Pure and Applied Mathematics Quarterly

Volume 20 (2024)

Number 1

Special Issue dedicated to Corrado De Concini

Guest Editors: Alberto De Sole, Nicoletta Cantarini, and Andrea Maffei

A basis for the cohomology of compact models of toric arrangements

Pages: 427 – 470

DOI: https://dx.doi.org/10.4310/PAMQ.2024.v20.n1.a9

Authors

Giovanni Gaiffi (Dipartimento di Matematica, Università di Pisa, Italy)

Oscar Papini (Istituto di Scienza e Tecnologie dell’Informazione “A. Faedo”, Consiglio Nazionale delle Ricerche, Pisa, Italy)

Viola Siconolfi (Dipartimento di Meccanica, Matematica e Management, Politecnico di Bari, Italy)

Abstract

In this paper we find monomial bases for the integer cohomology rings of compact wonderful models of toric arrangements. In the description of the monomials various combinatorial objects come into play: building sets, nested sets, and the fan of a suitable toric variety. We provide some examples computed via a SageMath program and then we focus on the case of the toric arrangements associated with root systems of type $A$. Here the combinatorial description of our basis offers a geometrical point of view on the relation between some Eulerian statistics on the symmetric group.

Keywords

toric arrangements, compact models, configuration spaces, Eulerian numbers

2010 Mathematics Subject Classification

05C30, 14N20

Full Text (PDF format)

A Corrado, con profonda stima e gratitudine

V.S. was supported by PRIN 2017 ‘Moduli and Lie Theory’, Dipartimento di Matematica, Università di Pisa.

Received 31 May 2022

Accepted 31 January 2023

Published 26 March 2024