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Mixed multiplicities and projective degrees of rational maps. (English) Zbl 07268487
Summary: We consider the notion of mixed multiplicities for multigraded modules by using Hilbert series, and this is later applied to study the projective degrees of rational maps. We use a general framework to determine the projective degrees of a rational map via a computation of the multiplicity of the saturated special fiber ring. As specific applications, we provide explicit formulas for all the projective degrees of rational maps determined by perfect ideals of height two or by Gorenstein ideals of height three.
13H15 Multiplicity theory and related topics
14E05 Rational and birational maps
13A30 Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics
13D02 Syzygies, resolutions, complexes and commutative rings
Full Text: DOI
[1] Bhattacharya, P. B., The Hilbert function of two ideals, Proc. Camb. Philos. Soc., 53, 568-575 (1957) · Zbl 0080.02903
[2] Brodmann, M. P.; Sharp, R. Y., Local Cohomology, Cambridge Studies in Advanced Mathematics, vol. 136 (2013), Cambridge University Press: Cambridge University Press Cambridge, An algebraic introduction with geometric applications
[3] Bruns, W.; Herzog, J., Cohen-Macaulay Rings, Cambridge Studies in Advanced Mathematics (1998), Cambridge University Press · Zbl 0909.13005
[4] Buchsbaum, D. A.; Eisenbud, D., Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension 3, Am. J. Math., 99, 3, 447-485 (1977) · Zbl 0373.13006
[5] Busé, L.; Cid-Ruiz, Y.; D’Andrea, C., Degree and birationality of multi-graded rational maps, Proc. Lond. Math. Soc., 121, 4, 743-787 (2020) · Zbl 07242515
[6] Cid-Ruiz, Y., Multiplicity of the saturated special fiber ring of height two perfect ideals, Proc. Am. Math. Soc., 148, 1, 59-70 (2020) · Zbl 1436.13007
[7] Cid-Ruiz, Y.; Mukundan, V., Multiplicity of the saturated special fiber ring of height three Gorenstein ideals, Acta Math. Vietnam. (2019), in press
[8] Cid-Ruiz, Y.; Simis, A., Degree of rational maps and specialization, Int. Math. Res. Not. (2019), in press
[9] Dolgachev, I. V., Classical Algebraic Geometry (2012), Cambridge University Press: Cambridge University Press Cambridge, A modern view · Zbl 1252.14001
[10] Eisenbud, D., Commutative Algebra with a View Towards Algebraic Geometry, Graduate Texts in Mathematics, vol. 150 (1995), Springer-Verlag
[11] Eisenbud, D.; Harris, J., The Geometry of Schemes, Graduate Texts in Mathematics, vol. 197 (2000), Springer-Verlag: Springer-Verlag New York · Zbl 0960.14002
[12] Görtz, U.; Wedhorn, T., Algebraic Geometry I, Advanced Lectures in Mathematics (2010), Vieweg + Teubner: Vieweg + Teubner Wiesbaden, Schemes with examples and exercises
[13] Grayson, D. R.; Stillman, M. E., Macaulay2, a software system for research in algebraic geometry, Available at
[14] Harris, J., Algebraic Geometry, Graduate Texts in Mathematics, vol. 133 (1995), Springer-Verlag: Springer-Verlag New York, A first course, Corrected reprint of the 1992 original
[15] Herrmann, M.; Hyry, E.; Ribbe, J.; Tang, Z., Reduction numbers and multiplicities of multigraded structures, J. Algebra, 197, 2, 311-341 (1997) · Zbl 0931.13002
[16] Herzog, J.; Hibi, T., Monomial Ideals, Graduate Texts in Mathematics, vol. 260 (2011), Springer-Verlag London, Ltd.: Springer-Verlag London, Ltd. London · Zbl 1206.13001
[17] Huneke, C.; Swanson, I., Integral Closure of Ideals, Rings, and Modules, Vol. 13 (2006), Cambridge University Press
[18] Hyry, E., The diagonal subring and the Cohen-Macaulay property of a multigraded ring, Trans. Am. Math. Soc., 351, 6, 2213-2232 (1999) · Zbl 0916.13005
[19] Jayanthan, A. V.; Verma, J. K., Grothendieck-Serre formula and bigraded Cohen-Macaulay Rees algebras, J. Algebra, 254, 1, 1-20 (2002) · Zbl 1094.13504
[20] Katz, D.; Mandal, S.; Verma, J. K., Hilbert functions of bigraded algebras, (Commutative Algebra. Commutative Algebra, Trieste, 1992 (1994), World Sci. Publ.: World Sci. Publ. River Edge, NJ), 291-302 · Zbl 0927.13021
[21] Kleiman, S.; Thorup, A., A geometric theory of the Buchsbaum-Rim multiplicity, J. Algebra, 167, 1, 168-231 (1994) · Zbl 0815.13012
[22] Kustin, A.; Polini, C.; Ulrich, B., Blowups and fibers of morphisms, Nagoya Math. J., 224, 1, 168-201 (2016) · Zbl 1408.13013
[23] Michałek, M.; Sturmfels, B.; Uhler, C.; Zwiernik, P., Exponential varieties, Proc. Lond. Math. Soc. (3), 112, 1, 27-56 (2016) · Zbl 1345.14048
[24] Miller, E.; Sturmfels, B., Combinatorial Commutative Algebra, Graduate Texts in Mathematics, vol. 227 (2005), Springer-Verlag: Springer-Verlag New York · Zbl 1090.13001
[25] Trung, N. V.; Verma, J. K., Hilbert functions of multigraded algebras, mixed multiplicities of ideals and their applications, J. Commut. Algebra, 2, 4, 515-565 (2010) · Zbl 1237.13048
[26] Trung, N. V., Reduction exponent and degree bound for the defining equations of graded rings, Proc. Am. Math. Soc., 101, 2, 229-236 (1987) · Zbl 0641.13016
[27] Trung, N. V., Positivity of mixed multiplicities, Math. Ann., 319, 1, 33-63 (2001) · Zbl 0979.13023
[28] Van der Waerden, B. L., On Hilbert’s function, series of composition of ideals and a generalization of the theorem of Bezout, Proc. R. Acad. Amst., 31, 749-770 (1929) · JFM 54.0190.02
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