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

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 = $\mathrm{P}\mathrm{r}\mathrm{o}\mathrm{j}$\left(\oplus_{n \in $\mathbb{N}$} \overline{I^n}t^n\right)$\mathrm{.}Weprovesomeconsequencesofthevanishingof$$\mathrm{H}$^2(X, \mathscr{O}X)$,whoselengthequalsthetheconstantterm$\bar e_3(I)$ofthenormalHilbertpolynomialof$I$\mathrm{.}Firstly,$X$ is Cohen-Macaulay. Secondly, if the extended Rees ring $A := \oplus{n \in $\mathbb{Z}$} \overline{I^n}t^n$isnotCohen-Macaulay,andeither$R$ is equicharacteristic or $\bar{I}=\mathfrak{m}$, then $\bar e_2(I) - $\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}$_R\left(\frac{\overline{I^2}}{I\overline{I}}\right) \geq 3$; this estimate is proved using Boij-S"oderberg 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).