Computational Algebraic Geometry [Lecture notes] by Thomas Markwig Keilen

By Thomas Markwig Keilen

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We then say that ϕ parametrises its image. g. 25 If f ∈ K[x] we call Xf = X \ V(f) a basic open subset of X. Show that every open subset of X is a union of finitely many basic open subsets. One says that the basic open subsets form a basis of the Zariski topology. 26 Let X be an irreducible affine algebraic variety and U ⊆ X a non-empty open subset of X. e. its topological closure is all of X. 27 Show that any two non-empty open subsets of AnK have a non-empty intersection. 28 If X = V(x1 x2 − x3 x4 ) ⊂ A4C and U = X \ V(x1 , x3 ) then the function f : U −→ C : (x1 , x2 , x3 , x4 ) → x2 , x3 x4 , x1 if x3 = 0, if x1 = 0 is well-defined and regular.

This all amounts to the fact that the collection of K-algebras OX (U) and restriction maps resU,V , where U and V run over all open subsets of X such that V ⊆ U, forms a sheaf of K-algebras. We call it the structure sheaf of X and denote it by OX . In modern algebraic geometry the language of sheaves is vital. In this minicourse, however, we will avoid it since it is rather technical and not that important for computational questions. 14 (Morphisms) Let X ⊆ AnK and Y ⊆ Am K be two affine algebraic varieties.

G Figure 15. ) n If ϕ : X −→ Am K with X ⊆ AK is any morphism, then its graph is the set Graph(ϕ) = p, ϕ(p) ∈ AKn+m p ∈ X . It actually is an affine algebraic variety, namely the one defined by the ideal f1 , . . , fk , y1 − g1 , . . , ym − gm ✂ K[x1 , . . , xn , y1 , . . , ym ] if X = V(f1 , . . , fk ) and ϕ = (g1 , . . , gm ). 32 Moreover, it is clear that ϕ(X) = πn,m Graph(ϕ) , and we can thus compute the closure of the image of ϕ as the projection of the graph of ϕ. g. consider the morphism and its graph Graph(ϕ) = ϕ : A1R −→ A2R : t → t, ϕ(t) t∈R = 1 − t2 2t , 1 + t2 1 + t2 t, 1 − t2 2t , 1 + t 2 1 + t2 t∈R .

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