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**Example text**

30. The quadric of rank 2 is the union of two diﬀerent lines, and the quadric of rank 1 is a line. 3. , if the rank of a matrix of q is maximal. 8) we will deﬁne in general when a prevariety is smooth and see that for quadrics the general deﬁnition coincides with the deﬁnition given here. We see that if Q is a quadric of rank r > 2 and dimension d, then Q is a cone ˜ Λ of a smooth quadric Q ˜ of dimension r − 2 with respect to a d − r + 1-dimensional Q, subspace Λ. The cases r = 1 and r = 2 are “degenerate”: The quadric Q ∼ = V+ (X02 ) = V+ (X0 ) is n a hyperplane in P (k).

Show that V consists of three irreducible components and determine the corresponding prime ideals. 6. Let f ∈ k[X1 , . . , Xn ] be a non-constant polynomial. Write f = i=1 fini with irreducible polynomials fi such that (fi ) = (fj ) for all i = j and integers ni ≥ 1. Show that rad(f ) = (f1 · · · fr ) and that the irreducible components of V (f ) ⊆ An (k) are the closed subsets V (fi ), i = 1, . . , 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 , .

In particular, (D(f ), OX |D(f ) ) is an aﬃne variety. Proof. 40 we have OX (D(f )) = Γ(X)f . As two aﬃne varieties are isomorphic if and only if their coordinate rings are isomorphic, it suﬃces to show that (D(f ), OX |D(f ) ) is an aﬃne variety. Let X ⊆ An (k) and a = I(X) ⊆ k[T1 , . . , Tn ] be the corresponding radical ideal. We consider k[T1 , . . , Tn ] as a subring of k[T1 , . . , Tn+1 ] and denote by a ⊆ k[T1 , . . , Tn+1 ] the ideal generated by a and the polynomial f Tn+1 −1. Then the aﬃne coordinate ring of Y is Γ(Y ) = Γ(X)f ∼ = k[T1 , .