Affine and Projective Varieties

Varieties are the geometric solution sets of polynomial equations, studied in affine space and, by adding points at infinity, in the more uniform setting of projective space.

Definition

An affine variety is the common zero set in affine space of a collection of polynomials; a projective variety is the analogous zero set of homogeneous polynomials in projective space, where the geometry is compact and intersection theory is well behaved.

Scope

This topic develops affine varieties as zero loci of polynomials, the Zariski topology, and the correspondence between varieties and radical ideals furnished by Hilbert's Nullstellensatz. It introduces the coordinate ring and function field, regular and rational maps, and the passage to projective space and projective varieties where Bézout's theorem and the absence of exceptional behavior at infinity hold. Dimension, irreducibility, and singular versus smooth points are treated as the basic geometric invariants.

Core questions

• How does the Nullstellensatz make the correspondence between varieties and ideals precise?
• Why is projective space the natural home for varieties, and what does adding points at infinity fix?
• How are the coordinate ring and function field of a variety its algebraic shadows?
• What distinguishes smooth points from singular points, and how is dimension defined algebraically?

Key concepts

• Affine varieties and the Zariski topology
• Hilbert's Nullstellensatz and the ideal-variety correspondence
• Coordinate ring, function field, and rational maps
• Projective space and projective varieties
• Dimension, irreducibility, and smooth versus singular points

Clinical relevance

Varieties are the basic objects studied throughout algebraic geometry and its applications, from elliptic curves in cryptography and number theory to the projective models used in computer vision and the solution sets analyzed in algebraic statistics.

History

The study of curves and surfaces by polynomial equations dates to the 19th century; Hilbert's Nullstellensatz (1893) and Zariski's introduction of rigorous topological and algebraic tools in the 1930s and 1940s established the variety as a precise object, the starting point of the modern subject.

Key figures

• David Hilbert
• Oscar Zariski
• Robin Hartshorne

Seminal works

• hartshorne1977
• eisenbud1995

Frequently asked questions

What does Hilbert's Nullstellensatz say?

Over an algebraically closed field, it establishes a bijection between affine varieties and radical ideals of the polynomial ring, so geometric containment and intersection correspond exactly to algebraic operations on ideals.

Why work in projective space rather than affine space?

Projective space compactifies affine space by adding points at infinity, which makes varieties compact, removes special cases (parallel lines meet), and yields clean intersection results such as Bézout's theorem.