Tricki
a repository of mathematical know-how

Algebraic geometry front page

Quick description

Algebraic geometry is the study of the geometric properties of sets of solutions to algebraic equations. Many well-known manifolds can be described in this way (e.g. the 2-sphere is the set of solutions in \R^3 to the equation x^2 + y^2 + z^2 = 1), and so in principle algebraic geometry has a lot of overlap with other areas of geometry. However, two characteristic features of algebraic geometry are: one very often works over an algebraically closed field, such as \C (so that parts of algebraic geometry have quite a lot of overlap with the theory of complex manifolds, or complex analytic geometry); one very often considers projective varieties rather than affine varieties. (Thus the 2-sphere is the set of real points of an affine variety; the set of complex solutions of the associated projective variety is a quadric surface.)

Algebraic geometry can use both geometric methods and algebraic methods. When it does use algebraic methods, it often interprets them in geometric terms. This opens up the possibility of using geometric ideas, and geometric intuition, even when studying algebraic problems over the integers or rational numbers, or even over finite fields. Thus algebraic geometry plays an important role in the study of Diophantine equations.

Algebraic geometry articles

Counting constants

Dimension counting via incidence varieties

How to compute the dimensions of the fibres of a map of varieties

Use finite fields

Comments

Post new comment

(Note: commenting is not possible on this snapshot.)

Before posting from this form, please consider whether it would be more appropriate to make an inline comment using the Turn commenting on link near the bottom of the window. (Simply click the link, move the cursor over the article, and click on the piece of text on which you want to comment.)

snapshot
Notifications