Computational Algebraic Geometry [Lecture notes] by Thomas Markwig Keilen

By Thomas Markwig Keilen

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A morphism is an isomorphism if it is bijective and its inverse in a morphism as well. ) If f1 , . . , fm ∈ K[X] then we get a morphism X −→ Am K : p → f1 (p), . . , fm (p) by just taking the fi as component functions. This works since composing a rational function with a polynomial gives a rational function. The following theorem states that actually this is the only way to get a morphism. ) If X ⊆ AnK and Y ⊆ Am K are affine algebraic varieties and K = K, then the pull-back Mor(X, Y) −→ HomK−alg (K[Y], K[X]) : ϕ → ϕ∗ is a bijection.

P) · xn x1 xn = Ker ∂f ∂f (p), . . , (p) x1 xn the tangent space of X. This leads us to the following generalisation of the notion of tangent space. 13 (The tangent space of X at p) Let X ⊆ AnK be an affine algebraic variety with I(X) = f1 , . . , fk and let p ∈ X. Then ∼ Ker Df(p) mp /m2 = p as K-vector spaces, where   Df(p) =  ∂f1 (p) ∂x1 ... . ∂fk (p) . . ∂x1 is the Jacobian matrix of f = (f1 , . . , fk ) at p. In particular, the vector space ∂f1 (p) ∂xn   ..  . ∂fk (p) ∂xn Tp (X) = Ker Df(p) is independent of the chosen generators of I(X).

Fk generate I(X) and f = (f1 , . . , fk ), then p is regular if and only if dim(X, p) = dimK Tp (X) = n − rank Df(p) . This can be generalised even if we do not know that the fi generate the vanishing ideal of X. 18 (Jacobian Criterion) Let X = V(f1 , . . , fk ) ⊆ AnK with K = K and f = (f1 , . . , fk ). Then p ∈ X is regular if and only if dim(X, p) ≥ n − rank Df(p) . Idea of the proof: Choose polynomials g1 , . . , gl such that the vanishing ideal is I(X) = f1 , . . , fk , g1 , . . , gl .

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