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By Ulrich Gortz, Torsten Wedhorn

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R. 7. Let f ∈ k[T1 ] be a non-constant polynomial. Show that X1 := V (T2 − f ) ⊂ A2 (k) is isomorphic to A1 (k) and show that X2 := V (1 − f T2 ) ⊂ A2 (k) is isomorphic to A1 (k) \ {x1 , . . , xn } for some n ≥ 1. Show that X1 and X2 are not isomorphic (look at the invertible elements of their coordinate rings). 8. Show that the affine algebraic set V (Y 2 − X 3 − X) ⊂ A2 (k) is irreducible and in particular connected. Sketch the set { (x, y) ∈ R2 ; y 2 = x3 + x } and show that it is connected.

Xn ] = R[X0 , . . , Xn ]d . 58. Let i ∈ {0, . . , n} and d ≥ 0. There is a bijective R-linear map (d) ∼ Φi = Φi : R[X0 , . . , Xn ]d → { g ∈ R[T0 , . . , Ti , . . , Tn ] ; deg(g) ≤ d }, f → f (T0 , . . , 1, . . , Tn ). ) Proof. We construct an inverse map. Let g be a polynomial in the right hand side set d and let g = j=0 gj be its decomposition into homogeneous parts (with respect to T for = 0, . . , n, = i). Define 27 d Xid−j gj (X0 , . . , Xi , . . , Xn ). Ψi (g) = j=0 It is easy to see that Φi and Ψi are inverse to each other (as both maps are R-linear, it suffices to check this on monomials).

We define the cone X, p of X over p by X, p = qp. q∈X 32 1 Prevarieties This is a closed subvariety of Pn (k): Indeed, after a change of coordinates we may assume H = V+ (Xn ) and p = (0 : . . : 0 : 1). Then we have X = V+ (f1 , . . , fm ) ⊆ Pn−1 (k) = H for fi ∈ k[X0 , . . , Xn−1 ]. Let f˜i be the polynomial fi considered as an element of k[X0 , . . , Xn ]. Then we obtain X, p = V+ (f˜1 , . . , f˜m ). , we have Λ ∩ Ψ = ∅ and the smallest linear subspace of Pn (k) that contains Λ and Ψ is Pn (k) itself).

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