We compute the Euler obstruction and Mather’s Chern class of the discriminant hypersurface of a very ample linear system on a nonsingular variety. Comparing the codimension-1 and 2 terms of this and other characteristic classes of the discriminant leads to a quick computation of the degrees of the loci of cuspidal and binodal sections of a very ample line bundle on a smooth variety, and of the, tacnodal locus for linear systems on a surface. We also compute explicitly all terms in the Schwartz-MacPherson’s classes of strata of the discriminant in the P9of of cubic plane curves, and of the discriminant of∣O(d)∣ on P1.