Exercises to Algebraic Geometry

Georg Hein, Angela Ortega

Wintersemester 2008/09

Exercises to Algebraic Geometry Exercise 14.1. Let K/Q be a number field, and OK be the integral closure of Z in K. Show that ΩOK /Z is finite. Exercise 14.2. Compute the Z[i]-module ΩZ[i]/Z . Exercise 14.3. Compute the ΩP2 /k (P2 ). Hint: As in the lectures use the standard covering and look for the glueing condtions. Exercise 14.4. We consider the curve C = V (Y 2 Z − X 3 − XZ 2 ) ⊂ P2k . Compute ΩC/k (C). Hint: First show that the two standard charts U2 = (Z 6= 0), and U1 = (Y 6= 0) give an affine cover of C, and proceed like in the lecture.

These exercises are discussed on Wednesday February 4, 2009



Exercises to Algebraic Geometry Exercise 13.1. Let B be a ring and M a B-module. Set A = B ⊕ M and define a B-algebra structure on A by (b, m) · (b′ , m′ ) = (b · b′ , bm′ + b′ m). Compute ΩA/B . Exercise 13.2. Let k be a field of characteristic zero, f (x) ∈ k[x] an irreducible polynomial and consider the ring A = k[x, y]/(y 2 − f (x)). Show that the A-module ΩA/k is freely generated by one element. Exercise 13.3. Let k be a field and consider the ring A = k[x, y]/(y 2 − x3 ). Show that the A-module ΩA/k is not freely generated by one element. Exercise 13.4. Three lines meeting in one point We consider the k-algebras A1 = k[x, y]/(x2 y + xy 2 ) and A2 = k[x, y, z]/(xy, yz, zx). Show that the spectra Spec(Ai ) have both three irreducible components which are all isomorphic to A1k which intersect in one point. Compute the modules ΩAi /k , and the minimal numbers of generators.

These exercises are discussed on Friday January 30, 2009



Exercises to Algebraic Geometry Exercise 12.1. the quadratic field √ Let d ∈ Z \ {0, 1} a square free integer, and consider √ extension Q[ d]/Q. Compute the algebraic closure of Z in Q[ d]. Exercise 12.2. Normalization of irreducible curves We consider the two rings A1 = k[x, y]/(y 2 −x3 ) and A2 = k[x, y]/(y 2 −x3 −x2 ). Compute the algebraic closure of Ai in its fraction field and give the morphism Spec(A¯i ) → Spec(Ai ). Give the dimensions of Ai . Exercise 12.3. Let G be a finite subgroup of the automorphism group of the ring A. By AG we denote the subring AG = {a ∈ A | τ (a) = a for all τ ∈ G}. (i) Show that the AG ⊂ A is integral. (ii) Let p ∈ Spec(AG ), and let Mp = {p1 , p2 , . . . } be the set of all prime ideals in A with AG ∩ pi = p. Show that G acts transitivly on Mp. Exercise 12.4. A Noetherian domain with maximal ideals of different heights We consider the following Noetherian domain A = k[[x]][y]. Show that the two ideals m1 = (x, y) and m2 = (xy − 1) are maximal and compute their heights.

These exercises are discussed on Wednesday January 21, 2009



Exercises to Algebraic Geometry ¯ and f ∈ k[x, y] a prime element. We set K = Frac(k[x, y]/(f )), Exercise 11.1. Let k = k, and Xmax = {(a, b) ∈ k 2 | f (a, b) = 0}. We equip Xmax with the topology of the finite complement and define for all non empty open subsets U ⊂ Xmax the ring OXmax (U) = {g ∈ K | g is regular at all points P ∈ U}. (i) Show that (Xmax , OXmax ) is a ringed space. (ii) Compare it with Spec(k[x, y]/(f )). Show that there exists a local morphism (Xmax , OXmax ) → Spec(k[x, y]/(f )) which gives an bijection of open sets which is not invertible. (iii) Show furthermore, that (Xmax , OXmax ) is not a scheme. Exercise 11.2. Compute the Krull dimension of Rn with the standard topology.

Exercise 11.3. Compute the dimension of k[x, y]. Exercise 11.4. Let f, g ∈ k[x, y] be two prime elements. Compute the dimension of the rings Am,n = k[x, y]/(f m · g n ).




Exercises to Algebraic Geometry Exercise 10.1. The direct image sheaf Let f : X → Y be a continuous map between topological spaces and F be a sheaf on X. For any open subset U ⊂ Y we define f∗ F (U) := F (f −1 U). Show that f∗ F is a sheaf on Y . This sheaf is called the direct image sheaf of F under f . Exercise 10.2. “The other definition of a morphism of ringed spaces” Show that to give a morphism of ringed spaces (X, OX ) → (Y, OY ) is equivalent to give a continuous f : X → Y and a morphism of sheaf of rings ψ : OY → f∗ OX . Exercise 10.3. Compute the direct image sheaf f∗ OX in the following examples: f (i) X = Spec(k[x]) −→ Spec(k) given by k → k[x]. (ii)

f

X = Spec(k) −→ Spec(k[x]) given by k[x] → k with x 7→ 0 ∈ k. f

(iii) X = Spec(k[x, y]/(xy − 1)) −→ Spec(k[x]) given by k[x] → k[x, y]/(xy − 1) where x 7→ x. (iv) In examples (i)–(iii) compute the stalks of f∗ OX at all points the target. Exercise 10.4. The inverse image sheaf Let f : X → Y be a continuous map between topological spaces and F be a sheaf on Y . For any open subset U ⊂ X we define f˜−1 F (U) := limV ⊂f (U ) F (V ). Show that f˜−1 F is −→ −1 a presheaf on X. The sheaf f F associated to the presheaf f˜−1 F is called the inverse image sheaf of F under f . We consider the following ring morphisms B = k → A = k[x] ⊕ k[y] given by λ 7→ (λ, λ). For the associated morphism f : Spec(A) → Spec(B) show that f˜−1 OSpec(B) is not a sheaf and compute f −1 OSpec(B) (Spec(A)). Compute the stalk of f −1 OSpec(B) in some point x ∈ Spec(A).




Exercises to Algebraic Geometry Exercise 9.1. Which of the four presheaves of exercise 8.1 are sheaves? Exercise 9.2. Compute the sheafication of the three presheaves of 9.1 which are no sheaves. Exercise 9.3. We consider the local ring A = { ab ∈ Q | a ∈ Z , b ∈ Z \ 2Z} ⊂ Q. We set X = Spec(A). (i) Give all points of X. (ii) Give all specialization relations x1 x0 of points in X. (iii) Give all open subsets U ⊂ X. (iv) Give all rings OX (U) for U ⊂ X open. Exercise 9.4. Let A be a ring and X = Spec(A). Take U ⊂ X open and p ∈ U. We have a natural map OX (U) → OX,p = Ap. Show that these natural maps define a ring homomorphism Y ϕ : OX (U) → Ap . p∈U

(i) Show that ϕ is injective. (ii) Give an example for A and U ⊂ Spec(A) such that ϕ is not surjective. (iii) Show that the image of ϕ consists of all (fp)p∈U such that for each p ∈ U there exists a basic open subset D(g) ∋ p such that there exist a f ∈ Ag such that f 7→ fp for all p ∈ D(g). Q (iv) Use (i) and (iii) to give a new definition of OX (U) as a subring of p∈U Ap. Show that with this definition the sheaf condition holds obviously.

These exercises are discussed on Wednesday December 17, 2008



Exercises to Algebraic Geometry Exercise 8.1. Let X be a topological space. For open subsets V ⊂ U ⊂ X and a function f : U → R we denote by ρUV yhe restriction of f to V . Decide which of the following examples give a presheaf (with the above restriction) of sets on X. (1) F1 (U) = {f : U → R | f is continuous}. (2) F2 (U) = {f : U → R | f is not continuous}. (3) F3 (U) = {f : U → R | f is continuous in almost all points of U}. (4) F4 (U) = {f : U → R | f is constant}. (5) F5 (U) = {f : U → R | f has finite image}. (6) F6 (U) = {f : U → R | f has infinite image}. Exercise 8.2. Functions regular on all points but one, part I We consider the ring A = k[X, Y ]/(X 3 − X − Y 2 ). Let m be the maximal ideal generated by the images of X, and Y in A. We set X = Spec(A) and U = X \ {m}. (i) Find an element of OX (U) which is not in the image of OX (X) → OX (U). (ii) Is U a basic open subset D(f ) for some f ∈ A? Justify your answer! (iii) Write down the ring OX (U) explicitly. Exercise 8.3. Functions regular on all points but one, part II Let 0 6= f ∈ k[X, Y ] be a irreducible polynomial contained in the maximal ideal (X, Y ). We consider the ring A = k[X, Y ]/(f ) and the maximal ideal m generated by the image of (X, Y ) in A. We set X = Spec(A) and U = X \ {m}. Show that OX (X) $ OX (U). Exercise 8.4. Functions regular on all points but one, part III We consider the ring A = k[X, Y ], the maximal ideal m = (X, Y ), X = Spec(A), and U = X \ {m}. Is ρX U : OX (X) → OX (U) a bijection? Justify your answer!




Exercises to Algebraic Geometry This series of exercise is devoted to the proof of Hasse’s theorem for the elliptic curve C = V (Y 2 Z − X 3 − aXZ 2 ) over the field Fp where p ≡ 1 mod 4 is a prime number, and a ∈ F∗p . The case p ≡ 3 mod 4 works as in the previous exercise 6.3.

Exercise 7.1. Remember p ≡ 1 mod 4 is a prime and a ∈ F∗p . Show that: X xy X xy X xy X x(x + a) =0 = 2p−2 = 2−2p = −1 p p p p ∗ ∗ ∗ x,y∈F x∈F p

x, y ∈ Fp with x2 = y 2

x, y ∈ Fp with x2 6= y 2

(ii) (iii)

p−1 −1

x3 +ax . x∈Fp p Show that the curve C = V (Y 2 Z − X 3 − aXZ 2 ) ⊂ P2Fp has p + 1 + S(a) points. Show that for λ ∈ F∗p we have an equality S(λ2 a) = λp S(a). 2 2 Deduce that S 2 (a) takes only two values one Ssq for a square, and one Snsq for a

Exercise 7.2. We define the integer S(a) := (i)

p

P

not a square. Exercise 7.3. Show the following equations. (i) X x2 + a y 2 + a a∈Fp

p

p

(ii)

X

a∈Fp

a∈Fp

when x2 = y 2 when x2 = 6 y2

S 2 (a) = 2p(p − 1)

(iii) X

=

S 2 (a) =

p−1 2 2 (Ssq + Snsq ) 2

2 2 (iv) Ssq + Snsq = 4p. Hints: For (i) use the transformation a′ = a + y 2 and exercise 7.1. For (ii) use X xy x2 + a y 2 + a X xy X x2 + a y 2 + a X 2 S (a) = = p p p p a∈F p p a,x,y∈F x,y∈F a∈F p

p

p

p

and exercise 7.1 nad 7.3 (i). For part (iii) have a look at 7.2 (iii). Exercise 7.4. Let Np be number of Fp -rational points of the projective curve C = V (Y 2 Z − X 3 − aXZ 2 ) ⊂ P2Fp for p ≡ 1 mod 4 and a ∈ F∗p √ (i) Show the inequality |Np − 1 − p| < 2 p (this is a special case of Hasse’s theorem) (ii) Show that p is the sum of two squares.




Exercises to Algebraic Geometry Exercise 6.1. Points of order two on elliptic curves Show that on a elliptic curve Ca,b = V (y 2 − x3 − ax − b) over an algebraically closed field k there are exactly three points of order two. Give these points for the elliptic curve C−1,0 = V (y 2 − x3 + x). Exercise 6.2. The addition formula Give a formula for the sum of two points P1 = (x1 , y1 ) and P2 = (x2 , y2 ) on the elliptic curve C = V (y 2 − x3 + x − 1). Use your formula to compute the sum (0, 1) + (−1, −1).

Exercise 6.3. Here we consider the elliptic curve C = V (ZY 2 − X 3 + XZ 2 ) over the finite field k = Fp for a prime number p ≡ 3 mod 4. Prove that C has exactly p + 1 points in P2k with coordinates in k. Here are some hints: Show that C ′ = V (y 2 − x3 + x) ⊂ A2k has p points. Let n be the number of k-rational points of C ′ = V (y 2 − x3 + x) ⊂ A2k . First show that 1 ≤ n ≤ 2p − 1. Now show that n ≡ 0 mod p, by using the following trick: 3 For each x ∈ k the number of possible y is x p−x + 1, so we can compute modulo p 3p−3

n≡

X x3 − x

x∈Fp

p

≡

X

x∈Fp

(x3 − x)

p−1 2

≡

2 X X

x∈Fp k= p−1 2

ak xk ≡ −ap−1 = 0

Of course YOU have to explain why the above line make sense! Exercise 6.4. Determine the group structure of the group of F23 -rational points on the elliptic curve C = V (ZY 2 − X 3 + XZ 2 ) ⊂ P2F23 .

These exercises are discussed on Wednesday November 26, 2008



Exercises to Algebraic Geometry Exercise 5.1. We consider three lines l1 , l2 , l3 in P2 . Let the lines li be given by li = V (ai1 X + ai2 Y + ai3 Z). Find a number ρ depending on the aij with the property ρ = 0 ⇐⇒ (l1 ∩ l2 ∩ l3 ) 6= ∅. Exercise 5.2. The unification of ellipse, hyperbola, and parabola We consider the conic C = V (4X 2 − 5XY + 4XZ + Y 2 + Z 2 ) ⊂ P2R . Show that the intersection of C with the three standard charts U0 = (X 6= 0), U1 = (Y 6= 0), and U2 = (Z 6= 0) gives ellipse, hyperbola, and parabola. Give the right order! If we denote the lines li = P2 \ Ui , then investigate the number of real points of li ∩ C. Exercise 5.3. We assume that k is a algebraic closed field not of characteristic 2. We let f (x) ∈ k[X] be a monic polynomial. We consider the affine curve C = V (f (x) −y 2 ) ⊂ A2k . (i) Show that C is nonsingular ⇐⇒ the polynomial f has no multiple zeroes. (ii) Is the homogenization C˜ ⊂ P2k of C nonsingular? Hint: That may depend on the degree of f . Exercise 5.4. We consider a nonsingular curve C = V (Y 2 Z − X 3 + Z 3 ) over a algebraic closed field k not of characteristic 2 or three. Let ω a primitive third root of unity. (i) Show that the map P2 → P2 with (A : B : C) 7→ (ω · A : −B : C) defines a projective transformation which induces an isomorphism on C. Give the order of this transformation. (ii) What are the fixed points of this isomorphism? (iii) Compute the degree of the polar curve! (iv) Show that the map which assigns each point P ∈ C its tangent line is a bijection of C and its polar curve. (v) Is the polar curve nonsingular? Hint: Check it at the tangent line of (0 : 1 : 0).




Exercises to Algebraic Geometry Exercise 4.1. Suppose char(k) 6= 2. Show that the curve C = V (x3 − x − y 2) ⊂ A2k has no singular points. Give all points P ∈ C where the rational function X ∈ K(C) has Y value ∞. Exercise 4.2. Rational functions in singular points I Let C = V (X 3 + X 2 − Y 2 ) ⊂ A2R . We want to investigate the rational function f = X . Show that there exists sequences of Y points (an , bn ) ∈ C with bn 6= 0 such that limn→∞ (an , bn ) = (0, 0) but limn→∞ abnn is not defined. Hint: Remember the parameterization t 7→ (t2 − 1, t3 − t) of this curve!

the node t*t-1, t*t*t-t

6 4 2 0 -2 -4 -6 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3

Exercise 4.3. Rational functions in singular points II Let C = V (X 3 − Y 2 ) ⊂ A2k . We want to investigate the rational functions f = X k Y m for k, m ∈ Z. Show that the value of these functions is well defined even in the point P = (0, 0) and give this value for all pairs (k, m) ∈ Z2 . Hint: Remember the parameterization t 7→ (t2 , t3 ) of this curve!

the cusp 8 t*t, t*t*t 6 4 2 0 -2 -4 -6 -8 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4

Exercise 4.4. Let C = V (f ) ⊂ A2k be irreducible of degree d = deg(f ) ≥ 2. (i) Suppose C has two points of multiplicity m1 and m2 . Show that m1 + m2 ≤ d. (ii) Show that a irreducible cubic cannot have two singular points. (iii) Suppose now d ≥ 3, and that C has five points with multiplicities m1 , m2 , m3 , m4 , m5 . Show that m1 + m2 + m3 + m4 + m5 ≤ 2d. (iv) Can you give a irreducible curve C ⊂ A2k with 5 singular points?




Exercises to Algebraic Geometry Exercise 3.1. Homogeneous polynomials I. A polynomial X im f= ai1 ,i2 ,...,im X1i1 X2i2 · · · Xm ∈ k[X1 , X2 , . . . , Xm ] i1 ≥0,...,im ≥0

is called homogeneous of degree n, if ai1 ,i2 ,...,im = 0 unless i1 + i2 + · · · + im = n. (i) Show that the homogeneous polynomials of degree n in k[X1 , X2 , . . . , Xm ] form a vector space of dimension n+m−1 . m−1 (ii) Show that any f ∈ k[X1 , X2 , . . . , Xm ] possess a unique decomposition f = f0 + f1 + f2 + . . . with fk homogeneous of degree k. Explain the relation with the degree of f . (iii) Show Euler’s identity for a homogeneous polynomial f ∈ k[X1 , X2 , . . . , Xm ] of m X ∂ Xi · degree n: f = n · f. ∂X i i=1 Exercise 3.2. Homogeneous polynomials II. (i) Show the following implication f ∈ k[X1 , X2 , . . . , Xm ] is homogeneous of degree n =⇒ f (λX1, λX2 , . . . , λXm ) = λn f (X1 , X2 , . . . , Xm ) for all λ ∈ k. (ii) Suppose that k is an infinite field. Show that f (λX1, λX2 , . . . , λXm ) = λn f (X1 , X2 , . . . , Xm ) for all λ ∈ k implies, that f ∈ k[X1 , X2 , . . . , Xm ] is homogeneous of degree n. (iii) Give a counter-example for (ii) for a polynomial f ∈ F5 [X1 , X2 ].

Exercise 3.3. Let C = V (f ) ⊂ A2k be curve given by f ∈ k[X, Y ]. Let P = (a, b) ∈ k 2 be a point of C. Using a coordinate transform Xa = X − a, and Yb = Y − b we can express f as an element of k[Xa , Yb ]. Decomposing f into homogeneous components, we obtain f = fm + fm+1 + · · · + fd with fm 6= 0 and fd 6= 0, and fk homogeneous of degree k in k[Xa , Yb]. The tangent cone of C in P is defined to be the set of lines V (A · Xa + B · Yb ) such that fm (A, B) = 0. (i) Show that m is the multiplicity of f in P , and d is the degree of f . (ii) Show that there are at most m lines in the tangent cone to C at P . (iii) Show that for a nonsingular point P of C the tangent cone of C at P consists of exactly one line, the tangent line defined before. (iv) Compute the tangent cones at P = (0, 0) of the following two curves C1 = V (x3 − y 2), and C2 = V (x3 + x2 − y 2 ). Exercise 3.4. Tangent lines to the cusp (i) Recall that the discriminant of the cubic polynomial f (X) = X 3 + aX 2 + bX + c equals −27 c2 + 18 cab + a2 b2 − 4 a3 c − 4 b3 . (ii) Use (i) to compute the tangent lines T (C) of the irreducible curve C = V (x3 −y 2 ). It is enough to consider the lines of the form V (ax + y + b) parameterized by (a, b) ∈ A2k . (iii) Decompose the equation for T (C) into irreducible components. Give a geometric interpretation of each component. Show that all components are rational curves. (iv) Which component gives the lines from the tangent cones?




Exercises to Algebraic Geometry Exercise 2.1. Pythagorean triples and Euclid’s formula Give a parameterization of the unit circle C = V (x2 + y 2 − 1) ⊂ A2k by assigning λ ∈ k the intersection point of the line V (λx − y + λ) with C which is different from (−1, 0). Do we get all points of C this way? Use the formula obtained to give all integer solutions (a, b, c) of a2 + b2 = c2 . 20082 + 629852 = 630172 Exercise 2.2. We fix a prime p. We consider the equation a2 + b2 = p · c2 for integers (a, b, c) ∈ Z3 . Show the statement: If the equation has one solution (different from (0, 0, 0)), then it has infinitely many coprime solutions. (Here we call a solution (a, b, c) coprime when the greatest common divisor of a, b, and c is 1.) For which primes p do there exist solutions? Hint for the second question: Have look on the Algebra II exercise 1.2. Exercise 2.3. Compute for the three conics C1 = V (y − x2 ), C2 = V (x2 + y 2 − 1), and C3 = V (x2 − y 2 + 1) all tangent lines not parallel to the y-axis. Hint: Consider the lines to be given by V (ax + y + b) with two parameters (a, b) ∈ k 2 .

Exercise 2.4. Show that for a irreducible conic C ⊂ A2k the tangent lines not parallel to the y-axis form a irreducible conic in the space of lines not parallel to the y-axis. Is any conic in the space of lines coming from a conic?

These exercises are discussed on Wednesday October 29, 2008



Exercises to Algebraic Geometry Exercise 1.1. The polar line I Let C = V (x2 + y 2 − 1) be the unit circle in A2k for a field k of characteristic char(k) 6= 2. We assume in this exercise that k is algebraically closed. Suppose that there exist exactly two tangent lines to C through P . The line connecting the two points of contact of tangents on C through P is called the polar line of P with respect to C. For a point P = (a, b) 6= (0, 0) not on C show that there exist two tangent lines to C through P . (ii) Give the equation for the polar line of P = (a, b) 6∈ C. (P 6= (0, 0)) Hint: You do NOT need to give a formula for the two points of contact. (iii) Use the equation established in (ii) to define the polar line for points P ∈ C. Which line is it? (i)

Exercise 1.2. The polar line II Let C = V (x2 + y 2 − 1) be the unit circle in A2k for a field k of characteristic char(k) 6= 2. We don’t assume that k is algebraically closed. For a point P ∈ A2k we define the polar line (with respect to C - as always) to be the line defined by the formula in (ii) of the exercise before. We assume here P 6= (0, 0). Show that this definition assigns a point in A2k a line defined by coefficients in k. Describe the lines given by a linear equation with coefficients in k which are not polar lines of points P ∈ A2k . (iii) Would it be nice to have a polar line l∞ to (0, 0) and the polar lines to the points of l∞ give the missing lines from (ii)? (Yes/No is enough)

(i) (ii)

Exercise 1.3. The polar line III Let C = V (x2 + y 2 − 1) be the unit circle in A2R . We assume here P = (a, b) 6= (0, 0) is a point with real coordinates inside C, that is a2 + b2 < 1. Show that the polar line l of P can be described as the set of all points Q ∈ A2R such that the polar line of Q passes through P . Exercise 1.4. Use an algebraic parameterization of the unit circle C = V (x2 +y 2−1) ⊂ R2 Z 3 5 dx √ to compute the integral . 1 − x2 0

These exercises are discussed on Wednesday October 22, 2008