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© 2000-2017  Gérard P. Michon, Ph.D.

Algebraic Geometry

[Algebraic geometry]  seems to have acquired the reputation of being esoteric, exclusive and very abstract, with adherents who are secretly plotting to take over all the rest of mathematics.  In one respect this last point is accurate.
David B. Mumford  (b. 1937),  1975

Related articles on this site:

Related Links (Outside this Site)

Pre-Introduction to Algebraic Geometry, in Pictures  Donu Arapura  (2001).
Introduction to Algebraic Geometry  by  Donu Arapura  (Purdue, Fall 2001).
Why Study Algebraic Geometry?  by  Javier Alvarez  (2012-12-13).
Algebraic Geometry  at  Wikipedia.
Elliptic Curves  by  Eric W. Weisstein  (MathWorld).

Videos :

Historical Development of Algebraic Geometry (1972-03-03)  Jean Dieudonné.
Algebraic Geometry For Physicists  (October 10-14, 2011)  by  Ugo Bruzzo
Sheaves & Cohomology   |   Varieties & Bundle   |   Moduli Spaces of Bundles


Algebraic Geometry

Algebraic geometry is essentially the study of the  sets of solutions  to systems of polynomial equations in several variables over a field  K.

K  is understood to be  algebraically closed  unless otherwise specified.  In  classical  algebraic geometry,  K  is merely the field of complex numbers...

(2016-11-15)   Bézout's Theorem

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 still working on this one...

Wikipedia :   Bezout's Theorem

(2016-11-15)   Ideals and  Riemann Surfaces

Ramification points Victor Puiseux (1820-1883).

Riemann invents algebraic topology on Riemann surfaces.

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Wikipedia :   Riemann surfaces   |   Algebraic topology   |   Riemann-Roch theorem (Riemann 1857, Roch 1865)

(2005-09-28)   Ideals and  Hilbert's Nullstellensatz
Two key properties of algebraically closed fields involve  ideals.

K[x1 , ..., xn ] is the ring R of polynomials with n variables whose range is K.

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Wikipedia :   Algebraic Geometry   |   Ideal

(2005-09-25)   Basic "Elimination Theory"
Resultant of two polynomials.

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Wikipedia :   Elimination Theory   |   Bézoutian

(2005-09-25)   Schemes

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Wikipedia :   Scheme

(2017-01-01)   The Index Theorem   (Atiyah & Singer, 1963)
Equating the  algebraic index  and the  topological index.

 Ascending and Descending

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Index theorem   |   Michael Atiyah (1929-)   |   Isadore Singer (1924-)

(2005-09-23)   The Language of Algebraic Geometry

Consider an algebraically closed field K.  The set Kn is called an affine space

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  • Affine Space:  The affine space
  • Affine Geometry: 
  • Algebraic Subset: 
  • Algebraic Map: 
  • Algebraic Group: 
  • Birational: 
  • Coordinate Ring: 
  • Fibers: 
  • Homogeneous Space: 
  • Ideal: 
  • Motives: 
  • Projective Geometry: 
  • Schemes: 
  • Sheaf: 
  • Topos: 

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Glossary of Algebraic Geometry  by  Donu Arapura  (Purdue University).



k is an algebraically closed field.

n dimensional affine space An over k is the set kn. The components of a vector are its coordinates. A1 and A2 are called the affine line and plane respectively.

n dimensional projective space Pn over k is the set of lines through the origin in kn+1. P1 and P2 are called the projective line and plane respectively. The coordinates of a vector in kn+1-{0} are called the homogeneous coordinates of line spanned by it. There is (noncanonical) decomposition of Pn as a union of An and Pn-1 where the Pn-1 corresponds to the set where the last coordinate is 0. When n=1, we can also see this as follows: every line in k2 is determined by it's slope which is either finite (in A1) or infinity.

Given a set S of polynomials (respectively homogeneous polynomials) in k[x1,...xn] (respectively k[x0,...xn]), let V(S) (respectively VP(S)) be the set of zeros of these polynomials in An (respectively Pn). An algebraic subset of n-dimensional affine (respectively projective) space is a set of the form V(S) (VP(S)). These are also called closed sets since they are precisely closed sets for the Zariski topology.

When X is a subset of An, let I(X) be the set of polynomials in k[x1,...xn] which vanish on X. I(X) is always a radical ideal. The Hilbert Nullstellensatz states that I and V are inverse operations yielding a bijection between the collection of algebraic subsets of An and radical ideals in k[x1,... xn]. As a corollary, we have the weak Nullstellensatz V(I) is empty if and only if I = (1).

The set of complements of algebraic sets in affine or projective space forms a topology called the Zariski topology. Any subset gets an induced topology which goes by the same name. If k is a topological field such as C, then affine and projective space carries a second topology called the usual topology which is finer than the Zariski topology.

An affine or projective algebraic set is called a variety if it is irreducible in its Zariski topology i.e. if it cannot be written as the union of two proper closed sets. An algebraic subset X of An is a variety if and only if I(X) is prime.

A quasiprojective variety is an open subset of a projective variety. Projective varieties are clearly quasiprojective. Affine varieties are also quasiprojective, X in An is an open subset of its closure in Pn.

An algebraic map or regular map or morphism of quasiprojective varieties is a map of whose graph is closed. The collection of quasiprojective varieties and morphisms forms a category. An isomorphism of quasiprojective is an isomorphism in this category; more concretely it is a bijection for which it and its inverse are both morphisms.

If X is affine variety in An, then its coordinate ring A(X) = k[x1,...xn]/I(X). The functor X -> A(X) is an antiequivalence of the full subcategory of affine varieties and the category of affine domains over k (finitely generated k-algebras which are integral domains). In particular, a morphism of affine varieties X -> Y is an isomorphism if and only if the induced homomorphism A(Y) -> A(X) is an isomorphism of algebras.

The category of quasiprojective varieties has products which refines the set theoretic product. The product of An and Am is isomorphic to An+m. The product of two affine varieties is affine, and its coordinate ring is the the tensor product of the coordinate rings of the factors.

The product of projective varieties is projective thanks to the Segre embedding PnxPm -> Pnm+n+m. The map sends a pair to the product of homogeneous coordinates of the pair.

An algebraic group is a group G in the category of quasiprojective varieties i.e. G is simulateneously a group and variety and the group multiplication G x G -> G and inversion G -> G are morphisms.

A homogenous space is a variety X such that there is an algebraic group G and a transitive action on X for which GxX -> X is a morphism.

A nonconstant polynomial f in k[x1...xn] defines a hypersurface V(f) in An. A point a of V(f) is called a singular point if all the partial derivatives of f vanish at a. The definition can be extended to all varieties and it can be made more intrinsic. In particular an isomorphism takes a singular point to a singular point. When k=C, a nonsingular variety (i.e. a variety whose points are all nonsingular) is a complex manifold.

The function field of an affine variety is the quotient field of its coordinate ring. An element of the function field, called a rational function, can be represented by a regular function on a nonempty open set. This leads to a defintion of function field for quasiprojective varieties.

Two varieties are birational if they have isomorphic function fields. Equivalently, two varieties are birational if they have isomorphic open sets. Such and isomorphism is called a birational equivalence.

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