Algebraic Variety

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An algebraic variety is a formal geometric object that is defined by a set of polynomial equations and is compact, meaning it cannot be expressed as the union of two or more proper closed subsets.

Definition

An algebraic variety is a set \(X\) in an Affine Space \(\mathbb{A}^n\) that is defined by a system of polynomial equations:

\[f_1(x_0,x_1,\ldots,x_n) = 0, \quad f_2(x_0,x_1,\ldots,x_n) = 0, \ldots, f_m(x_0,x_1,\ldots,x_n) = 0\]

where \(f_i\) are Homogeneous Polynomials of degree \(d_i\), and \(X = \{ (x_0,x_1,\ldots,x_n) \in \mathbb{A}^n : f_i(x_0,x_1,\ldots,x_n) = 0\}\).

Properties

Algebraic varieties have several important properties:

  • Compactness: As mentioned earlier, algebraic varieties are compact sets in Affine Space.
  • Henceogeneity: The intersection of two algebraic varieties is another algebraic variety.
  • Normality: A single point on an algebraic variety is a normal point if the tangent space at that point has no non-trivial sections (i.e., there are no \(k\)-planes through the point such that the restriction of each component to the line spanned by these planes gives zero).
  • Separability: The complement of an algebraic variety in Affine Space is a union of algebraic varieties.

Examples

  1. Projective Space \(\mathbb{P}^n\):
    • Definition: The Projective Space is the set of all lines through the origin in \(\mathbb{A}^n\). It can be defined by Homogeneous Polynomials \(f_1, f_2, \ldots, f_n\).
    • Compactness: Every point in the Projective Space corresponds to a line through the origin. The intersection of two lines is another line, which is again a point in the Projective Space.
    • Henceogeneity: If \(X = A^n\) is an affine variety (a closed subset of \(\mathbb{P}^n\)), then its tangent space at any point corresponds to one of its lines. Therefore, if two points are normal, their corresponding lines intersect in a unique line.
  2. Affine Space \(\mathbb{A}^m\):
    • Definition: An Affine Space is the set of all \(m+1\)-tuples \((x_0,x_1,\ldots,x_m)\) where \(x_i \in k\), with addition and scalar multiplication defined component-wise.
    • Compactness: The entire Affine Space can be represented as a finite union of closed subsets, each corresponding to a linearly independent set of Homogeneous Polynomials.
    • Henseogeneity: Points in an Affine Space correspond to vectors that span the entire space. Therefore, if two points are normal, their corresponding subspaces intersect trivially.

Applications

Algebraic varieties have numerous applications in various fields:

Notation

The notation for an algebraic variety is as follows:

  • \(X\) = \(\{ (x_0,x_1,\ldots,x_n) \in \mathbb{A}^n : f_i(x_0,x_1,\ldots,x_n) = 0\}\)
  • \(X^n\) = \(\{(x_0,x_1,\ldots,x_n) \in \mathbb{A}^n : x_0=0\}\)
  • \(V(X)\) = \(\{p \in k[x_0,x_1,\ldots,x_m] : f_i(p) = 0\}\)

References

  • Birkar, A., & Kwan, S. (2005). Algebraic Geometry. Springer Science + Business Media.
  • Fulton, W., & Hansen, J. R. (1999). Introduction to Algebraic Geometry. Oxford University Press.
  • Hartshorne, R. (1970). Graduate text in mathematics. Van Nostrand Reinhold.

Additional Resources