# Computational Algebraic Geometry [Lecture notes] by Thomas Markwig Keilen PDF By Thomas Markwig Keilen

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Example text

That is, we can compute the closure of the image of an affine algebraic variety under a projection by intersecting a defining ideal with a subalgebra. 21 using the concept of block orderings. In Singular one can instead use the built-in command eliminate which computes the intersection of an ideal with a subalgebra by eliminating the variables one wants to get rid of. > ring r=0,(x,y,z),dp; > ideal I=y2-xz,x2y-z2,x3-yz; > eliminate(I,x); _=y5-z4 The example shows that if we project the space curve V(I) to the yz-plane we get a curve with the equation y5 − z4 = 0 (see Figure 15).

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 (p) ∂xn Tp (X) = Ker Df(p) is independent of the chosen generators of I(X). We call it the tangent space of X at p. 14 Consider the affine algebraic variety X = V(x2 + y2 − z) and p = (1, 0, 1). lib"; > J=substitute(J,x,1,y,0,z,1); > print(J); 2,0,-1 > print(syz(J)); 0,1, 1,0, 0,2 We have thus computed the tangent space of X at p to be     0 1     = V(2x − z). lib to have the command substitute at hand which allows to substitute values for the variables x, y and z all at once. Moreover, we have used the command jacob in order to compute the Jacobian matrix of the generators of I and we have used the command syz in order to compute generators of the kernel of the Jacobian matrix.