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Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical problems. Classically, it studies zeros of multivariate polynomials ; the modern approach generalizes this in a few different aspects.
Algebraic geometry is the place where the algebra involved in solving systems of simultaneous multivariable polynomial equations meets the geometry of curves, surfaces, and higher dimensional algebraic varieties.
This is a list of algebraic geometry topics, by Wikipedia page. Classical topics in projective geometry. Affine space; Projective space; Projective line, cross-ratio;
The twisted cubic is a projective algebraic variety. Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Modern definitions generalize this concept in ...
In mathematics, specifically algebraic geometry, a scheme is a structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations x = 0 and x 2 = 0 define the same algebraic variety but different schemes) and allowing "varieties" defined over any commutative ring (for example, Fermat curves are defined over the integers).
Real algebra is the part of algebra which is relevant to real algebraic (and semialgebraic) geometry. It is mostly concerned with the study of ordered fields and ordered rings (in particular real closed fields ) and their applications to the study of positive polynomials and sums-of-squares of polynomials .
In the above example, a connection with classical Galois theory can be seen by regarding ^ as the profinite Galois group Gal(F /F) of the algebraic closure F of any finite field F, over F. That is, the automorphisms of F fixing F are described by the inverse limit, as we take larger and larger finite splitting fields over F .
An algebraic manifold is a cycle of projective space, in other words a formal linear combination of irreducible subvarieties. Algebraic manifolds may have singularities, so their underlying topological spaces need not be manifolds in the sense of differential topology. Semple & Roth (1949, p.14–15) meet The meet of two sets is their intersection.