Given a projective surface and a generic projection to the plane,the braid monodromy factorization（and thus,the braid monodromy type）of the complement of its branch curve is one of the most important topological inv...Given a projective surface and a generic projection to the plane,the braid monodromy factorization（and thus,the braid monodromy type）of the complement of its branch curve is one of the most important topological invariants,stable on deformations.From this factorization,one can compute the fundamental group of the complement of the branch curve,either in C<sup>2</sup> or in CP<sup>2</sup>.In this article,we show that these groups,for the Hirzebruch surface F<sub>1</sub>,（a,b）,are almost-solvable.That is, they are an extension of a solvable group,which strengthen the conjecture on degeneratable surfaces.展开更多
Denoting by T the complex projective torus, we can embed the surface CP^1 × T in CP^5. In this paper we compute the fundamental group of the complement of the branch curve of this surface. Since the embedding is ...Denoting by T the complex projective torus, we can embed the surface CP^1 × T in CP^5. In this paper we compute the fundamental group of the complement of the branch curve of this surface. Since the embedding is not ＂ample enough＂, the embedded surface does not belong to the classes of surfaces where the fundamental group is virtually solvable： a property which holds for these groups for ＂ample enough＂ embeddings. On the other hand, as it is the first example of this computation for non simply-connected surfaces, the structure of this group （as shown in this paper） give rise to the extension of the conjecture regarding the structure of those fundamental groups of any surface.展开更多
基金This work was supported by the Emmy Noether Institute Fellowship(by the Minerva Foundation of Germany)Israel Science Foundation(Grant No.8008/02-3)
文摘Given a projective surface and a generic projection to the plane,the braid monodromy factorization（and thus,the braid monodromy type）of the complement of its branch curve is one of the most important topological invariants,stable on deformations.From this factorization,one can compute the fundamental group of the complement of the branch curve,either in C<sup>2</sup> or in CP<sup>2</sup>.In this article,we show that these groups,for the Hirzebruch surface F<sub>1</sub>,（a,b）,are almost-solvable.That is, they are an extension of a solvable group,which strengthen the conjecture on degeneratable surfaces.
基金supported by DAADEU-network HPRN-CT-2009-00099(EAGER)+2 种基金The Emmy Noether Research Institute for Mathematicsthe Minerva Foundation of GermanyThe Israel Science Foun dation grant #8008/02-3 (Excellency Center "Group Theoretic Methods in the Study of Algebraic Varieties")
文摘Denoting by T the complex projective torus, we can embed the surface CP^1 × T in CP^5. In this paper we compute the fundamental group of the complement of the branch curve of this surface. Since the embedding is not ＂ample enough＂, the embedded surface does not belong to the classes of surfaces where the fundamental group is virtually solvable： a property which holds for these groups for ＂ample enough＂ embeddings. On the other hand, as it is the first example of this computation for non simply-connected surfaces, the structure of this group （as shown in this paper） give rise to the extension of the conjecture regarding the structure of those fundamental groups of any surface.
