Applications of Intersection Theory on Surfaces

I will state the basic theorems of intersection theory in the context of nonsingular surfaces. I will then apply these theorems to prove two non-trivial facts about algebraic curves: (1) the Weil conjectures for curves over a finite field and (2) the finiteness of the automorphism group of a curve of genus greater than one over an algebraically closed field. The talk will be fairly elementary in nature and should be accessible even to those just learning algebraic geometry.