On projective wonderful models for toric arrangements and their cohomology
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On projective wonderful models for toric arrangements and their cohomology Corrado De Concini1 · Giovanni Gaiffi2
· Oscar Papini2
In reverent and grateful memory of S¸ tefan Papadima Received: 15 March 2019 / Revised: 24 December 2019 / Accepted: 17 May 2020 © Springer Nature Switzerland AG 2020
Abstract This paper is divided into two parts. The first part is a brief survey, accompanied by concrete examples, on the main results of the papers (De Concini and Gaiffi in Adv Math 327:390–09, 2018; Algebr Geom Topol 19(1):503–532, 2019): the construction of projective models of toric arrangements and the presentation of their cohomology rings by generators and relations. In the second part we focus on the notion of wellconnected building set that appears in the cohomological computations mentioned above: we explore some of its properties in the more general context of arrangements of subvarieties of a variety X . Keywords Toric arrangements · Compact models · Configuration spaces Mathematics Subject Classification 14N20
B
Giovanni Gaiffi [email protected] Corrado De Concini [email protected] Oscar Papini [email protected]
1
Department of Mathematics, Università di Roma “La Sapienza”, Piazzale Aldo Moro 5, 00185 Rome, Italy
2
Department of Mathematics, Università di Pisa, Largo Pontecorvo 5, 56127 Pisa, Italy
123
C. De Concini et al.
1 Introduction Let T (C∗ )n be an n-dimensional torus and let X ∗ (T ) be its group of characters; a layer in T is a subvariety of T of the form K(, φ) ..= t ∈ T | χ (t) = φ(χ ) for all χ ∈ where < X ∗ (T ) is a split direct summand and φ : → C∗ is a homomorphism. A toric arrangement A is a (finite) set of layers {K1 , . . . , Kr } in T . We remark that in literature the usual definition of “toric arrangement” requires the layers to be 1codimensional; instead we allow layers of any codimension and use the term divisorial arrangement for the case where all layers are 1-codimensional. Toric arrangements have been studied since the early 1990s, and over the last two decades several aspects have been investigated: in particular, as far as the topology of the complement is concerned, De Concini and Procesi [12] determined the generators of the cohomology modules over C in the divisorial case, as well as the ring structure in the case of totally unimodular arrangements; d’Antonio and Delucchi, generalizing an algebraic complex first introduced by Moci and Settepanella [25], provided a presentation of the fundamental group for the complement of a divisorial complexified arrangement [5,6]; Callegaro, Delucchi and Pagaria computed the graded cohomology ring with integer coefficients (see [3,4,26]); the cohomology ring itself was computed by Callegaro et al. [2]. The problem of studying wonderful models for toric arrangement was first addressed by Moci [24], where he described a construction of a non-projective model. Wonderful models for subspace arrangements were introduced by De Concini and Procesi [10,11], where they provided both a projective and a no
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