MATH 765/865 (ALGEBRAIC GEOMETRY) – FALL 2019 – Take home midterm exam Solve all 6 problems. This is a strictly individual assignment, you should not discuss it with others. You can use the textbook and notes. Please present your arguments clearly, quoting theorems when you use them. Please write legibly and avoid excessive corrections. Return only the final versions of your solutions. The solutions are due in class on Wednesday, October 9. Important: Solutions of different problems should be on separate pages. Please use only one side of a page. Carefully staple the pages of your exam. Write down your name on the top of each page. Problem 1 (5 points) This problem is concerned with the ellipse x 2 a2 + y 2 b2 = 1 in R2 (a, b > 0). Let ℓt be a non-vertical line with a slope t passing through the point (−a, 0). Let (x(t), y(t)) be the point of intersection (other than (−a, 0)) of ℓt with the ellipse. Show that x(t) and y(t) are rational functions of t and derive a rational parametric representation of the ellipse. Problem 2 (5 points) (1) Find a system of polynomial equations in R[x1, x2, x3, x4, x5] describ- ing the affine algebraic variety in R5 parametrized by x1 = 2t− 5u, x2 = t+ 2u, x4 = −t+ u, x5 = t+ 3u. (2) Find a system of polynomial equations in R[x, y, z] describing the affine algebraic variety in R3 parametrized by x = t, y = t4, z = t3. Problem 3 (5 points) Let V = V(f1, . . . fs) and W = V(g1, . . . gt) be affine algebraic varieties in kn. (1) Prove that V ∪W and V ∩W are affine algebraic varieties. (2) Fix a point (a01, a 0 2, . . . , a 0 n) in the affine space k n. Let V 0 := {(a1 + a 0 1, a2 + a 0 2, . . . , an + a 0 n) | (a1, a2, . . . , an) ∈ V } be the subset of kn obtained by shifting all points of V by (a01, a 0 2, . . . , a 0 n). Prove that V 0 is an affine algebraic variety. (3) Let σ be a permutation of {1, 2 . . . , n}, i.e., an element of the symmet- ric group Sn. Let T : k n → kn be a map defined by T (a1, a2, . . . , an) = (aσ(1), aσ(2), . . . , aσ(n)). Prove that T (V ), the image of V under T , is an affine algebraic variety. In all parts, describe (in terms of fi, gk) the polynomials that determine these varieties. 1 2Problem 4 (5 points) (1) Prove that the ideal 〈x3 − 1, x2 + x〉 ⊂ R[x] is R[x] itself. (2) Find a polynomial in R[x, y] that does not belong to the ideal 〈x, y〉. (3) Prove the equality of ideals 〈x+xy, y+xy, x2, y2〉 = 〈x, y〉 in R[x, y]. Problem 5 (5 points) Consider the monomial ideal I = 〈x3, x2y, xy2, y3〉 ⊂ R[x, y]. Determine if the following polynomials belong to I: (1) x4 + x2y2 + y4, (2) x2 + xy + y2, (3) a polynomial f ∈ R[x, y] such that f(0, 0) = 1. Problem 6 (5 points) (1) Give an example of monomials f, g, h ∈ R[x, y] such that f >lex g >lex h but h >grlex g >grlex f, where >lex is the Lex order and >grlex is the Graded Lex order (with x >lex y). (2) Prove that for monomials f and g in R[x, y] we have f >grlex g if and only if g >grevlex f, where >grlex is the Graded Reverse Lex order. Is this conclusion true in R[x, y, z]? 1 2 3 4 5 6 TOTAL
