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Homaloidal nets and ideals of fat points I

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

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 

References

Alberich-Carramiñana, M., Geometry of the Plane Cremona Maps , Lecture Notes in Mathematics 1769 (Springer, Berlin–Heidelberg, 2002).Google Scholar
Bisi, C., Calabri, A. and Mella, M., ‘On plane Cremona transformations of fixed degree’, J. Geom. Anal. 25 (2015) 1108–1131.Google Scholar
Blanc, J. and Calabri, A., ‘On degenerations of plane Cremona transformations’, Math. Z. 282 (2016) 223–245.CrossRefGoogle Scholar
Ciliberto, C., Cueto, M. A., Mella, M., Ranestad, K. and Zwiernik, P., ‘Cremona linearizations of some classical varieties’, Proceedings of the Conference “Homage to Corrado Segre”, Torino, Italy, 2014, Trends in the History of Science (Birkhäuser, Basel) to appear, arXiv:1403.1814v1 [math.AG].Google Scholar
Ciliberto, C., Russo, F. and Simis, A., ‘Homaloidal hypersurfaces and hypersurfaces with vanishing Hessian’, Adv. Math. 218 (2008) 1759–1805.CrossRefGoogle Scholar
Costa, B. and Simis, A., ‘New constructions of Cremona maps’, Math. Res. Lett. 20 (2013) 629–645.CrossRefGoogle Scholar
Davis, E. D., Geramita, A. V. and Maroscia, P., ‘Perfect homogeneous ideals: Dubreil’s theorems revisited’, Bull. Sci. Math., 2e série 108 (1984) 143–185.Google Scholar
Doria, A., Hassanzadeh, H. and Simis, A., ‘A characteristic free criterion of birationality’, Adv. Math. 230 (2012) 390–413.Google Scholar
Dumnicki, M., Szemberg, T. and Tutaj-Gasińska, H., ‘A vanishing theorem and symbolic powers of planar point ideals’, LMS J. Comput. Math. 16 (2013) 373–387.CrossRefGoogle Scholar
Geramita, A. V., Gimigliano, A. and Pitteloud, Y., ‘Graded Betti numbers of some embedded rational n-folds’, Math. Ann. 301 (1995) 363–380.CrossRefGoogle Scholar
Geramita, A. V. and Orecchia, F., ‘Minimally generating ideals defining certain tangent cones’, J. Algebra 78 (1982) 36–57.CrossRefGoogle Scholar
Guardo, E. and Harbourne, B., ‘Resolutions of ideals of six fat points in ℙ2 ’, J. Algebra 318 (2007) 619–640.CrossRefGoogle Scholar
Harbourne, B., ‘Free resolutions of fat point ideals on ℙ2 ’, J. Pure Appl. Algebra 125 (1998) 213–234.CrossRefGoogle Scholar
Hassanzadeh, S. H. and Simis, A., ‘Plane Cremona maps: saturation, regularity and fat ideals’, J. Algebra 371 (2012) 620–652.Google Scholar
Miranda, R., ‘Linear systems of plane curves’, Notices Amer. Math. Soc. 46 (1999) 192–202.Google Scholar
Nagata, M., ‘On rational surfaces II, Memoirs of the College of Sciences’, Univ. Kyoto, Ser. A 33 (1960) 271–293.Google Scholar
Pan, I., ‘Sur le multidegré des transformations de Cremona’, C. R. Acad. Sci. Paris 330 (2000) no. Série I, 297–300.Google Scholar
Pan, I., ‘Les transformations de Cremona stellaires’, Proc. Amer. Math. Soc. 129 (2001) 1257–1262.CrossRefGoogle Scholar
Pan, I. and Russo, F., ‘Cremona transformations and special double structures’, Manuscripta Math. 117 (2005) 491–510.Google Scholar
Pan, I. and Simis, A., ‘Cremona maps of de Jonquières type’, Canad. J. Math. 67 (2015) 923–941.CrossRefGoogle Scholar
Simis, A., ‘Cremona transformations and some related algebras’, J. Algebra 280 (2004) no. 1, 162–179.Google Scholar
Simis, A. and Tohǎneanu, S., ‘The ubiquity of Sylvester forms in almost complete intersections’, Collect. Math. 66 (2015) 1–31.Google Scholar
Simis, A. and Vasconcelos, W. V., ‘The syzygies of the conormal module’, Amer. J. Math. 103 (1981) 203–224.Google Scholar
Simis, A. and Villarreal, R. H., ‘Linear syzygies and birational combinatorics’, Results Math. 48 (2005) no. 3-4, 326–343.Google Scholar
Simis, A. and Villarreal, R. H., ‘Combinatorics of Cremona monomial maps’, Math. Comp. 81 (2012) 1857–1867.Google Scholar