subvarieties

Subvarieties are fundamental objects in algebraic geometry. Given an algebraic variety X over a field k, a subvariety of X is a subset V of X that is itself an algebraic variety with the induced structure. In the common affine or projective setting, subvarieties are typically described as irreducible closed subsets of X. Equivalently, V is the common zero locus of a prime ideal in the coordinate ring of X (or a homogeneous ideal in the projective case). The vanishing ideal I_V encodes the equations defining V, and V is recovered as the zeros of I_V inside X.

In practice, subvarieties arise as the solutions to systems of equations. In affine space, a linear subspace defined by linear equations is a subvariety; a hypersurface defined by a single equation f = 0 is a subvariety of codimension one; a curve on a surface is a one-dimensional subvariety. In projective space, hyperplanes and more generally projective varieties cut out by homogeneous equations are subvarieties of projective space. The defining ideal can be described in terms of regular functions or, in the projective setting, by homogeneous ideals.

Subvarieties inherit the Zariski topology from the ambient variety and have dimension less than or equal to that of X. If V is irreducible, it is called an irreducible subvariety; otherwise it may be reducible and decomposes into irreducible components. The concept extends to closed subschemes via the ideal sheaf I_V, encoding a richer algebraic structure than mere sets.

Relation to ambient geometry: Every closed subset of X is a finite union of irreducible subvarieties (its irreducible components). The study of subvarieties includes questions about smoothness, dimension, intersections, and how subvarieties sit inside X, forming a foundation for stratifications and intersection theory.