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

Abstract

Let $(R,\mathfrak{m},\mathbb{k})$ be a Noetherian three-dimensional Cohen-Macaulay analytically unramified ring and $I$ an $\mathfrak{m}$-primary $R$-ideal. Write $X=\mathrm{Proj}\left({\oplus }_{n\in \mathbb{N}}\overline{I^n}{t}^n\right)$. We prove some consequences of the vanishing of ${\mathrm{H}}^2(X,{\mathcal{O}}_X)$, whose length equals the the constant term ${\bar{e}}_3(I)$ 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 ${\bar{e}}_2(I)-{\mathrm{length}}_R\left(\frac{\overline{I^2}}{I\overline{I}}\right)\ge 3$; 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 (J. Algebra, 1992). Thirdly, ${\mathrm{H}}_E^2(X,I^m{\mathcal{O}}_X)=0$ for all integers $m$, where $E$ is the exceptional divisor in $X$. Finally, if additionally $R$ is regular and $X$ is pseudo-rational, then the adjoint ideals $\widetilde{I^n},n\ge 1$ satisfy $\widetilde{I^n}=I\widetilde{I^{n-1}}$ for all $n\ge 3$. The last two results are related to conjectures of J. Lipman (Math. Res. Lett., 1994).

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