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MAT 248A Homework 1
Please write the homework solutions in connected sentences and explain your work. Mark the answers to each question. Scan or take pictures of your homework and upload it to Gradescope before due time.
In all problems we work over an arbitrary field k unless stated otherwise.
1. Recall that the projective space P n is defined as the set of (n + 1)-tuples [x0 : . . . : xn] such that not all xi vanish, modulo equivalence relation [x0 : . . . : xn] ∼ [λx0 : . . . : λxn] for λ ̸= 0. a) Let Ui = {xi ̸= 0}. Prove that any point in Ui has a unique representative with xi = 1, and that Ui is isomorphic to the affine space A n with coordinates xj/xi for all j ̸= i. b) Prove that P n \ Ui is an algebraic set in P n isomorphic to P n−1 . c) Consider Z = {x 2 0 = x1x2} ⊂ P
2 . Describe the intersections of Z with the charts U0, U1, U2. 2. a) Let f(x1, . . . , xn) be a degree d polynomial. Prove that there is a unique homogeneous degree d polynomial F(x0, x1, . . . , xn) such that F(1, x1, . . . , xn) = f(x1, . . . , xn). Such F is usually called a homogenization of f. b) Let Z be an algebraic set in A n . Use part (a) to constuct an algebraic set Z in P n such that Z ∩ U0 = Z. Such Z is called the projective closure of Z and the difference Z \ Z is called the ”set of points of Z at infinity”. c) Find the projective closure and the set of points at infinity for the parabola Z1 = {y = x 2} and hyperbola Z2 = {xy = 1}. d) Assume k = C. Find the set of points at infinity for Z = {(x − a) 2 + (y − b) 2 = R2}.
3. a) Prove that there is a unique line through any two distinct points in A n . b) Prove that there is a unique line through any two distinct points in P n . c) Prove that a line in A n (resp. P n ) is an affine (resp. projective) algebraic set. d) Prove that any two distinct lines in P 2 intersect at exactly one point.
4. a) Let f(x1, . . . , xn) be a degree d polynomial, and Z = {f = 0} ⊂ A n . Prove that any line in A n is either completely contained in Z or interesects Z in at most d points. b) Let f(x0, x1, . . . , xn) be a degree d homogeneous polynomial, and Z = {f = 0} ⊂ P n . Prove that any line in P n is either completely contained in Z or interesects Z in at most d points.
5. a) Consider the circle Z = {x 2 + y 2 = 1} and the point N = (0, 1). Let ℓm be the line of slope m through N. Prove that ℓm intersects Z at another point P(m) and find the coordinates of P(m). b) Prove that m is rational if and only if P(m) has rational coordinates. c) Find all integer solutions of the equation a 2+b 2 = c 2 . Hint: rewrite it as a c 2 + b c 2 =