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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
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Abstract: Let $ R$ 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 $ I$. Interestingly, this formula is equal to an elementary symmetric polynomial in terms of the degrees of the syzygies of $ I$. Applying ideas introduced by Busé, D'Andrea, and the author, we obtain the value of the $ j$-multiplicity of $ I$ and an effective method for determining the degree and birationality of rational maps defined by homogeneous generators of $ I$.


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Additional Information

Yairon Cid-Ruiz
Affiliation: Department de Matemàtiques i Informàtica, Facultat de Matemàtiques i Informàtica, Universitat de Barcelona, Gran Via de les Corts Catalanes, 585; 08007 Barcelona, Spain
Email: ycid@ub.edu

DOI: https://doi.org/10.1090/proc/14693
Keywords: Saturated special fiber ring, rational and birational maps, $j$-multiplicity, syzygies, Rees algebra, symmetric algebra, special fiber ring, multiplicity, Hilbert-Burch theorem, local cohomology
Received by editor(s): July 12, 2018
Received by editor(s) in revised form: April 11, 2019
Published electronically: July 10, 2019
Additional Notes: The author was funded by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 675789
Communicated by: Claudia Polini
Article copyright: © Copyright 2019 American Mathematical Society