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

Abstract

Let $\left(R,\mathfrak{𝔪},\mathbb{𝕜}\right)$ be a three-dimensional Cohen–Macaulay analytically unramified local ring and $I$ an $\mathfrak{𝔪}$-primary $R$-ideal. Write $X=\text{Proj}\left({\oplus }_{n\in \mathbb{N}}\overline{I^n}{t}^n\right)$. We prove some consequences of the vanishing of ${\text{H}}^2\left(X,{\mathcal{𝒪}}_X\right)$, whose length equals the constant term ${\overline{e}}_3(I)$ of the normal Hilbert polynomial of $I$. Firstly, $X$ is Cohen–Macaulay. Secondly, if the extended Rees ring $A:={\oplus }_{n\in \mathbb{Z}}\overline{I^n}{t}^n$ is not Cohen–Macaulay, and either $R$ is equicharacteristic or $\overline{I}=\mathfrak{m}$, then ${\overline{e}}_2(I)-{\text{length}}_R\left(\overline{I^2}/\left(I\overline{I}\right)\right)\ge 3$; this estimate is proved using Boij–Söderberg theory of coherent sheaves on ${\mathbb{P}}_{\mathbb{𝕜}}^2$. The two results above are related to a conjecture of S. Itoh (1992). Thirdly, ${\text{H}}_E^2\left(X,I^m{\mathcal{𝒪}}_X\right)=0$ 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 $\widetilde{I^n}$, $n\ge 1$ satisfy $\widetilde{I^n}=I\widetilde{I^{n-1}}$ for every $n\ge 3$. The last two results are related to conjectures of J. Lipman (1994).