Download e-book for iPad: Computational Algebraic Geometry [Lecture notes] by Thomas Markwig Keilen

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Additional info for Computational Algebraic Geometry [Lecture notes]

Example text

Xk and let p ∈ X. We define the dimension of X locally at p as dim(X, p) = max{dim(Xi ) | p ∈ Xi } the maximal dimension of an irreducible component of X containing p. 6 (The dimension locally at a point) If X is an affine algebraic variety over K = K and p ∈ X, then dim(X, p) = dim OX,p = dim K[X]mp , the dimension of X locally at p is the Krull dimension of the local ring of X at p. Idea of the proof: When localising at p all components which do not pass through p are lost. 7 Consider X = V(xz, yz) = V(z) ∪ V(x, y) and p = (0, 0, 1) ∈ V(x, y).

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 . Setting F = (f1 , . . , gl ) we have by definition dimK Tp (X) = n − rank DF(p) ≤ n − rank Df(p) . 19 (Jacobian Criterion for hypersurfaces) If X = V(f) ⊆ AnK with K = K is a hypersurface and f ∈ K[x] is squarefree, then p is a singular point of X if and only if f(p) = ∂f ∂f (p) = . . = (p) = 0.

T. ∀ q ∈ V : f(q) = g(q) . h(q) By OX (U) we denote all regular functions on U, and we call the functions in OX (X) global regular functions. Note that with the usual operations OX (U) is a K-algebra. 8 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. However, it is impossible to write f as a quotient of two polynomials on the whole of U! 9 (Elements in the coordinate ring as regular functions) Every polynomial f ∈ K[x] defines a regular function f : X → K : p → f(p) on an affine algebraic variety X, and two polynomials f and g define the same function on X if their difference is in I(X).

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Computational Algebraic Geometry [Lecture notes] by Thomas Markwig Keilen


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