It is one of the deep problems in algebraic geometry to determine which cohomology classes on a projective variety can be realized as Chern classes of vector bundles. In low dimensions the answer is known. On a curve X any class c1 ∈ H2(X,ℤ) can be realized as the first Chern class of a vector bundle of prescribed rank r. In dimension two the existence of bundles is settled by Schwarzenberger's result, which says that for given cohomology classes c1 ∈ H2(X,ℤ) ∩ H1,1(X) and c2 ∈ H4(X,ℤ) ≅ ℤ on a complex surface X there exists a vector bundle of prescribed rank ≥ 2 with first and second Chern class c1 and c2, respectively.

The next step in the classification of bundles aims at a deeper understanding of the set of all bundles with fixed rank and Chern classes. This naturally leads to the concept of moduli spaces.

The case r = 1 is a model for the theory. By means of the exponential sequence, the set Picc1 (X) of all line bundles with fixed first Chern class c1 can be identified, although not canonically for c1 ≠ 0, with the abelian variety H1(X,Ox)/H1(X,ℤ). Furthermore, over the product Picc1 (X) × X there exists a ‘universal line bundle’ with the property that its restriction to [L] × X is isomorphic to the line bundle L on X. The following features are noteworthy here: Firstly, the set of all line bundles with fixed Chern class carries a natural scheme structure, such that there exists a universal line bundle over the product with X.