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

Abstract

Let $(R, \mathfrak{m}, \Bbbk)$ 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 $\mathrm{H}^2(X, \mathscr{O}_X)$, 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 $\bar e_2(I) - \mathrm{length}_R\left(\frac{\overline{I^2}}{I\overline{I}}\right) \geq 3$; this estimate is proved using Boij-S\"oderberg theory of coherent sheaves on $\mathbb{P}^2_\Bbbk$. The two results above are related to a conjecture of S. Itoh (J. Algebra, 1992). Thirdly, $\mathrm{H}^2_E(X, I^m\mathscr{O}_X) = 0$ 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 $\widetilde{I^n}, n \geq 1$ satisfy $\widetilde{I^n} = I\widetilde{I^{n-1}}$ for all $n \geq 3$. The last two results are related to conjectures of J. Lipman (Math. Res. Lett., 1994).