Multiplicity of the saturated special fiber ring of height two perfect ideals

Cid-Ruiz, Yairon

doi:10.1090/proc/14693

Multiplicity of the saturated special fiber ring of height two perfect ideals

Author: Yairon Cid-Ruiz
Journal: Proc. Amer. Math. Soc. 148 (2020), 59-70
MSC (2010): Primary 13A30; Secondary 14E05, 13D02, 13D45
DOI: https://doi.org/10.1090/proc/14693
Published electronically: July 10, 2019
MathSciNet review: 4042830
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a polynomial ring and let $I \subset R$ be a perfect ideal of height two minimally generated by forms of the same degree. We provide a formula for the multiplicity of the saturated special fiber ring of . Interestingly, this formula is equal to an elementary symmetric polynomial in terms of the degrees of the syzygies of . Applying ideas introduced by Busé, D'Andrea, and the author, we obtain the value of the -multiplicity of and an effective method for determining the degree and birationality of rational maps defined by homogeneous generators of .

[1] Rüdiger Achilles and Mirella Manaresi, Multiplicity for ideals of maximal analytic spread and intersection theory, J. Math. Kyoto Univ. 33 (1993), no. 4, 1029–1046. MR 1251213, https://doi.org/10.1215/kjm/1250519127
[2] Nicolás Botbol, Laurent Busé, Marc Chardin, Seyed Hamid Hassanzadeh, Aron Simis, and Quang Hoa Tran, Effective criteria for bigraded birational maps, J. Symbolic Comput. 81 (2017), 69–87. MR 3594324, https://doi.org/10.1016/j.jsc.2016.12.001
[3] M. P. Brodmann and R. Y. Sharp, Local cohomology, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 136, Cambridge University Press, Cambridge, 2013. An algebraic introduction with geometric applications. MR 3014449
[4] Winfried Bruns and Jürgen Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR 1251956
[5] Laurent Busé, Yairon Cid-Ruiz, and Carlos D'Andrea, Degree and birationality of multi-graded rational maps, ArXiv e-prints (2018-05). arXiv:1805.05180.
[6] Marc Chardin, Regularity of ideals and their powers, Prépublication, Institut de Mathématiques de Jussieu 364 (2004).
[7] Yairon Cid-Ruiz, A -module approach on the equations of the Rees algebra, to appear in J. Commut. Algebra (2017). arXiv:1706.06215.
[8] Yairon Cid-Ruiz and Aron Simis, Degree of rational maps via specialization, ArXiv e-prints (2019-01). arXiv:1901.06599.
[9] David A. Cox, Equations of parametric curves and surfaces via syzygies, Symbolic computation: solving equations in algebra, geometry, and engineering (South Hadley, MA, 2000) Contemp. Math., vol. 286, Amer. Math. Soc., Providence, RI, 2001, pp. 1–20. MR 1874268, https://doi.org/10.1090/conm/286/04751
[10] A. V. Doria, S. H. Hassanzadeh, and A. Simis, A characteristic-free criterion of birationality, Adv. Math. 230 (2012), no. 1, 390–413. MR 2900548, https://doi.org/10.1016/j.aim.2011.12.005
[11] David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960
[12] David Eisenbud and Bernd Ulrich, Row ideals and fibers of morphisms, Michigan Math. J. 57 (2008), 261–268. Special volume in honor of Melvin Hochster. MR 2492452, https://doi.org/10.1307/mmj/1220879408
[13] H. Flenner, L. O’Carroll, and W. Vogel, Joins and intersections, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1999. MR 1724388
[14] Daniel R. Grayson and Michael E. Stillman, Macaulay2, a software system for research in algebraic geometry. Available at http://www.math.uiuc.edu/Macaulay2/.
[15] Seyed Hamid Hassanzadeh and Aron Simis, Plane Cremona maps: saturation and regularity of the base ideal, J. Algebra 371 (2012), 620–652. MR 2975417, https://doi.org/10.1016/j.jalgebra.2012.08.022
[16] S. H. Hassanzadeh and A. Simis, Bounds on degrees of birational maps with arithmetically Cohen-Macaulay graphs, J. Algebra 478 (2017), 220–236. MR 3621670, https://doi.org/10.1016/j.jalgebra.2017.01.010
[17] J. Herzog, A. Simis, and W. V. Vasconcelos, Koszul homology and blowing-up rings, Commutative algebra (Trento, 1981) Lecture Notes in Pure and Appl. Math., vol. 84, Dekker, New York, 1983, pp. 79–169. MR 686942
[18] Klaus Hulek, Sheldon Katz, and Frank-Olaf Schreyer, Cremona transformations and syzygies, Math. Z. 209 (1992), no. 3, 419–443. MR 1152265, https://doi.org/10.1007/BF02570843
[19] C. Huneke and M. Rossi, The dimension and components of symmetric algebras, J. Algebra 98 (1986), no. 1, 200–210. MR 825143, https://doi.org/10.1016/0021-8693(86)90023-2
[20] Jack Jeffries and Jonathan Montaño, The 𝑗-multiplicity of monomial ideals, Math. Res. Lett. 20 (2013), no. 4, 729–744. MR 3188029, https://doi.org/10.4310/MRL.2013.v20.n4.a9
[21] Jack Jeffries, Jonathan Montaño, and Matteo Varbaro, Multiplicities of classical varieties, Proc. Lond. Math. Soc. (3) 110 (2015), no. 4, 1033–1055. MR 3335294, https://doi.org/10.1112/plms/pdv005
[22] Andrew Kustin, Claudia Polini, and Bernd Ulrich, Degree bounds for local cohomology, ArXiv e-prints (2015-05). arXiv:1505.05209.
[23] Andrew R. Kustin, Claudia Polini, and Bernd Ulrich, Blowups and fibers of morphisms, Nagoya Math. J. 224 (2016), no. 1, 168–201. MR 3572752, https://doi.org/10.1017/nmj.2016.34
[24] Andrew R. Kustin, Claudia Polini, and Bernd Ulrich, The equations defining blowup algebras of height three Gorenstein ideals, Algebra Number Theory 11 (2017), no. 7, 1489–1525. MR 3697146, https://doi.org/10.2140/ant.2017.11.1489
[25] Koji Nishida and Bernd Ulrich, Computing 𝑗-multiplicities, J. Pure Appl. Algebra 214 (2010), no. 12, 2101–2110. MR 2660901, https://doi.org/10.1016/j.jpaa.2010.02.008
[26] Ivan Pan and Aron Simis, Cremona maps of de Jonquières type, Canad. J. Math. 67 (2015), no. 4, 923–941. MR 3361019, https://doi.org/10.4153/CJM-2014-037-3
[27] Claudia Polini and Yu Xie, 𝑗-multiplicity and depth of associated graded modules, J. Algebra 379 (2013), 31–49. MR 3019244, https://doi.org/10.1016/j.jalgebra.2013.01.001
[28] Francesco Russo and Aron Simis, On birational maps and Jacobian matrices, Compositio Math. 126 (2001), no. 3, 335–358. MR 1834742, https://doi.org/10.1023/A:1017572213947
[29] Aron Simis, Cremona transformations and some related algebras, J. Algebra 280 (2004), no. 1, 162–179. MR 2081926, https://doi.org/10.1016/j.jalgebra.2004.03.025
[30] Bernd Ulrich and Wolmer V. Vasconcelos, The equations of Rees algebras of ideals with linear presentation, Math. Z. 214 (1993), no. 1, 79–92. MR 1234599, https://doi.org/10.1007/BF02572392
[31] Wolmer V. Vasconcelos, Arithmetic of blowup algebras, London Mathematical Society Lecture Note Series, vol. 195, Cambridge University Press, Cambridge, 1994. MR 1275840