Non-planar degenerations and related fundamental groups

In this paper we establish a preliminary study of non-planar degenerations. We study two primary cases, the tetrahedron and the double tetrahedron. We calculate the fundamental groups and signatures of the Galois covers of surfaces that degenerate to these two objects. We find that the fundamental group of the Galois cover related to the tetrahedron is trivial, while the one related to the double tetrahedron is $\mathbb{Z}_2^4$.Comment: 19 pages, 8 figure

Fundamental group of Galois covers of degree 66 surfaces

In this paper we consider the Galois covers of algebraic surfaces of degree 6, with all associated planar degenerations. We compute the fundamental groups of those Galois covers, using their degeneration. We show that for 8 types of degenerations the fundamental group of the Galois cover is non-trivial and for 20 types it is trivial. Moreover, we compute the Chern numbers of all the surfaces with this type of degeneration and prove that the signatures of all their Galois covers are negative. We formulate a conjecture regarding the structure of the fundamental groups of the Galois covers based on our findings. With an appendix by the authors listing the detailed computations and an appendix by Guo Zhiming classifying degree 6 planar degenerations.Comment: Main paper is 21 pages and contains 7 figures. The appendix is 65 pages and contains 42 figures. arXiv admin note: substantial text overlap with arXiv:1610.0961