Nearly free curves and arrangements: a vector bundle point of view

Autor(es)

Marchesi, Simone ^[1] ; Valles, Jean ^[2]

Afiliação do(s) autor(es)

^[1] Univ Estadual Campinas, IMECC, Rua Sergio Buarque de Holanda 651, BR-13083859 Campinas, SP - Brazil

^[2] Univ Pau & Pays Adour, LMAP UMR CNRS 5142, Ave Univ BP 576, F-64012 Pau - France

• Tipo de documento: Artigo Científico
• Fonte: MATHEMATICAL PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY; v. 170, n. 1, p. 51-74, JAN 2021.
• Citações Web of Science: 4

Resumo

Over the past forty years many papers have studied logarithmic sheaves associated to reduced divisors, in particular logarithmic bundles associated to plane curves. An interesting family of these curves are the so-called free ones for which the associated logarithmic sheaf is the direct sum of two line bundles. Terao conjectured thirty years ago that when a curve is a finite set of distinct lines (i.e. a line arrangement) its freeness depends solely on its combinatorics, but this has only been proved for sets of up to 12 lines. In looking for a counter-example to Terao's conjecture, the nearly free curves introduced by Dimca and Sticlaru arise naturally. We prove here that the logarithmic bundle associated to a nearly free curve possesses a minimal non-zero section that vanishes on one single point, P say, called the jumping point, and that this characterises the bundle. We then give a precise description of the behaviour of P. Based on detailed examples we then show that the position of P relative to its corresponding nearly free arrangement of lines may or may not be a combinatorial invariant, depending on the chosen combinatorics. (AU)

• Processo FAPESP: 18/08524-2 - Arrangements e fibrados vetoriais
• Beneficiário: Simone Marchesi
• Modalidade de apoio: Auxílio à Pesquisa - Pesquisador Visitante - Internacional