Abstract
As we explained before, Riemann did not study algebraic curves embedded in the projective plane by themselves, but only up to birational transforms. Nevertheless, his techniques, passing through the associated Riemann surface or through the abelian integrals of the first kind attached to them, allow one to prove properties of such curves. For instance, one has
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Notes
- 1.
I am grateful to Walter Neumann for the English translation of the fragments quoted in this chapter.
- 2.
They are called the cyclic points , with homogeneous coordinates [1: ±i: 0]. All the circles of the euclidean plane with coordinates (x, y) pass through them, a fact which motivates their name.
References
A. Clebsch, Ueber diejenigen ebenen Curven, deren Coordinaten rationale Functionen eines Parameters sind. J. Reine Angew. Math. 64, 43–65 (1865)
A. Clebsch, Sur les surfaces algébriques. C.R. Acad. Sci. Paris 67, 1238–1239 (1868)
M. Noether, Rationale Ausführungen der Operationen in der Theorie der algebraischen Funktionen. Math. Ann. 23, 311–358 (1883)
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Popescu-Pampu, P. (2016). Clebsch and the Choice of the Term “Genus”. In: What is the Genus?. Lecture Notes in Mathematics(), vol 2162. Springer, Cham. https://doi.org/10.1007/978-3-319-42312-8_20
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DOI: https://doi.org/10.1007/978-3-319-42312-8_20
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