Intersection Multiplicity on Blow-ups

A conjecture on vanishing and non-negativity of intersection multiplicity on the blow-up of a regular local ring at its closed point has been proposed. Proofs of vanishing, several special cases of non-negativity and a sufficient condition for non-negativity of this conjecture are described. The topic on intersection multiplicity on vector bundles is also addressed. Introduction. Let X be a regular scheme of finite type over a field or an excellent discrete valuation ring and F ,G be two coherent OX -modules such that l(H i(X,TorX j (F ,G))) < ∞, for i, j ≥ 0. We define χX (F ,G) as Σ(−1)i+jl(Hi(X,TorX j (F ,G))). Note that this is an extension of Serre’s definition of intersection multiplicity on regular local rings ([Se]). First, we would propose the following conjecture: Conjecture. Let (R,m,K) be a regular local ring of dimension n and of essentially finite type over a field or an excellent discrete valuation ring. Let X̃ denote the blow-up of X = SpecR at the closed point s = [m]. Let Ỹ , Z̃ be any two subvarieties of X̃ such that Ỹ ∩ Z̃ ⊂ E = P K . The following statements hold: i) if dim Ỹ + dim Z̃ < dim X̃, then χX̃ (OỸ ,OZ̃) = 0 ii) if dim Ỹ +dim Z̃ = dim X̃ and if at least one of Ỹ and Z̃ is not contained in E , then χX̃ (OỸ ,OZ̃) ≥ 0. AMS Subject Classification: Primary 13H15, 14C17; secondary 13D25, 14C40