Asymptotic linearity of regularity and $a^*$-invariant of powers of ideals

Let $X = \proj R$ be a projective scheme over a field $k$, and let $I \subseteq R$ be an ideal generated by forms of the same degree $d$. Let $\pi: \bx \rightarrow X$ be the blowing up of $X$ along the subscheme defined by $I$, and let $\phi: \bx \rightarrow \ix$ be the projection given by the divisor $dE_0 - E$, where $E$ is the exceptional divisor of $\pi$ and $E_0$ is the pullback of a general hyperplane in $X$. We investigate how the asymptotic linearity of the regularity and the $a^*$-invariant of $I^q$ (for $q \gg 0$) is related to invariants of fibers of $\phi$.

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Published 28 February 2011