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MATH 176: ALGEBRAIC GEOMETRY HW 3 DAGAN KARP (1) (Reid 3.5) Let J = (xy, xz, yz) ⊆ k[x, y, z]. Find Z(J) ⊆ A3k . Is it irreducible? Is it true that J = I(Z(J))? Prove that J cannot be generated by two elements. Now let J 0 = (xy, (x − y)z). √ Find Z(J 0 ) and calculate J 0 . (2) (Reid 3.6) Let J = (x2 + y2 − 1, y − 1). Find f ∈ I(Z(J)) \ J. (3) Let (X, TX ) and (Y, TY ) be topological spaces. The product topology on X × Y is defined by the basis B = {U × V ⊆ X × Y : U ∈ TX , V ∈ TY }. (a) Prove that the product topology is indeed a topology on X × Y. (b) Is the Zariski topology on A2 the same as the product topology on A1 × A1 , where each copy of A1 is under the Zariski topology? (4) Portfolio Problem. (a) Let Y ⊆ A3k be the set Y = {(t, t2 , t3 ) : t ∈ k}. (i) Is Y is an algebraic set? (ii) Is Y irreducible? (iii) Find the generators of I(Y). (iv) Find the quotient ring k[x, y, z]/I(Y). (b) Prove or disprove: a polynomial f ∈ R[x, y] is irreducible if and only if its zero set Z(f) ⊂ A2R is irreducible. Date: Due Wed. Oct 5, 2016. Following Hartshorne, Section 1, Chapter 1. 1