As an example of the usefulness of the canonical bundle in making calculations, one can compute the genus of a projective algebraic plane curve, say over the complex numbers. Then the genus is actually the topological genus of the real surface underlying the complex curve. In general the (degree of the) canonical class on a curve of genus g equals 2g-2, i.e. the negative of the euler characteristic.
By computing an actual 2 form one sees that the canonical class of the plane itself is O(-3), i.e. a standard 2 form has a triple pole along a line, and no zeroes. Then there is a wonderful formula for how canonical divisors restrict, the adjunction formula. This says that for a curve of degree d in the plane, the canonical class on the curve equals O(d-3).
Hence for a line we get -2, agreeing with the fact that the 1-form dz has no zeroes in the affine plane and one double pole at infinity, under the substitution z = 1/w. For a plane conic we get O(-1) but this means the divisor is cut out on a conic by intersecting with a curve of degree 1, i.e. a line. Since a line meets a conic twice we again get O(-2) on the conic itself, which agrees with the fact that a conic is isomorphic via projection with a line, and both are homeomorphic to the sphere, with genus zero. On a plane cubic we get O(0), or the trivial class, agreeing with the fact that a smooth cubic is homeomorphic to a torus of genus one, and euler characteristic zero. As exercise check that a (smooth) plane curve of degree4 has genus 3.
We can also compute the genus of a curve cut by two surfaces in P^3, complex projective 3 space, using the higher adjunction formula, that the curve cut by surfaces of degree d and e, has canonical class of degree equal to: de(d+e-4). Thus two quadratic surfaces cut a curve of degree 4 and class zero, hence again a torus. Indeed projecting such a curve to the plane from one point of itself lowers the degree by one and gives an isomorphism with a plane cubic.
Intersecting two surfaces of degrees 2 and 3 gives (exercise) a curve of genus 4.
If we pass two cubic surfaces through a quartic space curve C of genus g = one, the full intersection has degree 9 hence consists of that quartic plus some other curve C’ of degree 5. We can even compute the genus of that residual curve C’ by the formula (obtained by subtracting two adjunction formulas):
2(g’-g) = (5-4)(3+3-4) = 2, so the other curve C’ has genus g’ = g+1 = 2, and is hence a space quintic of genus 2. Note that projecting this curve to the plane from a point of itself gives a plane quartic which should have genus 3. Since it has only genus 2, the plane projection must cross itself once, lowering the genus. This means that every point of the space quintic, that we might choose to project from, must lie on a trisecant, something not obvious to me at the moment. It also suggests that the canonical class on C' of degree 2g'-2 = 2, is swept out residually on the space quintic by the pencil of planes through such a trisecant. I.e. each such plane cuts the quintic in 3 fixed points on the trisecant, and further in two moving points. I don't see why this follows from adjunction either at the moment, but it is true for general reasons.
These calculations are all manifestations of the concept of differential n - form, i.e. the "determinant" of the space of one forms dual to tangent vectors. These computations were known to European geometers close to 150 years ago, but seem simpler today through the systematic use of the canonical bundle of n - forms. In particular I found this last "residual genus" formula mysteriously presented in a wonderful old book by Semple and Roth on classical algebraic geometry, and was able to derive it myself (just yesterday) this way.
