Projective curves and Riemann–Roch

Write hⁱ(L) for the k-dimension of the cohomology of a line bundle L, and define the genus as h¹(O_C). Properness makes these dimensions finite. Since C is geometrically connected and smooth over the algebraically closed field, h⁰(O_C)=1. Riemann–Roch says that the Euler characteristic h⁰(L)−h¹(L) equals deg L+1−g. The left side is an alternating count, not simply the number of sections whenever higher cohomology is present.

For a divisor D, use L=O(D) and write l(D)=h⁰(O(D)). The canonical bundle is the sheaf of regular differentials on the smooth curve, and its degree is 2g−2. A canonical divisor K is a divisor representing that bundle. Different representatives of K are linearly equivalent, so replacing K by another canonical divisor does not change the dimensions in the formula. This is a statement about divisor classes and line bundles rather than a preferred rational differential.

Serre duality identifies H¹(L) with the dual of H⁰(ω_C⊗L⁻¹), which gives the correction term in Riemann–Roch. A line bundle of negative degree has no nonzero global section: a nonzero section would define an effective zero divisor of that negative degree, impossible. Therefore if deg D exceeds 2g−2, the complementary bundle has negative degree and the correction vanishes. In that range l(D) is exactly deg D+1−g.

At lower degrees the correction can be nonzero, and degree alone generally does not determine the section dimension. The canonical bundle itself is an instructive case: its complementary bundle is trivial, so the correction is one. On a positive-genus curve, blindly using the high-degree formula at the canonical degree undercounts the canonical sections. The theorem measures this failure precisely rather than promising that every line bundle of a fixed small degree behaves identically.

When l(D) is positive, the complete linear system has projective dimension l(D)−1. Its points are nonzero sections modulo scalar multiplication, equivalently the effective divisors linearly equivalent to D. A chosen basis supplies a candidate map to projective space, but base-point freeness is needed for a morphism on all of C. Separating points and tangent directions requires additional conditions before one obtains an embedding. A numerical section count alone does not perform these geometric tests.

The projective line checks the formula explicitly: its genus is zero, the canonical degree is minus two, and O(n) has n+1 sections for nonnegative n, agreeing with the previous Čech calculation. More general curves need not have rational parameterizations even when some high-degree line bundle has many sections. Riemann–Roch is a bridge from local pole constraints to global linear data, and it supports further geometric arguments without replacing them. Always keep the curve hypotheses and the vanishing argument beside the numerical result.