of a function f. Any section over U of can be written as s/f, where s is a holomorphic function on U. Let η be a section over U of ωX. The Poincaré residue is the map that is, it is formed by applying the vector field ∂/∂f to the volume form η, then multiplying by the holomorphic function s. If U admits local coordinates z1, ..., zn such that for some i, ∂f/∂zi ≠ 0, then this can also be expressed as Another way of viewing Poincaré residue first reinterprets the adjunction formula as an isomorphism On an open set U as before, a section of is the product of a holomorphic function s with the form . The Poincaré residue is the map that takes the wedge product of a section of ωD and a section of . Inversion of adjunction The adjunction formula is false when the conormal exact sequence is not a short exact sequence. However, it is possible to use this failure to relate the singularities of X with the singularities of D. Theorems of this type are called inversion of adjunction. They are an important tool in modern birational geometry. The Canonical Divisor of a Plane Curve Let be a smooth plane curve cut out by a degree homogeneous polynomial . We claim that the canonical divisor is where is the hyperplane divisor. First work in the affine chart . The equation becomes where and . We will explicitly compute the divisor of the differential At any point either so is a local parameter or so is a local parameter. In both cases the order of vanishing of at the point is zero. Thus all contributions to the divisor are at the line at infinity, . Now look on the line . Assume that so it suffices to look in the chart with coordinates and . The equation of the curve becomes Hence so with order of vanishing . Hence which agrees with the adjunction formula. Applications to curves The genus-degree formula for plane curves can be deduced from the adjunction formula. Let C ⊂ P2 be a smooth plane curve of degree d and genus g. Let H be the class of a hyperplane in P2, that is, the class of a line. The canonical class of P2 is −3H. Consequently, the adjunction formula says that the restriction of to C equals the