Sharpness and semistar operations in Pruefer-like domains

Let $\star$ be a semistar operation on a domain $D$, ${\star }_f$ the finite-type semistar operation associated to $\star$, and $D$ a Pr"ufer $\star$-multiplication domain (P$\star$MD). For the special case of a Pr"ufer domain (where $\star$ is equal to the identity semistar operation), we show that a nonzero prime $P$ of $D$ is sharp, that is, that $D_P \nsupseteq \bigcap D_M$,wheretheintersectionistakenoverthemaximalideals$M$of$D$ that do not contain $P$, if and only if two closely related spectral semistar operations on $D$ differ. We then give an appropriate definition of ${\star }_f$-sharpness for an arbitrary P$\star$MD $D$ and show that a nonzero prime $P$ of $D$ is ${\star }_f$-sharp if and only if its extension to the $\star$-Nagata ring of $D$ is sharp. Calling a P$\star$MD ${\star }_f$-sharp (${\star }_f$-doublesharp) if each maximal (prime) ${\star }_f$-ideal of $D$ is sharp, we also prove that such a $D$ is ${\star }_f$-doublesharp if and only if each $(\star ,t)$-linked overring of $D$ is ${\star }_f$-sharp.