  We introduce a geometric method to study additive combinatorial problems.
Using equivariant cohomology we reprove the Dias da Silva-Hamidoune theorem. We
improve a result of Sun on the linear extension of the Erd\H{o}s-Heilbronn
conjecture. We generalize a theorem of G. K\'os (the Grashopper problem) which
in some sense is a simultaneous generalization of the Erd\H{o}s-Heilbronn
conjecture. We also prove a signed version of the Erd\H{o}s-Heilbronn
conjecture and the Grashopper problem. Most identities used are based on
calculating the projective degree of an algebraic variety in two different
ways.
