Powers of Ideals and Fibers of Morphisms

By David Eisenbud, J. Harris. 2008

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Let X\subset PP^n be a projective scheme over a field, and let phi:X --> Y be a finite morphism. Our main result is a formula in terms of global data for the maximum of the Castelnuovo-Mumford regularity of the fibers of \phi, considered as subschemes of \PP^n. From an algebraic point of view, our formula is related to the theorem of Cutkosky-Herzog-Trung and Kodiyalam showing that for any homogeneous ideal I in a standard graded algebra S, the regularity of I^t can be written as dt+\epsilon for some non-negative integers d, \epsilon, and all large t. In the special case where I contains a power of S_+ and is generated by forms of a single degree, our formula gives an interpretation of \epsilon: it is one less than the maximum regularity of a fiber of the morphism associated to I. These formulas have strong consequences for ideals generated by generic forms.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/b1a117f5e4eaed9708332e04a7fed8a6ba9188b5

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Citations in the corpus

Regularity of powers of forests and cycles (2014, background): “[4, 8, 14, 15, 19]), even in simple situations such as when I is generated by general linear forms (cf.”
Rational normal scrolls and the defining equations of Rees algebras (2008, background): “The integer e has been determined in [12].”
Powers of ideals and the cohomology of stalks and fibers of morphisms (2010, background): “Putting this together provides a collection of results that covers the results obtained in [Eisenbud and Harris 2010; Hà 2011].”
Regularity of Squarefree Monomial Ideals (2013, unknown): “[4, 10, 16, 17, 24]).”