Algebraic geometry

Not to be confused withGeometric algebra,an application of Clifford algebra to geometry.

ThisTogliatti surfaceis analgebraic surfaceof degree five. The picture represents a portion of its reallocus

Algebraic geometryis a type ofmathematics,studyingpolynomialequations. Modern algebraic geometry is based on more abstract techniques ofabstract algebra,especiallycommutative algebra,with the language and the problems ofgeometry.

Aims

The main objects of study in algebraic geometry arealgebraic varieties,which are geometricmanifestationsof sets of solutions ofsystems of polynomial equations.Examples of the most studied classes of algebraic varieties are:plane algebraic curves,which includelines,circles,parabolas,ellipses,hyperbolas,cubic curveslikeelliptic curvesand quartic curves likelemniscates,andCassini ovals.A point of the plane belongs to an algebraic curve if itscoordinatesmatch a given polynomial equation. Simple questions involve the study of points of special interest like thesingular points,theinflection pointsand thepoints at infinity.More difficult questions involve thetopologyof the curve and relations between the curves given by different equations.

Algebraic geometry takes a central place in modern mathematics. The concepts it uses connects it to diverse fields such ascomplex analysis,topologyandnumber theory.At the start, algebraic geometry was about studyingsystems of polynomial equationsin several variables. Algebraic geometry starts at the point whereequation solvingleaves off: In many cases, finding the properties of all the solutions in a given set of equations have, are more important than finding a particular solution: this leads into some of the deepest place in all of mathematics, bothconceptuallyand in terms of technique.

Developments in the 20th century

Much of the development 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 anambient coordinate space.The developments intopology,differentialandcomplex geometrycame much in the same way.

In the 20th century, algebraic geometry split into several subareas.

• The main stream 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.
• The study of the points of an algebraic variety with coordinates in the field of therational numbersor in anumber fieldbecamearithmetic geometry,a subfield ofalgebraic number theory.
• The study of the real points of analgebraic varietyis the subject ofreal algebraic geometry.
• A large part ofsingularity theoryis devoted to the singularities of algebraic varieties.
• Whencomputersbecame more common, a field named 'computational algebraic geometry' was developed.It looks at the intersection of algebraic geometry andcomputer algebra.It is concerned with the development ofalgorithmsandsoftwarefor studying and finding the properties of explicitly given algebraic varieties.

Scheme theory

One key achievement of this abstract algebraic geometry isGrothendieck'sscheme theory.This allows one to usesheaf theoryto study algebraic varieties in a way which is very similar to its use in the study ofdifferentialandanalytic manifolds.This is obtained by extending the notion of point: In classical algebraic geometry, a point of anaffine varietymay be identified, throughHilbert's Nullstellensatz,with amaximal idealof 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 sub variety.

Achievements of the scheme theory

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's proofof theFermat's last theoremis an example of the power of this approach.