By Andreas Gathmann

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The set Z(I) := {(a0 : · · · : an ) ∈ Pn ; f (a0 , . . 6. Subsets of Pn that are of the form Z(I) are called algebraic sets. If X ⊂ Pn is any subset, we call I(X) :=the ideal generated by { f ∈ k[x0 , . . , xn ] homogeneous ; f (a0 , . . , an ) = 0 for all (a0 : · · · : an ) ∈ X} ⊂ k[x0 , . . , xn ] the ideal of X; by definition this is a homogeneous ideal. If we want to distinguish between the affine zero locus Z(I) ⊂ An+1 and the projective zero locus Z(I) ⊂ Pn of the same (homogeneous) ideal, we denote the former by Za (I) and the latter by Z p (I).

E. f = ∑ f (d) with f (d) homogeneous of degree d for all d. 8. Let I ⊂ k[x0 , . . , xn ] be an ideal. The following are equivalent: (i) I can be generated by homogeneous polynomials. (ii) For every f ∈ I we have f (d) ∈ I for all d. An ideal that satisfies these conditions is called homogeneous. Proof. (i) ⇒ (ii): Let I = ( f1 , . . , fm ) with all fi homogeneous. Then every f ∈ I can be written as f = ∑i ai fi for some ai ∈ k[x0 , . . , xn ] (which need not be homogeneous). Restricting this equation to the degree-d part, we get f (d) = ∑i (ai )(d−deg fi ) fi ∈ I.

7. 11. ) Let fi (x0 , . . , xn ), 0 ≤ i ≤ N = n+d n − 1 be the set of all monomials in k[x0 , . . e. of the monomials of the form x00 · · · xnin with i0 + · · · + in = d. Consider the map F : Pn → PN , (x0 : · · · : xn ) → ( f0 : · · · : fN ). 9 this is a morphism (note that the monomials x0d , . . , xnd , which cannot be simultaneously zero, are among the fi ). 7 the image X = F(Pn ) is a projective variety. We claim that F : X → F(X) is an isomorphism. All we have to do to prove this is to find an inverse morphism.