I got a copy of Manin’s classic treatise ‘Cubic forms’ out the library today in order to run through the details of a pleasing episode in the study of del Pezzo surfaces. The 27 lines on a cubic surface (aka , the blowup of in 6 general points) are intimately connected with the root system and accompanying Lie-theoretic paraphernalia. Something similar is true for the other del Pezzo surfaces, at least when 2 or more points are blown up to ensure that the Picard lattice is big enough for something interesting to happen. More precisely, one can find a [usually irreducible] root system inside the Picard lattice of a del Pezzo surface whose Weyl group is isomorphic to the automorphism group of the incidence graph of the lines (or (-1)-curves) on the del Pezzo.
I worked out an example and stated the main result from Manin’s book with some discussion. I came across a paper by Iqbal, Neitzke, and Vafa this evening that seems to claim to interpret and enrich Manin’s result from a string theoretic point of view, which I plan to spend some time reading. I’ll keep you posted.