Vol. 11, No. 7, 2017

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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
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