2.1 Definition and examples of affine varieties

Affine varieties are the building blocks of algebraic geometry, defined by polynomial equations over algebraically closed fields. They represent geometric objects like lines, curves, and surfaces, forming a bridge between algebra and geometry.

Understanding affine varieties is crucial for grasping more complex concepts in algebraic geometry. We'll explore their definition, examples, and key properties, including the Zariski topology and the correspondence between varieties and ideals.

Affine varieties and algebraic sets

Definition of affine varieties

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• An affine variety is the set of solutions of a system of polynomial equations in $n$ variables over an algebraically closed field $k$
• The algebraic set of an affine variety $V$ is the set of all points $(x_1, \ldots, x_n)$ in $k^n$ that satisfy the polynomial equations defining $V$
  • For example, the algebraic set of the affine variety defined by the equation $x^2 + y^2 = 1$ over the field of complex numbers is the unit circle in the complex plane
• An affine variety $V$ is irreducible if it cannot be written as the union of two proper subvarieties
  • Equivalently, the ideal $I(V)$ is a prime ideal

Zariski topology and ideals

• The ideal $I(V)$ of an affine variety $V$ is the set of all polynomials $f(x_1, \ldots, x_n)$ that vanish on $V$, meaning $f(a_1, \ldots, a_n) = 0$ for all points $(a_1, \ldots, a_n)$ in $V$
  • For instance, the ideal of the affine variety defined by $x^2 + y^2 = 1$ is the ideal generated by the polynomial $x^2 + y^2 - 1$
• The Zariski topology on $k^n$ is defined by taking the closed sets to be the affine varieties (algebraic sets)
  • In this topology, the closure of a set is the smallest affine variety containing it
  • The Zariski topology is coarser than the usual Euclidean topology on $k^n$

Examples of affine varieties

Affine varieties in low dimensions

• In the affine plane (dimension 2), examples of affine varieties include lines, parabolas, ellipses, and hyperbolas
  • The line $y = mx + b$ is an affine variety defined by the linear equation $y - mx - b = 0$
  • The parabola $y = x^2$ is an affine variety defined by the quadratic equation $y - x^2 = 0$
• In 3-dimensional affine space, examples include planes, spheres, ellipsoids, and cubic surfaces
  • The plane $ax + by + cz + d = 0$ is an affine variety defined by a linear equation
  • The sphere $x^2 + y^2 + z^2 = r^2$ is an affine variety defined by a quadratic equation

Trivial and compound affine varieties

• The empty set and the entire affine space $k^n$ are both trivial examples of affine varieties in any dimension $n$
• The intersection of two affine varieties is always an affine variety
  • For example, the intersection of the plane $z = 0$ and the sphere $x^2 + y^2 + z^2 = 1$ is the unit circle in the $xy$-plane
• The union of two affine varieties is an affine variety if and only if one is contained in the other

Ideals and affine varieties

Correspondence between affine varieties and radical ideals

• There is a one-to-one correspondence (bijection) between affine varieties in $k^n$ and radical ideals in the polynomial ring $k[x_1, \ldots, x_n]$
  • Given an affine variety $V$, its ideal $I(V)$ is always a radical ideal
  • Conversely, given any radical ideal $I$ in $k[x_1, \ldots, x_n]$, the set $V(I)$ of points where all polynomials in $I$ vanish is an affine variety
• The correspondence $V \mapsto I(V)$ and $I \mapsto V(I)$ are inclusion-reversing
  • If $V_1 \subseteq V_2$ then $I(V_2) \subseteq I(V_1)$
  • If $I_1 \subseteq I_2$ then $V(I_2) \subseteq V(I_1)$

Hilbert's Nullstellensatz

• Hilbert's Nullstellensatz states that if $k$ is algebraically closed, the correspondence between affine varieties and radical ideals is a bijection
  • In other words, every radical ideal is the ideal of some affine variety, and every affine variety is the zero set of some radical ideal
  • This fundamental theorem establishes a deep connection between algebraic geometry and commutative algebra

Dimension and degree of affine varieties

Dimension of affine varieties

• The dimension of an affine variety $V$ is the Krull dimension of its coordinate ring $k[V] = k[x_1, \ldots, x_n] / I(V)$, which is the supremum of the lengths of chains of prime ideals in $k[V]$
  • Intuitively, the dimension measures the number of independent directions in which $V$ extends
• The dimension of $V$ is equal to the number of independent variables in a set of equations defining $V$, after eliminating redundant variables
  • For example, the parabola $y = x^2$ has dimension 1, since it can be described by a single independent variable $x$

Degree of affine varieties

• If $V$ is an affine variety of dimension $d$ in $k^n$, then almost all linear subspaces of $k^n$ of codimension $d$ intersect $V$ in a finite number of points, called the degree of $V$
  • The degree measures the size or complexity of $V$
  • For instance, a line has degree 1, a quadratic curve has degree 2, and a cubic surface has degree 3
• The degree of $V$ is equal to the leading coefficient of the Hilbert polynomial of $k[V]$, which describes the dimension of the vector space of polynomials on $V$ of a given degree
  • The Hilbert polynomial encodes important information about the structure of $V$ and its coordinate ring