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By Ulrich Gortz, Torsten Wedhorn

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Xn ] be homogeneous polynomials of the same degree such that for all y = (y0 : . . : yn ) ∈ Y there exists an index j such that fj (y) = 0. Then h : Y → Pm (k), y → (f0 (y) : . . : fm (y)) is a morphism of prevarieties. Another family g0 , . . , gm ∈ k[X0 , . . , Xn ] as above defines the same morphism h if and only if fi (y)gj (y) = fj (y)gi (y) for all y ∈ Y and all i, j ∈ {0, . . , m}. 33 (2) Conversely, let h : Y → Pm (k) be a morphism of prevarieties. Then there exists for every y ∈ Y an open neighborhood U of y in Y such that h|U is of the above form.

Xn ] is homogeneous of degree d if and only if f (λx0 , . . , λxn ) = λd f (x0 , . . , xn ) for all x0 , . . , xn ∈ k, 0 = λ ∈ K. (b) Let a ⊆ K[X0 , . . , Xn ] be an ideal. Show that the following assertions are equivalent. (i) The ideal a is generated by homogeneous elements. (ii) For every f ∈ a all its homogeneous components are again in a. (iii) We have a = d≥0 a ∩ K[X0 , . . , Xn ]d . An ideal satisfying these equivalent conditions is called homogeneous. (c) Show that intersections, sums, products, and radicals of homogeneous ideals are again homogeneous.

Xn )), and show that vd induces an isomorphism Pn (k) ∼ = V+ (a) of prevarieties. Is V+ (a) a linear subspace of PN (k)? (c) Let f ∈ k[X0 , . . , Xn ] be homogeneous of degree d. Show that vd (V+ (f )) is the intersection of V+ (a) and a linear subspace of PN (k). The morphism vd is called the d-Uple embedding or d-fold Veronese embedding. 40 2 Spectrum of a Ring Contents – Spectrum of a ring as topological space – Excursion: Sheaves – Spectrum of a ring as a locally ringed space In the first chapter we attached to a system of polynomials f1 , .

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