. Genus of plane curves with singularities An irreducible plane algebraic curve of degree d has (d − 1)(d − 2)/2 − g singularities, when properly counted. It follows that, if a curve has (d − 1)(d − 2)/2 different singularities, it is a rational curve and, thus, admits a rational parameterization. Riemann–Hurwitz formula The Riemann–Hurwitz formula concerning (ramified) maps between Riemann surfaces or algebraic curves is a consequence of the Riemann–Roch theorem. Clifford's theorem on special divisors Clifford's theorem on special divisors is also a consequence of the Riemann–Roch theorem. It states that for a special divisor (i.e., such that ) satisfying the following inequality holds: Proof Proof for algebraic curves The statement for algebraic curves can be proved using Serre duality. The integer is the dimension of the space of global sections of the line bundle associated to D (cf. Cartier divisor). In terms of sheaf cohomology, we therefore have , and likewise . But Serre duality for non-singular projective varieties in the particular case of a curve states that is isomorphic to the dual . The left hand side thus equals the Euler characteristic of the divisor D. When D = 0, we find the Euler characteristic for the structure sheaf is by definition. To prove the theorem for general divisor, one can then proceed by adding points one by one to the divisor and ensure that the Euler characteristic transforms accordingly to the right hand side. Proof for compact Riemann surfaces The theorem for compact Riemann surfaces can be deduced from the algebraic version using Chow's Theorem and the GAGA principle: in fact, every compact Riemann surface is defined by algebraic equations in some complex projective space. (Chow's Theorem says that any closed analytic subvariety of projective space is defined by algebraic equations, and the GAGA principle says that sheaf cohomology of an algebraic variety is the same as the sheaf cohomology of the analytic variety defined by the same equations). One may avoid the use of Chow's theorem by arguing identically to the proof in the case of algebraic curves, but replacing with the sheaf of meromorphic functions h such that all coefficients of the divisor are nonnegative. Here the fact that the Euler characteristic transforms as desired when one adds a point to the divisor can be read off from the long exact sequence induced by the short exact sequence where is the skyscraper