Hilbert scheme

The Hilbert scheme is an important concept in algebraic geometry that parametrizes subschemes of a given projective variety (or more generally, an algebraic scheme) in a systematic way. More precisely, for a projective variety \( X \), the Hilbert scheme \( \text{Hilb}^n(X) \) is a scheme that parametrizes all closed subschemes of \( X \) with a fixed length \( n \).