Mathematical Research Letters

Volume 17 (2010)

Number 2

Powers of ideals and fibers of morphisms

Pages: 267 – 273



David Eisenbud (University of California at Berkeley)

Joe Harris (Harvard University)


Let $X\subset \PP^n=\PP^n_F$ be a projective scheme over a field $F$, and let $\phi:X\to Y$ be a finite morphism. Our main result is a formula, in terms of global data, for the maximum of $\reg \phi^{-1}(y)$, the Castelnuovo-Mumford regularity of a fiber of $\phi$ over a point $y\in Y$, where the fiber $\phi^{-1}(y)$ is considered as a subscheme of $\PP^n$. For example, Cutkosky-Herzog-Trung \cite{CHT1999} and Kodiyalam \cite{Kodiyalam2000} showed that for any homogeneous ideal $I$ in a standard graded algebra $S$, $\reg I^t$ can be written as $ dt+\epsilon$ for some non-negative integers $d,\epsilon$ and all large $t$. If $I$ contains a power of $S_+$ and is generated by forms of a single degree, then it determines a morphism $\phi: X:= \Proj(S) \to Y :=\PP^m$ as above. In this situation our theorem states that the maximum of the regularities of the fibers of $\phi$ is precisely $\epsilon+1$. Our formula also gives new information about powers of ideals generated by generic forms in polynomial rings.

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