Retraction of the complement of smooth projective curves to a $2$-dimensional $Δ$-complex
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866916049223417856 |
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| author | Artal, E. Sancho, A. Larraya Buzunariz, M. A. Marco |
| author_facet | Artal, E. Sancho, A. Larraya Buzunariz, M. A. Marco |
| contents | 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 $2$-dimensional CW complex. However, no general method has been given to compute explicitly this complex. Here we give a explicit construction of a $2$- dimensional $Δ$-complex that is a strong deformation retract of the complement of a Fermat curve of degree $d$ in the complex projective space.
The retraction is performed in several steps, using the branched cover structure of the Fermat curves over the degree $1$ case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_26768 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Retraction of the complement of smooth projective curves to a $2$-dimensional $Δ$-complex Artal, E. Sancho, A. Larraya Buzunariz, M. A. Marco Algebraic Geometry Algebraic Topology Combinatorics 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 $2$-dimensional CW complex. However, no general method has been given to compute explicitly this complex. Here we give a explicit construction of a $2$- dimensional $Δ$-complex that is a strong deformation retract of the complement of a Fermat curve of degree $d$ in the complex projective space. The retraction is performed in several steps, using the branched cover structure of the Fermat curves over the degree $1$ case. |
| title | Retraction of the complement of smooth projective curves to a $2$-dimensional $Δ$-complex |
| topic | Algebraic Geometry Algebraic Topology Combinatorics |
| url | https://arxiv.org/abs/2605.26768 |