is called analgebraic set. TheV stands forvariety (a specific type of algebraic set to be defined below).
Given a subsetU ofAn, can one recover the set of polynomials which generate it? IfU isany subset ofAn, defineI(U) to be the set of all polynomials whose vanishing set containsU. TheI stands forideal: if two polynomialsf andg both vanish onU, thenf+g vanishes onU, and ifh is any polynomial, thenhf vanishes onU, soI(U) is always an ideal of the polynomial ringk[An].
Two natural questions to ask are:
Given a subsetU ofAn, when isU =V(I(U))?
Given a setS of polynomials, when isS =I(V(S))?
The answer to the first question is provided by introducing theZariski topology, a topology onAn whose closed sets are the algebraic sets, and which directly reflects the algebraic structure ofk[An]. ThenU =V(I(U)) if and only ifU is an algebraic set or equivalently a Zariski-closed set. The answer to the second question is given byHilbert's Nullstellensatz. In one of its forms, it says thatI(V(S)) is theradical of the ideal generated byS. In more abstract language, there is aGalois connection, giving rise to twoclosure operators; they can be identified, and naturally play a basic role in the theory; theexample is elaborated at Galois connection.
For various reasons we may not always want to work with the entire ideal corresponding to an algebraic setU.Hilbert's basis theorem implies that ideals ink[An] are always finitely generated.
An algebraic set is calledirreducible if it cannot be written as the union of two smaller algebraic sets. Any algebraic set is a finite union of irreducible algebraic sets and this decomposition is unique. Thus its elements are called theirreducible components of the algebraic set. An irreducible algebraic set is also called avariety. It turns out that an algebraic set is a variety if and only if it may be defined as the vanishing set of aprime ideal of thepolynomial ring.
Some authors do not make a clear distinction between algebraic sets and varieties and useirreducible variety to make the distinction when needed.
Just ascontinuous functions are the natural maps ontopological spaces andsmooth functions are the natural maps ondifferentiable manifolds, there is a natural class of functions on an algebraic set, calledregular functions orpolynomial functions. A regular function on an algebraic setV contained inAn is the restriction toV of a regular function onAn. For an algebraic set defined on the field of the complex numbers, the regular functions are smooth and evenanalytic.
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 anormaltopological space, where theTietze 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 onV form a ring, which we denote byk[V]. This ring is called thecoordinate ring of V.
Since regular functions on V come from regular functions onAn, there is a relationship between the coordinate rings. Specifically, if a regular function onV is the restriction of two functionsf andg ink[An], thenf − g is a polynomial function which is null onV and thus belongs toI(V). Thusk[V] may be identified withk[An]/I(V).
Using regular functions from an affine variety toA1, we can defineregular maps from one affine variety to another. First we will define a regular map from a variety into affine space: LetV be a variety contained inAn. Choosem regular functions onV, and call themf1, ...,fm. We define aregular mapf fromV toAm by lettingf = (f1, ...,fm). In other words, eachfi determines one coordinate of therange off.
IfV′ is a variety contained inAm, we say thatf is aregular map fromV toV′ if the range off is contained inV′.
The definition of the regular maps apply also to algebraic sets.The regular maps are also calledmorphisms, as they make the collection of all affine algebraic sets into acategory, where the objects are the affine algebraic sets and themorphisms are the regular maps. The affine varieties is a subcategory of the category of the algebraic sets.
Given a regular mapg fromV toV′ and a regular functionf ofk[V′], thenf ∘g ∈k[V]. The mapf →f ∘g is aring homomorphism fromk[V′] tok[V]. Conversely, every ring homomorphism fromk[V′] tok[V] defines a regular map fromV toV′. This defines anequivalence of categories between the category of algebraic sets and theopposite category of the finitely generatedreducedk-algebras. This equivalence is one of the starting points ofscheme theory.
In contrast to the preceding sections, this section concerns only varieties and not algebraic sets. On the other hand, the definitions extend naturally to projective varieties (next section), as an affine variety and its projective completion have the same field of functions.
IfV is an affine variety, its coordinate ring is anintegral domain and has thus afield of fractions which is denotedk(V) and called thefield of the rational functions onV or, shortly, thefunction field ofV. Its elements are the restrictions toV of therational functions over the affine space containingV. Thedomain of a rational functionf is notV but thecomplement of the subvariety (a hypersurface) where the denominator off vanishes.
As with regular maps, one may define arational map from a varietyV to a varietyV'. As with the regular maps, the rational maps fromV toV' may be identified to thefield homomorphisms fromk(V') tok(V).
Two affine varieties arebirationally equivalent if there are two rational functions between them which areinverse one to the other in the regions where both are defined. Equivalently, they are birationally equivalent if their function fields are isomorphic.
An affine variety is arational variety if it is birationally equivalent to an affine space. This means that the variety admits arational parameterization, that is aparametrization withrational functions. For example, the circle of equation is a rational curve, as it has theparametric equation
which may also be viewed as a rational map from the line to the circle.
The problem ofresolution of singularities is to know if every algebraic variety is birationally equivalent to a variety whose projective completion is nonsingular (see alsosmooth completion). It was solved in the affirmative incharacteristic 0 byHeisuke Hironaka in 1964 and is yet unsolved in finite characteristic.
Parabola (y =x2, red) and cubic (y =x3, blue) in projective space
Just as the formulas for the roots of second, third, and fourth degree polynomials suggest extending real numbers to the more algebraically complete setting of the complex numbers, many properties of algebraic varieties suggest extending affine space to a more geometrically complete projective space. Whereas the complex numbers are obtained by adding the numberi, a root of the polynomialx2 + 1, projective space is obtained by adding in appropriate points "at infinity", points where parallel lines may meet.
To see how this might come about, consider the varietyV(y −x2). If we draw it, we get aparabola. Asx goes to positive infinity, the slope of the line from the origin to the point (x, x2) also goes to positive infinity. Asx goes to negative infinity, the slope of the same line goes to negative infinity.
Compare this to the varietyV(y − x3). This is acubic curve. Asx goes to positive infinity, the slope of the line from the origin to the point (x, x3) goes to positive infinity just as before. But unlike before, asx goes to negative infinity, the slope of the same line goes to positive infinity as well; the exact opposite of the parabola. So the behavior "at infinity" ofV(y − x3) is different from the behavior "at infinity" ofV(y − x2).
The consideration of theprojective completion of the two curves, which is their prolongation "at infinity" in theprojective plane, allows us to quantify this difference: the point at infinity of the parabola is aregular point, whose tangent is theline at infinity, while the point at infinity of the cubic curve is acusp. Also, both curves are rational, as they are parameterized byx, and theRiemann-Roch theorem implies that the cubic curve must have a singularity, which must be at infinity, as all its points in the affine space are regular.
Thus many of the properties of algebraic varieties, including birational equivalence and all the topological properties, depend on the behavior "at infinity" and so it is natural to study the varieties in projective space. 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 these reasons, projective space plays a fundamental role in algebraic geometry.
Nowadays, theprojective spacePn of dimensionn is usually defined as the set of the lines passing through a point, considered as the origin, in the affine space of dimensionn + 1, or equivalently to the set of the vector lines in a vector space of dimensionn + 1. When a coordinate system has been chosen in the space of dimensionn + 1, all the points of a line have the same set of coordinates, up to the multiplication by an element ofk. This defines thehomogeneous coordinates of a point ofPn as a sequence ofn + 1 elements of the base fieldk, defined up to the multiplication by a nonzero element ofk (the same for the whole sequence).
A polynomial inn + 1 variables vanishes at all points of a line passing through the origin if and only if it ishomogeneous. In this case, one says that the polynomialvanishes at the corresponding point ofPn. This allows us to define aprojective algebraic set inPn as the setV(f1, ...,fk), where a finite set of homogeneous polynomials{f1, ...,fk} vanishes. Like for affine algebraic sets, there is abijection between the projective algebraic sets and the reducedhomogeneous ideals which define them. Theprojective varieties are the projective algebraic sets whose defining ideal is prime. In other words, a projective variety is a projective algebraic set, whosehomogeneous coordinate ring is anintegral domain, theprojective coordinates ring being defined as the quotient of the graded ring or the polynomials inn + 1 variables by the homogeneous (reduced) ideal defining the variety. Every projective algebraic set may be uniquely decomposed into a finite union of projective varieties.
The only regular functions which may be defined properly on a projective variety are the constant functions. Thus this notion is not used in projective situations. On the other hand, thefield of the rational functions orfunction field is a useful notion, which, similarly to the affine case, is defined as the set of the quotients of two homogeneous elements of the same degree in the homogeneous coordinate ring.
Real algebraic geometry is the study of real algebraic varieties.
The fact that the field of the real numbers is anordered field cannot be ignored in such a study. For example, the curve of equation is a circle if, but has no real points if. Real algebraic geometry also investigates, more broadly,semi-algebraic sets, which are the solutions of systems of polynomial inequalities. For example, neither branch of thehyperbola of equation is a real algebraic variety. However, the branch in the first quadrant is a semi-algebraic set defined by and.
One open problem in real algebraic geometry is the following part ofHilbert's sixteenth problem: Decide which respective positions are possible for the ovals of a nonsingularplane curve of degree 8.
One may date the origin of computational algebraic geometry to meeting EUROSAM'79 (International Symposium on Symbolic and Algebraic Manipulation) held atMarseille, France, in June 1979. At this meeting,
Daniel Lazard presented a new algorithm for solving systems of homogeneous polynomial equations with acomputational complexity which is essentially polynomial in the expected number of solutions and thus singly exponential in the number of the unknowns. This algorithm is strongly related toMacaulay'smultivariate resultant.
Since then, most results in this area are related to one or several of these items either by using or improving one of these algorithms, or by finding algorithms whose complexity is singly exponential in the number of the variables.
A body of mathematical theory complementary to symbolic methods callednumerical algebraic geometry has been developed over the last several decades. The main computational method ishomotopy continuation. This supports, for example, a model offloating-point computation for solving problems of algebraic geometry.
AGröbner basis is a system ofgenerators of a polynomialideal whose computation allows the deduction of many properties of the affine algebraic variety defined by the ideal.
Given an idealI defining an algebraic setV:
V is empty (over an algebraically closed extension of the basis field) if and only if the Gröbner basis for anymonomial ordering is reduced to {1}.
By means of theHilbert series, one may compute thedimension and thedegree ofV from any Gröbner basis ofI for a monomial ordering refining the total degree.
If the dimension ofV is 0, then one may compute the points (finite in number) ofV from any Gröbner basis ofI (seeSystems of polynomial equations).
A Gröbner basis computation allows one to remove fromV all irreducible components which are contained in a given hypersurface.
A Gröbner basis computation allows one to compute the Zariski closure of the image ofV by the projection on thek first coordinates, and the subset of the image where the projection is notproper.
More generally, Gröbner basis computations allow one to compute the Zariski closure of the image and thecritical points of a rational function ofV into another affine variety.
Gröbner basis computations do not allow one to compute directly theprimary decomposition ofI nor the prime ideals defining the irreducible components ofV, but most algorithms for this involve Gröbner basis computation. The algorithms which are not based on Gröbner bases useregular chains but may need Gröbner bases in some exceptional situations.
Gröbner bases are deemed to be difficult to compute. In fact they may contain, in the worst case, polynomials whose degree is doubly exponential in the number of variables and a number of polynomials which is also doubly exponential. However, this is only a worst-case complexity, and the complexity bound of Lazard's algorithm of 1979 may frequently apply.Faugère F5 algorithm realizes this complexity, as it may be viewed as an improvement of Lazard's 1979 algorithm. It follows that the best implementations allow one to compute almost routinely with algebraic sets of degree more than 100. This means that, presently, the difficulty of computing a Gröbner basis is strongly related to the intrinsic difficulty of the problem.
This theorem concerns the formulas of thefirst-order logic whoseatomic formulas are polynomial equalities or inequalities between polynomials with real coefficients. These formulas are thus the formulas which may be constructed from the atomic formulas by the logical operatorsand (∧),or (∨),not (¬),for all (∀), andthere exists (∃). Tarski's theorem asserts that, from such a formula, one may compute an equivalent formula without quantifiers (∀, ∃).
The complexity of CAD is doubly exponential in the number of variables. This means that CAD allows one, in theory, to solve every problem of real algebraic geometry which may be expressed by such a formula—this is almost every problem concerning explicitly given varieties and semi-algebraic sets.
While Gröbner basis computation has doubly exponential complexity only in rare cases, CAD has almost always this high complexity. This implies that, unless most polynomials appearing in the input are linear, it may not solve problems with more than four variables.
Since 1973, most of the research on this subject is devoted either to improving CAD or finding alternative algorithms in special cases of general interest.
As an example of the state of art, there are efficient algorithms to find at least one point in every connected component of a semi-algebraic set, and thus to test whether a semi-algebraic set is empty. On the other hand, CAD is yet, in practice, the best algorithm to count the number of connected components.
The basic general algorithms of computational geometry have a doubly exponential worst-casecomplexity. More precisely, ifd is the maximal degree of the input polynomials andn the number of variables, then their complexity is at mostd2cn for some constantc, and, for some inputs, the complexity is at leastd2c′n for another constantc′.
During the last 20 years of the 20th century, various algorithms were introduced to solve specific subproblems with a better complexity. Most of these algorithms have a complexity.[1]
Among those algorithms which solve a subproblem of the problems solved by Gröbner bases, one may citetesting whether an affine variety is empty andsolving nonhomogeneous polynomial systems which have a finite number of solutions. Such algorithms are rarely implemented because, on most entries,Faugère's F4 and F5 algorithms have a better practical efficiency and probably a similar or better complexity (probably because the evaluation of the complexity of Gröbner basis algorithms on a particular class of entries is a difficult task which has been done only in a few special cases).
The main algorithms of real algebraic geometry which solve a problem solved by CAD are related to the topology of semi-algebraic sets. One may citecounting the number of connected components,testing whether two points are in the same components, andcomputing aWhitney stratification of a real algebraic set. They have a complexity of, but the constant involved byO notation is so high that using them to solve any nontrivial problem effectively solved by CAD, is impossible even if one could use all the existing computing power in the world. Therefore, these algorithms have never been implemented and it is an active research area to search for algorithms with have together a good asymptotic complexity and a good practical efficiency.
The modern approaches to algebraic geometry redefine and effectively extend the range of basic objects in various levels of generality to schemes,formal schemes,ind-schemes,algebraic spaces,algebraic stacks, and so on. The need for this arises already from the useful ideas within the theory of varieties; for example, the formal functions of Zariski can be accommodated by introducingnilpotent elements in structure rings; considering spaces of loops and arcs, constructing quotients by group actions, and developing formal grounds for naturalintersection theory anddeformation theory lead to some of the further extensions.
Most remarkably, in the early 1960s,algebraic varieties were subsumed intoAlexander Grothendieck's concept of ascheme. Their local objects are affine schemes or prime spectra which arelocally ringed spaces which form a category which is antiequivalent to the category of commutative unital rings, extending the duality between the category of affine algebraic varieties over a fieldk, and the category of finitely generated reducedk-algebras. The gluing is along the Zariski topology; one can glue within the category of locally ringed spaces, but also, using theYoneda embedding, within the more abstract category ofpresheaves of sets over the category of affine schemes. The Zariski topology in the set-theoretic sense is then replaced by aGrothendieck topology. Grothendieck introduced Grothendieck topologies having in mind more exotic but geometrically finer and more sensitive examples than the crude Zariski topology, namely theétale topology, and the two flat Grothendieck topologies: fppf and fpqc; nowadays some other examples have become prominent, including theNisnevich topology. Sheaves can be furthermore generalized tostacks in the sense of Grothendieck, usually with some additional representability conditions leading toArtin stacks and, even finer,Deligne–Mumford stacks, both often calledalgebraic stacks.
Sometimes, other algebraic sites replace the category of affine schemes. For example,Nikolai Durov has introduced commutative algebraicmonads as a generalization of local objects in a generalized algebraic geometry. Versions of atropical geometry, of anabsolute geometry over afield of one element, and an algebraic analogue ofArakelov's geometry were realized in this setup.
Another formal generalization is possible touniversal algebraic geometry in which everyvariety of algebras has its own algebraic geometry. The termvariety of algebras should not be confused withalgebraic variety.
The language of schemes, stacks, and generalizations has proved to be a valuable way of dealing with geometric concepts and become cornerstones of modern algebraic geometry.
Algebraic stacks can be further generalized, and for many practical questions likedeformation theory andintersection theory, this is often the most natural approach. One can extend theGrothendieck site of affine schemes to ahigher-categorical site ofderived affine schemes, by replacing the commutative rings with aninfinity category ofdifferential graded commutative algebras, or of simplicial commutative rings or a similar category with an appropriate variant of a Grothendieck topology. One can also replace presheaves of sets with presheaves ofsimplicial sets (or ofinfinity groupoids). Then, in presence of an appropriate homotopic machinery, one can develop a notion of derived stack as such a presheaf on the infinity category of derived affine schemes, which satisfies certain infinite-categorical versions of sheaf axioms (and to be algebraic, inductively a sequence of representability conditions).Quillen model categories, Segal categories, andquasicategories are some of the most-used tools to formalize this, yielding thederived algebraic geometry, introduced by the school ofCarlos Simpson, including Andre Hirschowitz,Bertrand Toën, Gabrielle Vezzosi, Michel Vaquié, and others; and developed further byJacob Lurie,Bertrand Toën, andGabriele Vezzosi. Another (noncommutative) version of derived algebraic geometry, using A-infinity categories, has been developed from the early 1990s byMaxim Kontsevich and followers.
Some of the roots of algebraic geometry date back to the work of theHellenistic Greeks from the 5th century BC. TheDelian problem, for instance, was to construct a lengthx so that the cube of sidex contained the same volume as the rectangular boxa2b for given sidesa andb.Menaechmus (c. 350 BC) considered the problem geometrically by intersecting the pair of plane conicsay = x2 andxy = ab.[2] In the 3rd century BC,Archimedes andApollonius systematically studied additional problems onconic sections using coordinates.[2][3]Apollonius in the Conics further developed a method that is so similar to analytic geometry that his work is sometimes thought to have anticipated the work ofDescartes by some 1800 years.[4] His application of reference lines, adiameter, and atangent is essentially no different from our modern use of a coordinate frame, where the distances measured along the diameter from the point of tangency are the abscissas, and the segments parallel to the tangent and intercepted between the axis and the curve are the ordinates. He further developed relations between the abscissas and the corresponding coordinates using geometric methods like using parabolas and curves.[5][6][7] Medieval mathematicians, includingOmar Khayyam,Leonardo of Pisa,Gersonides andNicole Oresme in theMedieval Period,[8] solved certain cubic and quadratic equations by purely algebraic means and then interpreted the results geometrically. ThePersian mathematicianOmar Khayyám (born 1048 AD) believed that there was a relationship betweenarithmetic,algebra, andgeometry.[9][10][11] This was criticized by Jeffrey Oaks, who claims that the study of curves by means of equations originated with Descartes in the seventeenth century.[12]
Such techniques of applying geometrical constructions to algebraic problems were also adopted by a number ofRenaissance mathematicians such asGerolamo Cardano andNiccolò 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, notablyBlaise Pascal who argued against the use of algebraic and analytical methods in geometry.[13] The French mathematiciansFranciscus Vieta and laterRené Descartes andPierre de Fermat revolutionized the conventional way of thinking about construction problems through the introduction ofcoordinate geometry. They were interested primarily in the properties ofalgebraic curves, such as those defined byDiophantine equations (in the case of Fermat), and the algebraic reformulation of the classical Greek works on conics and cubics (in the case of Descartes).
During the same period, Blaise Pascal andGérard Desargues approached geometry from a different perspective, developing thesynthetic notions ofprojective geometry. Pascal and Desargues also studied curves, but from the purely geometrical point of view: the analog of the Greekruler-and-compass construction. Ultimately, theanalytic 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 ofNewton andLeibniz. However, by the end of the 18th century, most of the algebraic character of coordinate geometry was subsumed by thecalculus of infinitesimals ofLagrange andEuler.
It took the simultaneous 19th-century developments ofnon-Euclidean geometry andAbelian integrals in order to bring the old algebraic ideas back into the geometrical fold. The first of these new developments was seized up byEdmond Laguerre andArthur Cayley, who attempted to ascertain the generalized metric properties of projective space. Cayley introduced the idea ofhomogeneous polynomial forms, and more specificallyquadratic forms, on projective space. Subsequently,Felix Klein studied projective geometry (along with other types of geometry) from the viewpoint that the geometry on a space is encoded in a certain class oftransformations 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 theprojective linear transformations which were normally regarded as giving the fundamentalKleinian geometry on projective space, they concerned themselves also with the higher-degreebirational transformations. This weaker notion of congruence would later lead members of the 20th centuryItalian school of algebraic geometry to classifyalgebraic surfaces up tobirational isomorphism.
The second early-19th-century development, that of Abelian integrals, would leadBernhard Riemann to the development ofRiemann surfaces.
In the same period began the algebraization of the algebraic geometry throughcommutative algebra. The prominent results in this direction areHilbert's basis theorem andHilbert's Nullstellensatz, which are the basis of the connection between algebraic geometry and commutative algebra, andMacaulay'smultivariate resultant, which is the basis ofelimination theory. Probably because of the size of the computation which is implied by multivariate resultants, elimination theory was forgotten during the middle of the 20th century until it was renewed bysingularity theory and computational algebraic geometry.[a]
In the 1950s and 1960s,Jean-Pierre Serre andAlexander Grothendieck recast the foundations making use ofsheaf theory. Later, from about 1960, and largely led by Grothendieck, the idea ofschemes was worked out, in conjunction with a very refined apparatus ofhomological techniques. After a decade of rapid development the field stabilized in the 1970s, and new applications were made, both tonumber theory and to more classical geometric questions on algebraic varieties,singularities,moduli, andformal moduli.
An important class of varieties, not easily understood directly from their defining equations, are theabelian varieties, which are the projective varieties whose points form an abeliangroup. The prototypical examples are theelliptic curves, which have a rich theory. They were instrumental in theproof ofFermat's Last Theorem and are also used inelliptic-curve cryptography.
In parallel with the abstract trend of the algebraic geometry, which is concerned with general statements about varieties, methods for effective computation with concretely-given varieties have also been developed, which lead to the new area of computational algebraic geometry. One of the founding methods of this area is the theory ofGröbner bases, introduced byBruno Buchberger in 1965. Another founding method, more specially devoted to real algebraic geometry, is thecylindrical algebraic decomposition, introduced byGeorge E. Collins in 1973.
Ananalytic variety over the field of real or complex numbers is defined locally as the set of common solutions of several equations involvinganalytic functions. It is analogous to the concept ofalgebraic variety in that it carries a structure sheaf of analytic functions instead of regular functions. Anycomplex manifold is a complex analytic variety. Since analytic varieties may havesingular points, not all complex analytic varieties are manifolds. Over a non-archimedean field analytic geometry is studied viarigid analytic spaces.
Modern analytic geometry over the field of complex numbers is closely related to complex algebraic geometry, as has been shown byJean-Pierre Serre in his paperGAGA,[14] the name of which is French forAlgebraic geometry and analytic geometry. The GAGA results over the field of complex numbers may be extended to rigid analytic spaces over non-archimedean fields.[15]
^A witness of this oblivion is the fact thatVan der Waerden removed the chapter on elimination theory from the third edition (and all the subsequent ones) of his treatiseModerne algebra (in German).[citation needed]
^O'Connor, J. J.; Robertson, E. F."Omar Khayyam". School of Mathematics and Statistics, University of St Andrews. Archived fromthe original on November 12, 2017.Khayyam himself seems to have been the first to conceive a general theory of cubic equations.
^Oaks, Jeffrey (January 2016)."Excavating the errors in the "Mathematics" chapter of 1001 Inventions".Pp. 151-171 in: Sonja Brentjes, Taner Edis, Lutz Richter-Bernburd Edd., 1001 Distortions: How (Not) to Narrate History of Science, Medicine, and Technology in Non-Western Cultures.Archived from the original on 2021-02-27.
^Tannenbaum, Allen (1982).Invariance and Systems Theory: Algebraic and Geometric Aspects. Lecture Notes in Mathematics. Vol. 845. Springer-Verlag.ISBN9783540105657.
