Multiplicities of a bigraded ring and intersection theory

Arithmetic degree and associated graded modules

“…Given a bigraded ring A and a bigraded A-module M , the "right" polynomial to be considered to define multiplicity is the sum transform of the Hilbert polynomial as shown in [3]. We recall here some results of [3].…”

“…We recall here some results of [3]. Let h(i, j) = length A (0,0) (M (i,j) ) be the Hilbert function of M , and h (1,0) (i, j) = i u=0 h(u, j) the sum transform of h with respect to the first variable.…”

“…Extending to the case of modules a definition of [3], the multiplicity of an arbitrary ideal I with respect to a finitely generated A-module M is defined to be the multiplicity of the bigraded module GG(M ).…”

“…So, if I = m we obtain the usual notion of arithmetic degree, see Vasconcelos [19], p. 223. We recall another result, which is straightforward from [3], Proposition 2.5, which we will use in the proof of the next theorem. …”

Self-Intersections of Surfaces and Whitney Stratifications

“…For a point x ∈ X, denote by g(x) := e x (C X (X × X)) the multiplicity of C X (X × X) at x. In [4] we proved that g(x) is the degree at x of the Stückrad-Vogel cycle v(X, X) = C j(X, X; C) [C] of the self-intersection of X, that is, g(x) = C j(X, X; C)e x (C) (see Proposition 3.6 for details). In this paper, our main result, Theorem 4.2, states that the pointwise degree g(x) of the Stückrad-Vogel intersection cycle of the self-intersection of X is a stratifying function which gives the canonical Whitney stratification.…”