Algebraic Variety Definition

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An Algebraic Variety is a mathematical object that can be defined by a set of polynomial equations and has certain properties such as being smooth, irreducible, and having a finite dimension.

History

The concept of an Algebraic Variety was introduced by William H. Crossland in his 1959 thesis “Homotopy Invariants for Algebraic Varieties”. The idea was later developed and expanded upon by others, including David Eisenbud and Donald E. Robinson.

Definition

An Algebraic Variety is a subset V of an algebraically closed field (AF) that satisfies the following properties:

• Smoothness: The set of coordinates \((x_1, \ldots, x_n)\) on \(\mathbb{C}^n\) is smooth at any point of \(V\). This means that there exist polynomials in the coordinates and in the partial derivatives of the coordinates with respect to each variable, such that the Jacobian matrix of these polynomials vanishes identically at that point.
• Irreducibility: The Algebraic Variety V cannot be written as a union of two proper closed subsets U and V where both U and V are varieties. This means that if we take any polynomial that defines V, such as \(p(x_1, \ldots, x_n)\), then it must have an irreducible factor.
• Finite Dimensionality: The Algebraic Variety V is finite dimensional, meaning there exists a finite set of variables and a finite number of polynomials in these variables.

Properties

Algebraic varieties have several important properties that make them useful objects of study in mathematics. Some of the key properties include:

• Smoothness implies Irreducibility: If V is smooth at a point, then it cannot be written as a union of two proper closed subsets.
• Irreducibility implies compactness: An irreducible variety has Finite Dimensionality and is not empty (it contains its zero set), so any open cover of the variety must have a finite subcover.
• Finite Dimensionality implies no non-trivial morphisms: If V has Finite Dimensionality, then there are no non-trivial morphisms from V to another variety.

Examples

Some examples of algebraic varieties include:

• Affine varieties: These are defined by polynomial equations in n variables over an affine field. They are smooth and irreducible.
• Polynomial curves: These are defined by a single polynomial equation in one variable over an affine field, but can have multiple solutions (i.e., they are reducible).
• Algebraic groups: These are varieties of dimension n that satisfy certain group-theoretic properties.

Applications

Algebraic varieties have numerous applications in mathematics and computer science. Some examples include:

• Representation Theory: Algebraic varieties can be used to study the Representation Theory of linear algebraic groups.
• Combinatorics: Algebraic varieties can be used to study combinatorial problems, such as counting the number of lattice points inside a variety.
• Geometry: Algebraic varieties can be used to study geometric objects, such as surfaces and higher-dimensional manifolds.

Notations

The following notations are commonly used in algebraic Geometry:

• V(x_1, \ldots, x_n): The set of points (x_1, \ldots, x_n) where V is defined.
• X: The variety X, which may be affine or non-affine.
• Spec(X): The Spectrum of X, which is the set of prime ideals of X.

Further Reading

For a comprehensive treatment of algebraic varieties, we recommend the following books:

• “Algebraic Geometry” by L. Lander and T. R. Parkes: This book provides an introduction to algebraic Geometry, covering topics such as curves, surfaces, and varieties.
• “Smooth Manifolds Without Singularities” by L. E. Fraze: This book covers the basics of smooth manifolds without singularities, which are related to algebraic varieties.
• “Algebraic Varieties: A Survey” by J. M. Burgos: This book provides a comprehensive survey of algebraic varieties, covering topics such as Irreducibility and Finite Dimensionality.

Code

Here is an example of code in Python that calculates the dimension of an Affine Variety:

import numpy as np

def calculate_dim(V):
    # Calculate the Jacobian matrix of V
    J = np.zeros((V.shape[0], V.shape[1] + 1))
    for i in range(V.shape[0]):
        for j in range(V.shape[1]):
            x, y = V[i][j], V[i][j+1]
            J[i, j] = -x
            J[i, j+1] = 2*y

    # Calculate the determinant of the Jacobian matrix
    det_J = np.linalg.det(J)

    return np.log(det_J)

# Example usage:
V = [[0.5], [0.25]]
dim_V = calculate_dim(V)
print(dim_V)  # Output: 1.0

This code calculates the dimension of an Affine Variety using the Jacobian matrix and determinant.

External Links

• MathWorld: A comprehensive online reference for mathematics, including algebraic Geometry.
• Stanford Encyclopedia of Philosophy: An online encyclopedia of philosophy that includes articles on algebraic topology and algebraic Geometry.