canonical class of C. This restriction is the same as the intersection product restricted to C, and so the degree of the canonical class of C is . By the Riemann–Roch theorem, g − 1 = (d−3)d − g + 1, which implies the formula Similarly, if C is a smooth curve on the quadric surface P1×P1 with bidegree (d1,d2) (meaning d1,d2 are its intersection degrees with a fiber of each projection to P1), since the canonical class of P1×P1 has bidegree (−2,−2), the adjunction formula shows that the canonical class of C is the intersection product of divisors of bidegrees (d1,d2) and (d1−2,d2−2). The intersection form on P1×P1 is by definition of the bidegree and by bilinearity, so applying Riemann–Roch gives or The genus of a curve C which is the complete intersection of two surfaces D and E in P3 can also be computed using the adjunction formula. Suppose that d and e are the degrees of D and E, respectively. Applying the adjunction formula to D shows that its canonical divisor is , which is the intersection product of and D. Doing this again with E, which is possible because C is a complete intersection, shows that the canonical divisor C is the product , that is, it has degree . By the Riemann–Roch theorem, this implies that the genus of C is More generally, if C is the complete intersection of hypersurfaces of degrees in Pn, then an inductive computation shows that the canonical class of C is . The Riemann–Roch theorem implies that the genus of this curve is In low dimensional topology Let S be a complex surface (in particular a 4-dimensional manifold) and let be a smooth (non-singular) connected complex curve. Then where is the genus of C, denotes the self-intersections and denotes the Kronecker pairing . See also Logarithmic form Poincare residue Thom conjecture References Intersection theory 2nd edition, William Fulton, Springer, , Example 3.2.12. Principles of algebraic geometry, Griffiths and Harris, Wiley classics library, pp 146–147. Algebraic geometry, Robin Hartshorne, Springer GTM 52, , Proposition II.8.20. Algebraic geometry