Balanced normal cones and Fulton-MacPherson's intersection theory
  Let X be a subscheme of a reduced scheme Y. Then Y has a flat "degeneration to the normal cone" C_X Y of X, and this degeneration plays a key step in Fulton and MacPherson's "basic construction" in intersection theory. The intersection product has a canonical refinement as a sum over the components of C_X Y, for X and Y depending on the given intersection problem. The cone C_X Y is usually not reduced, which leads to the appearance of multiplicities in intersection formulae.   We describe a variant of this degeneration, due essentially to Samuel, Rees, and Nagata, in which Y flatly degenerates to the "balanced" normal cone \barC_X Y. This space is reduced, and has a natural map onto the reduction (C_X Y)_red of C_X Y. The multiplicity of a component now appears as the degree of this map. Hence intersection theory can be studied using only reduced schemes. Moreover, since the map \barC_X Y \to (C_X Y)_red may wrap multiple components of \barC_X Y around one component of C_X Y, writing the intersection product as a sum over the components of \barC_X Y gives a further canonical refinement. \\   In the case that X is a Cartier divisor in a projective scheme Y, we describe the balanced normal cone in homotopy-theoretic terms, and prove a useful upper bound on the Hilbert function of \barC_X Y. 
Allen Knutson