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 CXY 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 CXY, for X and Y depending on the given intersection problem. The cone CXY 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 CXY. This space is reduced, and has a natural map onto the reduction (CXY)red of CXY. 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 CXY → (CXY)red may wrap multiple components of CXY around one component of CXY, writing the intersection product as a sum over the components of CXY 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 CXY.