Degenerations and fundamental groups related to some special toric varieties. (English) Zbl 1148.14302
Consider a complex projective surface \(X\) in \({\mathbb P}^N\) and a generic projection to \({\mathbb P}^2\), with branch curve \(B\subset{\mathbb P}^2\). Moishezon and Teicher proposed the study of \(\pi_1({\mathbb P}^2\setminus B)\) as part of their programme to use braid monodromy as a tool for managing complex surfaces. But this implies obtaining a presentation of \(\pi_1({\mathbb P}^2\setminus B)\) rather than treating it as an abstract group. The generators are derived by degenerating \(X\) into a union of planes.
Teicher and her collaborators have developed various tools to control the combinatorics in this sort of situation. Amram and Ogata here take \(X\) to be one of four toric varieties and construct toric degenerations. This is explained early on in the present paper. The process itself is fairly well known to experts in toric geometry, but it is new to interpret Teicher’s diagrams in this context. It makes the whole procedure appear more methodical than in earlier cases, and then the braid group computation is carried out by using the toric degeneration. In the last case of the four, moreover, \(X\) is singular but the procedure still works.
Teicher and her collaborators have developed various tools to control the combinatorics in this sort of situation. Amram and Ogata here take \(X\) to be one of four toric varieties and construct toric degenerations. This is explained early on in the present paper. The process itself is fairly well known to experts in toric geometry, but it is new to interpret Teicher’s diagrams in this context. It makes the whole procedure appear more methodical than in earlier cases, and then the braid group computation is carried out by using the toric degeneration. In the last case of the four, moreover, \(X\) is singular but the procedure still works.
Reviewer: G. K. Sankaran (Bath) (MR2280496)
MSC:
| 14F35 | Homotopy theory and fundamental groups in algebraic geometry |
| 14M25 | Toric varieties, Newton polyhedra, Okounkov bodies |
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