Abstract
We saw in Chap. 28 that, in order to define a birationally invariant notion of genus for algebraic surfaces, Clebsch , Cayley and Noether considered certain surfaces passing through their singular locus. Those surfaces are analogs of the adjoint curves of a plane algebraic curve, passing in a controlled way through their singular points (see Chap. 22).
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Notes
- 1.
An ideal of a commutative ring is a subset which is closed under internal addition and exterior multiplication by elements of the ambient ring. An ideal of a polynomial ring is called homogeneous if it may be generated by homogeneous polynomials.
- 2.
She was Max Noether’s daughter.
- 3.
This ring is called the local ring of the algebraic surface S at the point O.
- 4.
Recall that modules are to rings what vector spaces are to fields. For instance, the ideals of a ring A are precisely its subsets which are A-modules.
- 5.
The difference of 1 comes from the facts that proportional polynomials define the same member of the linear system and that the dimension of a vector space exceeds by 1 the dimension of the associated projective space.
- 6.
I am grateful to Walter Neumann for having translated the following quotation into English.
References
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D. Hilbert, Ueber die Theorie der algebraischen Formen. Math. Ann. 36, 473–534 (1890)
E. Noether, Idealtheorie in Ringbereichen. Math. Ann. 83, 24–66 (1921)
P. Popescu-Pampu, Idéalisme radical. Images des Mathématiques (CNRS, 2011). http://images.math.cnrs.fr/Idealisme-radical.html
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O. Zariski, Polynomial ideals defined by infinitely near base points. Am. J. Math. 60 (1), 151–204 (1938)
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Popescu-Pampu, P. (2016). Hilbert’s Characteristic Function of a Module. In: What is the Genus?. Lecture Notes in Mathematics(), vol 2162. Springer, Cham. https://doi.org/10.1007/978-3-319-42312-8_36
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