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Homaloidal nets and ideals of fat points I

Part of: Birational geometry Theory of modules and ideals Local rings and semilocal rings Cycles and subschemes Special varieties Commutative algebra: Homological methods General commutative ring theory

Published online by Cambridge University Press:  01 March 2016

Zaqueu Ramos  and
Aron Simis
Zaqueu Ramos
Affiliation:
Departamento de Matemática, CCET, Universidade Federal de Sergipe, 49100-000 São Cristovão, Sergipe, Brazil email zaqueu.ramos@gmail.com
Aron Simis
Affiliation:
Departamento de Matemática, CCEN, Universidade Federal de Pernambuco, 50740-560 Recife, PE, Brazil Departamento de Matemática, CCEN, Universidade Federal da Paraíba, 58059-900 João Pessoa, PB, Brazil email aron@dmat.ufpe.br

Abstract

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We consider plane Cremona maps with proper base points and the base ideal generated by the linear system of forms defining the map. The object of this work is to study the link between the algebraic properties of the base ideal and those of the ideal of these points fattened by the virtual multiplicities arising from the linear system. We reveal conditions which naturally regulate this association, with particular emphasis on the homological side. While most classical numerical inequalities concern the three highest virtual multiplicities, here we emphasize also the role of one single highest multiplicity. In this vein we describe classes of Cremona maps for large and small values of the highest virtual multiplicity. We also deal with the delicate question as to when is the base ideal non-saturated and consider the structure of its saturation.

MSC classification

Primary: 13D02: Syzygies, resolutions, complexes 13H10: Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) 13H15: Multiplicity theory and related topics 14E05: Rational and birational maps 14E07: Birational automorphisms, Cremona group and generalizations 14M05: Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
Secondary: 13A02: Graded rings 13C14: Cohen-Macaulay modules 14C20: Divisors, linear systems, invertible sheaves
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 19 , Issue 1 , 2016 , pp. 54 - 77
Copyright
© The Author(s) 2016 

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