Algebra & Number Theory

The equations defining blowup algebras of height three Gorenstein ideals

Andrew Kustin, Claudia Polini, and Bernd Ulrich

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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.

Article information

Source
Algebra Number Theory, Volume 11, Number 7 (2017), 1489-1525.

Dates
Received: 19 June 2015
Revised: 17 October 2016
Accepted: 19 December 2016
First available in Project Euclid: 12 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.ant/1513096737

Digital Object Identifier
doi:10.2140/ant.2017.11.1489

Mathematical Reviews number (MathSciNet)
MR3697146

Zentralblatt MATH identifier
06775551

Subjects
Primary: 13A30: Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics
Secondary: 13D02: Syzygies, resolutions, complexes 13D45: Local cohomology [See also 14B15] 13H15: Multiplicity theory and related topics [See also 14C17] 14A10: Varieties and morphisms 14E05: Rational and birational maps

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

Citation

Kustin, Andrew; Polini, Claudia; Ulrich, Bernd. The equations defining blowup algebras of height three Gorenstein ideals. Algebra Number Theory 11 (2017), no. 7, 1489--1525. doi:10.2140/ant.2017.11.1489. https://projecteuclid.org/euclid.ant/1513096737


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References

  • I. M. Aberbach, S. Huckaba, and C. Huneke, “Reduction numbers, Rees algebras and Pfaffian ideals”, J. Pure Appl. Algebra 102:1 (1995), 1–15.
  • M. Artin and M. Nagata, “Residual intersections in Cohen–Macaulay rings”, J. Math. Kyoto Univ. 12 (1972), 307–323.
  • G. Boffi and R. Sánchez, “On the resolutions of the powers of the Pfaffian ideal”, J. Algebra 152:2 (1992), 463–491.
  • M. Boij, J. Migliore, R. M. Miró-Roig, U. Nagel, and F. Zanello, “On the weak Lefschetz property for Artinian Gorenstein algebras of codimension three”, J. Algebra 403 (2014), 48–68.
  • J. A. Boswell and V. Mukundan, “Rees algebras and almost linearly presented ideals”, J. Algebra 460 (2016), 102–127.
  • W. Bruns, A. Conca, and M. Varbaro, “Relations between the minors of a generic matrix”, Adv. Math. 244 (2013), 171–206.
  • W. Bruns, A. Conca, and M. Varbaro, “Maximal minors and linear powers”, J. Reine Angew. Math. 702 (2015), 41–53.
  • D. A. Buchsbaum and D. Eisenbud, “Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension $3$”, Amer. J. Math. 99:3 (1977), 447–485.
  • L. Burch, “On ideals of finite homological dimension in local rings”, Proc. Cambridge Philos. Soc. 64 (1968), 941–948.
  • L. Busé, “On the equations of the moving curve ideal of a rational algebraic plane curve”, J. Algebra 321:8 (2009), 2317–2344.
  • M. Chardin, D. Eisenbud, and B. Ulrich, “Hilbert functions, residual intersections, and residually ${\rm S}_2$ ideals”, Compositio Math. 125:2 (2001), 193–219.
  • A. Conca and G. Valla, “Betti numbers and lifting of Gorenstein codimension three ideals”, Comm. Algebra 28:3 (2000), 1371–1386.
  • T. Cortadellas Benítez and C. D'Andrea, “Rational plane curves parameterizable by conics”, J. Algebra 373 (2013), 453–480.
  • T. Cortadellas Benítez and C. D'Andrea, “Minimal generators of the defining ideal of the Rees algebra associated with a rational plane parametrization with $\mu=2$”, Canad. J. Math. 66:6 (2014), 1225–1249.
  • D. Cox, J. W. Hoffman, and H. Wang, “Syzygies and the Rees algebra”, J. Pure Appl. Algebra 212:7 (2008), 1787–1796.
  • E. De Negri and G. Valla, “The $h$-vector of a Gorenstein codimension three domain”, Nagoya Math. J. 138 (1995), 113–140.
  • S. J. Diesel, “Irreducibility and dimension theorems for families of height $3$ Gorenstein algebras”, Pacific J. Math. 172:2 (1996), 365–397.
  • D. Eisenbud and B. Ulrich, “Row ideals and fibers of morphisms”, Michigan Math. J. 57 (2008), 261–268.
  • D. Eisenbud, C. Huneke, and B. Ulrich, “The regularity of Tor and graded Betti numbers”, Amer. J. Math. 128:3 (2006), 573–605.
  • J. Elias and A. A. Iarrobino, “The Hilbert function of a Cohen–Macaulay local algebra: extremal Gorenstein algebras”, J. Algebra 110:2 (1987), 344–356.
  • F. Gaeta, “Ricerche intorno alle varietà matriciali ed ai loro ideali”, pp. 326–328 in Atti del Quarto Congresso dell'Unione Matematica Italiana (Taormina, 1951), vol. II, Casa Editrice Perrella, Roma, 1953.
  • A. V. Geramita and J. C. Migliore, “Reduced Gorenstein codimension three subschemes of projective space”, Proc. Amer. Math. Soc. 125:4 (1997), 943–950.
  • L. Gruson, R. Lazarsfeld, and C. Peskine, “On a theorem of Castelnuovo, and the equations defining space curves”, Invent. Math. 72:3 (1983), 491–506.
  • R. Hartshorne, “Geometry of arithmetically Gorenstein curves in $\mathbb P^4$”, Collect. Math. 55:1 (2004), 97–111.
  • J. Herzog, A. Simis, and W. V. Vasconcelos, “Koszul homology and blowing-up rings”, pp. 79–169 in Commutative algebra (Trento, 1981), edited by S. Greco and G. Valla, Lecture Notes in Pure and Appl. Math. 84, Dekker, New York, 1983.
  • J. Herzog, W. V. Vasconcelos, and R. Villarreal, “Ideals with sliding depth”, Nagoya Math. J. 99 (1985), 159–172.
  • J. Hong, A. Simis, and W. V. Vasconcelos, “On the homology of two-dimensional elimination”, J. Symbolic Comput. 43:4 (2008), 275–292.
  • C. Huneke, “Linkage and the Koszul homology of ideals”, Amer. J. Math. 104:5 (1982), 1043–1062.
  • C. Huneke and M. Rossi, “The dimension and components of symmetric algebras”, J. Algebra 98:1 (1986), 200–210.
  • A. A. Iarrobino, “Associated graded algebra of a Gorenstein Artin algebra”, pp. viii+115 Mem. Amer. Math. Soc. 514, 1994.
  • A. Iarrobino and V. Kanev, Power sums, Gorenstein algebras, and determinantal loci, Lecture Notes in Mathematics 1721, Springer, 1999.
  • M. R. Johnson, “Second analytic deviation one ideals and their Rees algebras”, J. Pure Appl. Algebra 119:2 (1997), 171–183.
  • J.-P. Jouanolou, “Résultant anisotrope, compléments et applications”, Electron. J. Combin. 3:2 (1996), art. id. #2.
  • J. P. Jouanolou, “Formes d'inertie et résultant: un formulaire”, Adv. Math. 126:2 (1997), 119–250.
  • K. Kimura and N. Terai, “Arithmetical rank of Gorenstein squarefree monomial ideals of height three”, J. Algebra 422 (2015), 11–32.
  • J. O. Kleppe and R. M. Miró-Roig, “The dimension of the Hilbert scheme of Gorenstein codimension $3$ subschemes”, J. Pure Appl. Algebra 127:1 (1998), 73–82.
  • A. R. Kustin and B. Ulrich, A family of complexes associated to an almost alternating map, with applications to residual intersections, vol. 95, Mem. Amer. Math. Soc. 461, 1992.
  • A. R. Kustin, C. Polini, and B. Ulrich, “Rational normal scrolls and the defining equations of Rees algebras”, J. Reine Angew. Math. 650 (2011), 23–65.
  • A. R. Kustin, C. Polini, and B. Ulrich, “Degree bounds for local cohomology”, preprint, 2015.
  • A. R. Kustin, C. Polini, and B. Ulrich, “Blowups and fibers of morphisms”, Nagoya Math. J. 224:1 (2016), 168–201.
  • A. R. Kustin, C. Polini, and B. Ulrich, “The Hilbert series of the ring associated to an almost alternating matrix”, Comm. Algebra 44:7 (2016), 3053–3068.
  • A. R. Kustin, C. Polini, and B. Ulrich, “The bi-graded structure of symmetric algebras with applications to Rees rings”, J. Algebra 469 (2017), 188–250.
  • A. R. Kustin, C. Polini, and B. Ulrich, “A matrix of linear forms which is annihilated by a vector of indeterminates”, J. Algebra 469 (2017), 120–187.
  • J. Madsen, “Equations of Rees algebras of ideals in two variables”, preprint, 2015.
  • J. C. Migliore and C. Peterson, “A construction of codimension three arithmetically Gorenstein subschemes of projective space”, Trans. Amer. Math. Soc. 349:9 (1997), 3803–3821.
  • S. Morey, “Equations of blowups of ideals of codimension two and three”, J. Pure Appl. Algebra 109:2 (1996), 197–211.
  • S. Morey and B. Ulrich, “Rees algebras of ideals with low codimension”, Proc. Amer. Math. Soc. 124:12 (1996), 3653–3661.
  • L. P. H. Nguyen, “On Rees algebras of linearly presented ideals”, J. Algebra 420 (2014), 186–200.
  • C. Polini and B. Ulrich, “Necessary and sufficient conditions for the Cohen–Macaulayness of blowup algebras”, Compositio Math. 119:2 (1999), 185–207.
  • B. Ulrich, “Artin–Nagata properties and reductions of ideals”, pp. 373–400 in Commutative algebra: syzygies, multiplicities, and birational algebra (South Hadley, MA, 1992), edited by W. J. Heinzer et al., Contemp. Math. 159, Amer. Math. Soc., Providence, RI, 1994.
  • W. V. Vasconcelos, “On the equations of Rees algebras”, J. Reine Angew. Math. 418 (1991), 189–218.
  • J. Watanabe, “A note on Gorenstein rings of embedding codimension three”, Nagoya Math. J. 50 (1973), 227–232.