Intersection theory on projective surfaces via cohomology
Cohomology contains a lot of numerical information. In this video, we see how the numerical theory of intersecting curves on a smooth projective surface X can be captured using the cohomology of various line bundles. Given curves C and D on X, there is an associated intersection number that reflects the number of intersection points of C and D, counted with multiplicity. We see in some simple examples, what this multiplicity mens, and how the intersection number can be defined quite easily in terms of Euler characteristics of associated line bundles O(-C), O(-D). This "intersection number" then lifts to a bilinear pairing on the group of divisors Div(X), and since the definition is in terms of the line bundles O(-C) and O(-D), it immediately follows that it depends only on the linear equivalence classes of C and D. We see how Bezout's theorem on intersecting plane curves is now easily recovered.