Dear Jeff, string theory has had a colossal influence on the renewal of enumerative geometry, a two century old branch of algebraic geometry inextricably linked to intersection theory. Here is a telling anecdote.
Ellingsrud and Strømme, two renowned specialists in Hilbert Scheme theory, had calculated the number of rational cubic curves on a general quintic threefold by arguments based on their paper
On the Chow ring of a geometric quotient, Annals of Math. 130 (1987) 159–187
Their result differed from that predicted by string theory. Of course everybody thought the mathematicians were right, but actually there had been a programming error in their calculations and the correct result was that of the physicists (which Ellingsrud and Strømme confirmed after fixing their bug).
This was the beginning of a long list of results predicted by string theorists and subsequently proved by mathematicians, a celebrated example being Kontsevich's formula for the number $N_d$ of degree $d$ rational curves in $\mathbb P^2$ passing through $3d-1$ points in general position.
You can read all about Kontsevich's formula in Kock and Vainsencher's free on-line book
And a pleasantly elementary general reference is Sheldon Katz's Enumerative Geometry and String Theory, published by the AMS in its Student Mathematical Library (vol. 32).