What is Algebraic geometry?
{{Distinguish2|[[Geometric algebra]], an application of Clifford algebra to geometry}} {{jargon|date=January 2024}} [[Image:Togliatti surface.png|thumb|right|This [[Togliatti surface]] is an [[algebraic surface]] of degree five. The picture represents a portion of its real [[locus (mathematics)|locus]]]] '''Algebraic geometry''' is a type of [[mathematics]], studying [[polynomial]] equations. Modern algebraic geometry is based on more abstract techniques of [[abstract algebra]], especially [[commutative algebra]], with the language and the problems of [[geometry]]. ==Aims== The main objects of study in algebraic geometry are [[algebraic variety|algebraic varieties]], which are geometric [[manifestations]] of sets of solutions of [[systems of polynomial equations]]. Examples of the most studied classes of algebraic varieties are: [[plane algebraic curve]]s, which include [[line (geometry)|lines]], [[circle]]s, [[parabola]]s, [[ellipse]]s, [[hyperbola]]s, [[cubic curve]]s like [[elliptic curve]]s and quartic curves like [[lemniscate of Bernoulli|lemniscate]]s, and [[Cassini oval]]s. A point of the plane belongs to an algebraic curve if its [[Coordinate system|coordinates]] match a given polynomial equation. Simple questions involve the study of points of special interest like the [[singular point of a curve|singular point]]s, the [[inflection point]]s and the [[point at infinity|points at infinity]]. More difficult questions involve the [[topology]] of 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 as [[complex analysis]], [[topology]] and [[number theory]]. At the start, algebraic geometry was about studying [[systems of polynomial equations]] in several variables. Algebraic geometry starts at the point where [[equation solving]] leaves off: In many cases, finding the properties of all the solutions in a given set of equations have, are more important than finding a p