Denoting by T the complex projective torus, we can embed the surface ℂℙ1× T in ℂℙ5. 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.
Bibliographical noteFunding Information:
Received September 13, 2006, Accepted May 21, 2008 The third author is partially supported by DAAD and EU-network HPRN-CT-2009-00099(EAGER); The Emmy Noether Research Institute for Mathematics and the Minerva Foundation of Germany; The Israel Science Foundation grant # 8008/02-3 (Excellency Center “Group Theoretic Methods in the Study of Algebraic Varieties”).
- Branch curve
- Curves and singularities
- Fundamental group
- Generic projection