Algebraic geometry

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This Togliatti surface is an algebraic surface of degree five. The picture represents a portion of its real 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[change | change source]

The main objects of study in algebraic geometry are 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 curves, which include lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves and quartic curves like lemniscates, and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates match a given polynomial equation. Simple questions involve the study of points of special interest like the singular points, the inflection points and the 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 particular solution: this leads into some of the deepest place in all of mathematics, both conceptually and in terms of technique.

Developments in the 20th century[change | change source]

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 an ambient coordinate space. The developments in topology, differential and complex geometry came much in the same way.

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

Scheme theory[change | change source]

One key achievement of this abstract algebraic geometry is Grothendieck's scheme theory. This allows one to use sheaf theory to study algebraic varieties in a way which is very similar to its use in the study of differential and analytic manifolds. This is obtained by extending the notion of point: In classical algebraic geometry, a point of an affine variety may be identified, through Hilbert's Nullstellensatz, with a maximal ideal of the coordinate 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[change | change source]

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 proof of the Fermat's last theorem is an example of the power of this approach.

References[change | change source]

  1. Lang, S. (2013). Algebraic number theory (Vol. 110). Springer Science & Business Media.
  2. Bochnak, J., Coste, M., & Roy, M. F. (2013). Real algebraic geometry (Vol. 36). Springer Science & Business Media.
  3. Bruce, J. W., & Giblin, P. J. (1992). Curves and Singularities: a geometrical introduction to singularity theory. Cambridge university press.
  4. Schenck, H. (2003). Computational algebraic geometry (Vol. 58). Cambridge University Press.