The geometric genus can be defined for non-singular complex projective varieties and more generally for complex manifolds as the Hodge number hn,0 (equal to h0,n by Serre duality), that is, the dimension of the canonical linear system plus one.
In other words, for a variety V of complex dimension n it is the number of linearly independent holomorphic n-forms to be found on V.[1] This definition, as the dimension of then carries over to any base field, when Ω is taken to be the sheaf of Kähler differentials and the power is the (top) exterior power, the canonical line bundle.
By the Riemann-Roch theorem, an irreducible plane curve of degree d has geometric genus where s is the number of singularities when properly counted.
If C is an irreducible (and smooth) hypersurface in the projective plane cut out by a polynomial equation of degree d, then its normal line bundle is the Serre twisting sheaf
(d), so by the adjunction formula, the canonical line bundle of C is given by The definition of geometric genus is carried over classically to singular curves C, by decreeing that is the geometric genus of the normalization C′.