Centering toric arrangements of maximal rank

Authors

  • Elia Saini Independent researcher

Abstract

The homotopy type of the complement manifold of a complexified toric arrangement has been investigated by d'Antonio and Delucchi in a paper that shows the minimality of such topological space. In this work we associate to a given toric arrangement a matrix that represents the arrangement over the integers. Then, we consider the family of toric arrangements for which this matrix has maximal rank. Our goal is to prove, by means of basic linear algebra arguments, that the complement manifold of the toric arrangements that belong to this family is diffeomorphic to that of centered toric arrangements and thus it is a minimal topological space, too.

References

Brändén, Petter, and Luca Moci. "The multivariate arithmetic Tutte polynomial." Trans. Amer. Math. Soc. 366, no. 10 (2014): 5523-5540.
##plugins.generic.googleScholarLinks.settings.viewInGS##

Callegaro, Filippo, and Emanuele Delucchi. "The integer cohomology algebra of toric arrangements." Adv. Math. 313 (2017): 746-802.
##plugins.generic.googleScholarLinks.settings.viewInGS##

D’Adderio, Michele, and Luca Moci. "Arithmetic matroids, the Tutte polynomial and toric arrangements." Adv. Math. 232 (2013): 335-367.
##plugins.generic.googleScholarLinks.settings.viewInGS##

d’Antonio, Giacomo, and Emanuele Delucchi. "A Salvetti complex for toric arrangements and its fundamental group." Int. Math. Res. Not. IMRN, no. 15 (2012): 3535-3566.
##plugins.generic.googleScholarLinks.settings.viewInGS##

d’Antonio, Giacomo, and Emanuele Delucchi. "Minimality of toric arrangements." J. Eur. Math. Soc. (JEMS) 17, no. 3 (2015): 483-521.
##plugins.generic.googleScholarLinks.settings.viewInGS##

De Concini, Corrado, and Giovanni Gaiffi. "A differential algebra and the homotopy type of the complement of a toric arrangement." Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 32, no. 1 (2021): 1-21.
##plugins.generic.googleScholarLinks.settings.viewInGS##

De Concini, Corrado, and Claudio Procesi. "On the geometry of toric arrangements." Transform. Groups 10, no. 3-4 (2005): 387-422.
##plugins.generic.googleScholarLinks.settings.viewInGS##

De Concini, Corrado, and Claudio Procesi. Topics in hyperplane arrangements, polytopes and box-splines. Springer: New York, 2010.
##plugins.generic.googleScholarLinks.settings.viewInGS##

Deligne, Pierre. "Théorie de Hodge. II." Sci. Publ. Math, no. 40 (1971): 5-57.
##plugins.generic.googleScholarLinks.settings.viewInGS##

Dimca, Alexandru, and Stefan Papadima. "Hypersurface complements, Milnor fibers and higher homotopy groups of arrangments." Ann. of Math. (2) 158, no. 2: 473-507.
##plugins.generic.googleScholarLinks.settings.viewInGS##

Lefschetz, Solomon. L’analysis situs et la géométrie algébrique. Gauthier-Villars: Paris, 1924.
##plugins.generic.googleScholarLinks.settings.viewInGS##

Lenz, Matthias. "Representations of weakly multiplicative arithmetic matroids are unique." Ann. Comb. 23, no. 2 (2019): 335-346.
##plugins.generic.googleScholarLinks.settings.viewInGS##

Looijenga, Eduard. "Cohomology of M3 and M13 ". In: Mapping class groups and moduli spaces of Riemann surfaces, 205-228. Vol. 150 of Contemporary Mathematics. Göttingen/Seattle/Providence RI: American Mathematical Society, 1993.
##plugins.generic.googleScholarLinks.settings.viewInGS##

Moci, Luca. "A Tutte polynomial for toric arrangements." Trans. Amer. Math. Soc. 364, no. 2 (2012): 1067-1088.
##plugins.generic.googleScholarLinks.settings.viewInGS##

Moci, Luca, and Simona Settepanella. "The homotopy type of toric arrangements." J. Pure Appl. Algebra 215, no. 8 (2011): 1980-1989.
##plugins.generic.googleScholarLinks.settings.viewInGS##

Newman, Morris. Integral Matrices. Vol. 45 of Monographs and Textbooks in Pure and Applied Mathematics. Washington, D.C.: Academic Press, 1972.
##plugins.generic.googleScholarLinks.settings.viewInGS##

Pagaria, Roberto. "Two examples of toric arrangements." J. Combin. Theory Ser. A 167 (2019): 389-402.
##plugins.generic.googleScholarLinks.settings.viewInGS##

Randell, Richard. "Morse theory, Milnor fibers and minimality of hyperplane arrangements." Proc. Amer. Math. Soc. 130, no. 9 (2002): 2737-2743.
##plugins.generic.googleScholarLinks.settings.viewInGS##

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Published

2024-11-19

How to Cite

Saini, E. (2024). Centering toric arrangements of maximal rank. Annales Universitatis Paedagogicae Cracoviensis Studia Mathematica, 23, 39–46. Retrieved from https://studmath.uken.krakow.pl/article/view/11524

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Vol. 23 (2024)

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