  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).
