Abstract
In his 1933 paper [65], Du Val said that an isolated singular point O of an algebraic surface S in ℙ3 does not affect the conditions of adjunction if it does not impose conditions on the surfaces adjoint to S. That is, if those surfaces are not obliged to pass through the point O.
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Notes
- 1.
Note that such singular points do not exist on plane algebraic curves, as shown by Noether’s Theorem 22.2.
- 2.
Compare this with the explanations given for curves in Chap. 22
- 3.
They are alternatively called Kleinian singularities or rational double points. The first name refers to the fact that they are also the singularities of the quotients of \(\mathbb{C}^{2}\) by finite subgroups of \(SL(2, \mathbb{C})\), which were studied by Klein in [117]. The second one refers to Artin’s characterization of Du Val singularities mentioned above.
References
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Popescu-Pampu, P. (2016). Singularities Which Do Not Affect the Genus. In: What is the Genus?. Lecture Notes in Mathematics(), vol 2162. Springer, Cham. https://doi.org/10.1007/978-3-319-42312-8_33
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DOI: https://doi.org/10.1007/978-3-319-42312-8_33
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