A lower bound on the essential dimension of a connected linear group

Abstract

Let G be a connected linear algebraic group defined over an algebraically closed field k and H be a finite abelian subgroup of G whose order does not divide char(k). We show that the essential dimension of G is bounded from below by rank(H) − rank CG(H)0, where rank CG(H)0 denotes the rank of the maximal torus in the centralizer CG(H). This inequality, conjectured by J.-P. Serre, generalizes previous results of Reichstein–Youssin (where char(k) is assumed to be 0 and CG(H) to be finite) and Chernousov–Serre (where H is assumed to be a 2-group).