functions on the curve whose poles at every point are not worse than the corresponding coefficient in D, the very same formula as above holds: where C is a projective non-singular algebraic curve over an algebraically closed field k. In fact, the same formula holds for projective curves over any field, except that the degree of a divisor needs to take into account multiplicities coming from the possible extensions of the base field and the residue fields of the points supporting the divisor. Finally, for a proper curve over an Artinian ring, the Euler characteristic of the line bundle associated to a divisor is given by the degree of the divisor (appropriately defined) plus the Euler characteristic of the structural sheaf . The smoothness assumption in the theorem can be relaxed, as well: for a (projective) curve over an algebraically closed field, all of whose local rings are Gorenstein rings, the same statement as above holds, provided that the geometric genus as defined above is replaced by the arithmetic genus ga, defined as (For smooth curves, the geometric genus agrees with the arithmetic one.) The theorem has also been extended to general singular curves (and higher-dimensional varieties). Applications Hilbert polynomial One of the important consequences of Riemann–Roch is it gives a formula for computing the Hilbert polynomial of line bundles on a curve. If a line bundle is ample, then the Hilbert polynomial will give the first degree giving an embedding into projective space. For example, the canonical sheaf has degree , which gives an ample line bundle for genus . If we set then the Riemann–Roch formula reads Giving the degree Hilbert polynomial of Because the tri-canonical sheaf is used to embed the curve, the Hilbert polynomial is generally considered while constructing the Hilbert scheme of curves (and the moduli space of algebraic curves). This polynomial is and is called the Hilbert polynomial of an genus g curve. Pluricanonical embedding Analyzing this equation further, the Euler characteristic reads as Since for , since its degree is negative for all , implying it has no global sections, there is an embedding into some projective space from the global sections of . In particular, gives an embedding into where since . This is useful in the construction of the moduli space of algebraic curves because it can be used as the projective space to construct the Hilbert scheme with Hilbert polynomial