Moduli

Moduli is the study of parameter spaces that classify mathematical objects up to isomorphism. In this framework a moduli problem asks for a space whose points correspond to isomorphism classes of objects of a given type, together with a way to organize families of such objects over varying bases. A moduli space represents this problem in the sense that its points encode objects, and nearby points encode deformations; a fine moduli space comes with a universal family, while a coarse moduli space only parametrizes isomorphism classes.

In many cases the objects admit nontrivial automorphisms, so a genuine moduli space in the traditional sense does not exist. Instead, one studies algebraic stacks or uses geometric invariant theory to construct coarse moduli spaces that capture the correct parameterization up to isomorphism.

Key examples include the moduli space Mg of smooth projective curves of genus g over a field. For g ≥ 2, Mg is a quasi-projective variety of dimension 3g − 3, and its Deligne–Mumford compactification M̄g adds stable nodal curves. The moduli space M1,1 of elliptic curves is one-dimensional and can be described analytically as the quotient of the upper half-plane by SL2(Z), with the j-invariant giving a global coordinate.

Another important family is the moduli space of stable vector bundles on a fixed smooth projective curve C, denoted M_C(n,d). For gcd(n,d)=1 this space is projective and has dimension (n^2 − 1)(g − 1). Moduli spaces also arise for polarized varieties, sheaves, and more, often requiring stability conditions and constructions via deformation theory, GIT, or stacks.

Moduli provide a framework for classification problems across algebraic geometry, number theory, and mathematical physics, linking deformations, invariants, and geometric structures.