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By Karl-Heinz Fieseler and Ludger Kaup

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E. , gr ∈ O(X) with ri=1 gi fi = 1. In particular regular functions without zeros are invertible. For convenience of notation we shall from now on denote the objects in T A simply by capital letters X, Y, ... and, in analogy to algebraic sets, O(X), O(Y ), ... their associated algebras of regular functions. 8. An object Y ∈ T A is called an affine variety (over the field k) if Y ∼ = X → k n with some Zariski closed set X in some affine n-space k n . We denote AV ⊂ T A the full subcategory with the affine varieties as objects.

Now endow X × Y with the π-quotient structure with respect to the natural map π : Z −→ X × Y , which on Ziµ is just the inclusion into X × Y . If we can show that π(Ziµ ) ⊂ X × Y is open and π|Ziµ : Ziµ −→ π(Ziµ ) an isomorphism, we are done. To that end let (Ziµ )jν := Uij × Vµν ⊂ Ziµ , considered as open (ringed) subspace of Ziµ , where Uij := Ui ∩ Uj and Vµν := Vµ ∩ Vν . We have to check that the identity map id (Ziµ )jν −→ (Zjν )iµ is an isomorphism. That can be done locally: Take a point (x, y) ∈ Uij × Vµν .

A complex analytic space without singular points is called nonsingular or smooth; its connected components form complex manifolds. 11. The singular locus S(X) of a complex analytic space X is a nowhere dense closed subset of X, locally given as the zero locus (:= set of zeros) of finitely many holomorphic functions. Its complement X \ S(X) is called the regular locus of X. 12. For the Neil parabola X := N (C2 ; T22 − T13 ) and the noose Y := N (C2 ; T22 − T12 (T1 + 1)) the only singular point of Xh resp.

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Algebraic Geometry [Lecture notes] by Karl-Heinz Fieseler and Ludger Kaup

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