Cyclic cover
In algebraic topology and algebraic geometry, a cyclic cover or cyclic covering is a covering space for which the set of covering transformations forms a cyclic group.[1][2] As with cyclic groups, there may be both finite and infinite cyclic covers.[3] Cyclic covers have proven useful in the descriptions of knot topology[1][3] and the algebraic geometry of Calabi–Yau manifolds.[2]
In classical algebraic geometry, cyclic covers are a tool used to create new objects from existing ones through, for example, a field extension by a root element.[4] The powers of the root element form a cyclic group and provide the basis for a cyclic cover. A line bundle over a complex projective variety with torsion index [math]\displaystyle{ r }[/math] may induce a cyclic Galois covering with cyclic group of order [math]\displaystyle{ r }[/math].
References
- ↑ 1.0 1.1 (in en) Seifert and Threlfall, A Textbook of Topology. Academic Press. 1980. p. 292. ISBN 9780080874050. https://archive.org/details/seifertthrelfall0000seif. Retrieved 25 August 2017. "cyclic covering."
- ↑ 2.0 2.1 Rohde, Jan Christian (2009). Cyclic coverings, Calabi-Yau manifolds and complex multiplication ([Online-Ausg.]. ed.). Berlin: Springer. pp. 59–62. ISBN 978-3-642-00639-5.
- ↑ 3.0 3.1 Milnor, John. "Infinite cyclic coverings". Conference on the Topology of Manifolds. Vol. 13. 1968.. http://www.maths.ed.ac.uk/~aar/papers/milncycl.pdf. Retrieved 25 August 2017.
- ↑ Ambro, Florin (2013). "Cyclic covers and toroidal embeddings". arXiv:1310.3951 [math.AG].
Further reading
- Fedorchuk, Maksym (2011-05-13). "Cyclic covering morphisms on M0,n". arXiv:1105.0655 [math.AG].
- Singh, Anurag K. (2002-08-28). "Cyclic covers of rings with rational singularities". arXiv:math/0208226.
- "what is the cyclic cover trick?". 19 June 2013. https://mathoverflow.net/q/134145.
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