to projective, but not necessarily smooth schemes is known as Verdier duality. Riemann–Roch theorem For a (smooth projective) curve X, H2 and higher vanish for dimensional reason and the space of the global sections of the structure sheaf is one-dimensional. Thus the arithmetic genus of X is the dimension of . By definition, the geometric genus of X is the dimension of H0(X, ωX). Serre duality thus implies that the arithmetic genus and the geometric genus coincide. They will simply be called the genus of X. Serre duality is also a key ingredient in the proof of the Riemann–Roch theorem. Since X is smooth, there is an isomorphism of groups from the group of (Weil) divisors modulo principal divisors to the group of isomorphism classes of line bundles. A divisor corresponding to ωX is called the canonical divisor and is denoted by K. Let l(D) be the dimension of . Then the Riemann–Roch theorem states: if g is a genus of X, for any divisor D on X. By the Serre duality, this is the same as: which can be readily proved. A generalization of the Riemann–Roch theorem to higher dimension is the Hirzebruch–Riemann–Roch theorem, as well as the far-reaching Grothendieck–Riemann–Roch theorem. Hilbert schemes Hilbert schemes parametrize all closed subvarieties of a projective scheme X in the sense that the points (in the functorial sense) of H correspond to the closed subschemes of X. As such, the Hilbert scheme is an example of a moduli space, i.e., a geometric object whose points parametrize other geometric objects. More precisely, the Hilbert scheme parametrizes closed subvarieties whose Hilbert polynomial equals a prescribed polynomial P. It is a deep theorem of Grothendieck that there is a scheme over k such that, for any k-scheme T, there is a bijection The closed subscheme of that corresponds to the identity map is called the universal family. For , the Hilbert scheme is called the Grassmannian of r-planes in and, if X is a projective scheme, is called the Fano scheme of r-planes on X. Complex projective varieties In this section, all algebraic varieties are complex algebraic varieties. A key feature of the theory of complex projective varieties is the combination of algebraic and analytic methods. The transition between these theories is provided by the following link: since any complex polynomial is also a holomorphic function, any complex variety X yields a complex analytic space, denoted