By Karl-Heinz Fieseler and Ludger Kaup
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Extra info for Algebraic Geometry [Lecture notes]
N ∈ O(X). 3. Let X ∈ T A and j : Y → k n be the inclusion of an algebraic set. Then a map ϕ : X −→ Y is a morphism iff j ◦ ϕ is. 4. A map ϕ : X −→ Y between embedded affine varieties X → k m and Y → k n is a morphism iff it is the restriction ϕ = ϕ|X of a polynomial map ϕ : k m −→ k n (then necessarily satisfying ϕ(X) ⊂ Y ). 5. Let X ∈ T A and Y → k n be an affine variety. Then the morphisms ϕ : X −→ Y correspond bijectively to the algebra homomorphisms σ : O(Y ) −→ O(X). , ϕn ) ∈ O(X)n (with ϕi = σ(Ti )).
For the third part take a point y ∈ Y . 14 and obtain a maximal ideal m → O(X) with m ∩ O(Y ) = my . But m = mx for some x ∈ X and thus y = ϕ(x). If A = ri=1 Bai , then A/my · A is generated by the residue classes ai + my · A over B/my ∼ = k, the same holds for O(ϕ−1 (y)) ∼ = A/I(ϕ−1 (y)), a factor ring of A/my · A. So O(ϕ−1 (y)) is a finite dimensional k-vector space of a dimension ≤ r, hence the affine variety ϕ−1 (y) is finite with at most r points. Finally let ϕ : X −→ Y be finite and Z → X be a closed subset.
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. Yh is the origin. The affine curves Z := N (C2 ; T22 − p(T1 )) induces a Riemann surface Zh . 13. 1. The singular locus S(Xh ) of a complex algebraic variety X is even a Zariski closed subset and can be defined without the passage to complex analytic spaces. Indeed, the definition works for any field k, but it is not literally the analogue of the definition for complex analytic spaces.