Multiplicities of discriminants

The discriminant of a homogenous polynomial F is another homogenous
polynomial  in the coefficients of the polynomial, which is zero if and
only if the  corresponding hypersurface F = 0 is singular. In case the
coefficients are in a  discrete valuation ring, the order of the
discriminant (if non-zero) measures  the bad reduction. We give some new
results on this order, and in particular tie  it to Bloch's conjecture/the
Kato-T.Saito formula on equality of localized Chern  classes and Artin
conductors. As an application, we can express the  multiplicities, in the
case of discriminants of ternary forms, in terms of the  geometry of the
singularities. This gives a partial generalization of Ogg's  formula for
elliptic curves.