E-Polytopes in Picard Groups of Smooth Rational Surfaces

Abstract

In this article, we introduce special divisors (root, line, ruling, exceptional system and rational quartic) in smooth rational surfaces and study their correspondences to subpolytopes in Gosset polytopes k(21). We also show that the sets of rulings and exceptional systems correspond equivariantly to the vertices of 2(k1) and 1(k2) via E-type Weyl action.