Computational Algebraic Geometry [Lecture notes] by Thomas Markwig Keilen

By Thomas Markwig Keilen

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9 (Local monomial orderings) a. Define the local lexicographical ordering >ls on Mon(x) by xα >ls xβ if ∃ i ∈ {1, . . , n} : α1 = β1 , . . , αi−1 = βi−1 , and αi < βi . b. Define the local degree reverse lexicographical ordering >ds on Mon(x) by xα >ds xβ if deg xα < deg xβ , or deg xα = deg xβ , but then ∃ i ∈ {1, . . , n} : αn = βn , . . , αi+1 = βi+1 , and αi < βi . c. , x21 >ds x1 x32 , but x1 x32 >ls x21 . ,xn . g. X = V(xz, yz) and p = (0, 0, 1) then the following Singular commands compute the dimension of X locally at p, where in the definition of the ring we use the local ordering ls.

Are all regular functions on U globally given by rational functions? If so, explain why, if not, give a counter example. t. the Zariski topology. 31 Let X ⊆ AnK be an affine algebraic variety with irreducible decomposition X = X1 ∪ . . ∪ Xk and let f ∈ K[x]. Show that the residue class of f is a zero-divisor in K[X] if and only if f vanishes identically on some Xi . 32 Let K be algebraically closed and U = A2K \ {0}. e. each regular function on U extends to a regular function on all of A2K . ) Show that the morphism ϕ : A1C −→ A2C : t → (t2 , t3 ) is bijective onto its image Y = V(x3 − y2 ), but it is not an isomorphism, since the pull back ϕ∗ : C[Y] = C[x, y]/ x3 − y2 −→ C[t] : x → t2 , y → t3 is not surjective — t is not in its image.

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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