On conjectures of Itoh and of Lipman on the cohomology of normalized blow-ups

Abstract

Let (R,𝔪,𝕜) be a three-dimensional Cohen–Macaulay analytically unramified local ring and I an 𝔪-primary R-ideal. Write X=Proj(⊕n∈ℕIn¯tn). We prove some consequences of the vanishing of H2(X,𝒪X), whose length equals the constant term e¯3(I) of the normal Hilbert polynomial of I. Firstly, X is Cohen–Macaulay. Secondly, if the extended Rees ring A := ⊕n∈ℤIn¯tn is not Cohen–Macaulay, and either R is equicharacteristic or I¯=m, then e¯2(I)−lengthRI2¯/II¯≥3; this estimate is proved using Boij–Söderberg theory of coherent sheaves on ℙ𝕜2. The two results above are related to a conjecture of S. Itoh (1992). Thirdly, HE2(X,Im𝒪X)=0 for all integers m, where E is the exceptional divisor in X. Finally, if additionally R is regular and X is pseudorational, then the adjoint ideals In ˜, n≥1 satisfy In ˜=IIn−1 ˜ for every n≥3. The last two results are related to conjectures of J. Lipman (1994).