Download MATH 176: ALGEBRAIC GEOMETRY HW 3 (1) (Reid 3.5) Let J = (xy

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.
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