Posted: February 12th, 2015

Algebraic Geometry

Paper, Order, or Assignment Requirements

The book “an invitation to algebraic geometry” by smith.
Complete proof is required.

There’s a typo on question 3(b) it should be x^2 – y^2 – z^2 + 1 =0 instead of x^2 + y^2 + z^2 + 1 = 0 on the second equation.

Problem 1 (a) Show that zero set of a non-zero polynomial in C[x, y] does not have interior points (with respect to the usual Euclidean topology on C 2 . (The same statement holds with similar proof in all dimensions.) (b) Show that the dimension of an affine algebraic variety is finite (c) Show that a radical ideal I in the ring k[x1, . . . , xn] is the intersection of all the maximal ideals. Problem 2 (a) Show that the Zariski topology on A 2 is not the product topology on A 1 × A 1 . (Hint: Consider the diagonal) Also show that every non-empty Zariski open set is dense in A n (in Zariski topology). (b) Find the Zariski closure of the graph of the function y = e x in the affine space C 2 . Problem 3 (a) Consider the twisted cubic curve C = {t 3 , t4 , t5}|t ∈ k. Prove that C is an irreducible algebraic set in A 3 of dimension 1. (b) Let k be a field of characteristic 6= 2. Decompose the algebraic set X ⊂ A 3 defined by the equations x 2 + y 2 + z 2 = 0 and x 2 + y 2 + z 2 + 1 = 0, into irreducible components.