Retraction of the complement of smooth projective curves to a $2$-dimensional $\Delta$-complex
Algebraic Geometry
2026-05-27 v1 Algebraic Topology
Combinatorics
Abstract
Due to a result by Andreotti and Frankel \cite{andreotti1959}, it can be seen that the complement of a complex projective curve has the homotopy type of a -dimensional CW complex. However, no general method has been given to compute explicitly this complex. Here we give a explicit construction of a - dimensional -complex that is a strong deformation retract of the complement of a Fermat curve of degree in the complex projective space. The retraction is performed in several steps, using the branched cover structure of the Fermat curves over the degree case.
Cite
@article{arxiv.2605.26768,
title = {Retraction of the complement of smooth projective curves to a $2$-dimensional $\Delta$-complex},
author = {E. Artal and A. Larraya Sancho and M. A. Marco Buzunariz},
journal= {arXiv preprint arXiv:2605.26768},
year = {2026}
}