The equations defining blowup algebras of height three Gorenstein ideals

Kustin, Andrew; Polini, Claudia; Ulrich, Bernd

doi:10.2140/ant.2017.11.1489

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The equations defining blowup algebras of height three Gorenstein ideals

Andrew R. Kustin, Claudia Polini and Bernd Ulrich

Vol. 11 (2017), No. 7, 1489–1525

DOI: 10.2140/ant.2017.11.1489

Abstract

We find the defining equations of Rees rings of linearly presented height three Gorenstein ideals. To prove our main theorem we use local cohomology techniques to bound the maximum generator degree of the torsion submodule of symmetric powers in order to conclude that the defining equations of the Rees algebra and of the special fiber ring generate the same ideal in the symmetric algebra. We show that the ideal defining the special fiber ring is the unmixed part of the ideal generated by the maximal minors of a matrix of linear forms which is annihilated by a vector of indeterminates, and otherwise has maximal possible height. An important step in the proof is the calculation of the degree of the variety parametrized by the forms generating the height three Gorenstein ideal.

Keywords

blowup algebra, Castelnuovo–Mumford regularity, degree of a variety, Hilbert series, ideal of linear type, Jacobian dual, local cohomology, morphism, multiplicity, Rees ring, residual intersection, special fiber ring

Mathematical Subject Classification 2010

Primary: 13A30

Secondary: 13D02, 13D45, 13H15, 14A10, 14E05

Milestones

Received: 19 June 2015

Revised: 17 October 2016

Accepted: 19 December 2016

Published: 7 September 2017

Authors

Andrew R. Kustin
Department of Mathematics University of South Carolina 1523 Greene Street Columbia, SC 29208 United States

Claudia Polini
Department of Mathematics University of Notre Dame 255 Hurley Hall Notre Dame, IN 46556-4618 United States

Bernd Ulrich
Department of Mathematics Purdue University 150 N University Street West Lafayette, IN 47907-2067 United States

Vol. 11, No. 7, 2017