Exercises

Exercises on Curves

1. Let k be an algebraically closed field. Classify the affine plane curves of degree two (conic curves) over k up to isomorphism.
2. Show that any two curves over an algebraically closed field are homeomorphic.

   Exercises on Hilbert's Theorems

3. Show that both forms of the Nullstellensatz fail over fields that are not algebraically closed.
4. Let J= < x² + y² + z², xy + xz + yz>. Identify Z(J) and I(Z(J)).
5. Let V = Z(xy = zw) Ì A⁴. Prove that A(V) is not a UFD.
6. Let Y Ì A³ be the set { (t, t², t³) : t Î k}. Find I(Y). Prove that A(Y) is isomorphic to a polynomial ring in one variable.
7. Let I and J be ideals of a ring R with I Ì Rad(J). Show that if I is finitely generated, then there exists a positive integer n such that Iⁿ Ì J.

   Exercises on Morphisms

8. Let k be a field of positive characteristic p. Consider the morphism f : A¹ ® A¹ given by f(t) = t^p. Show that f is a bijective morphism which is not an isomorphism.
9. Let k be an algebraically closed field, and let C be the algebraic plane curve over k defined by y² = x³. Let f : A¹ ® C be defined by f(t) = (t², t³). Show that f is a bijective morphism which is not an isomorphism.
10. Let f = (f₁, ¼, f_n): Aⁿ ® Aⁿ be a morphism, and let J = det[¶f_i / ¶ x_j] be the Jacobian polynomial of f. Prove that if f is an isomorphism, then J is a nonzero constant polynomial. (Note: the converse to this assertion is still an unsolved problem.)

    Exercises on Irreducibility

11. Let Y=Z(J) be an affine algebraic set. Show that if J can be generated by r elements, then every irreducible component of Y has dimension at least equal to n - r.
12. Let I be the ideal generated by <xz - y², yw - z²> in the ring k[x, y, z, w]. Show that the intersection of the zero set Z(I) with the affine space where w ¹ 0 consists of two irreducible pieces. Use this to show that I is the intersection of the ideals <y,z> and I₀ = <xz - y², yw - z², xw - yz>. Finally, show that the quotient k[x,y,z,w]/I₀ is isomorphic to a polynomial ring in two variables.
13. Let J=<xy, xz, yz> Î k[x, y, z]. Find the irreducible components of Z(J) Ì A³. Is J radical? prove that J cannot be generated by two elements. Now let J' = < xy, (x-y)z >, and find the radical of J'.
14. Prove that the product of two irreducible algebraic sets is again irreducible. (Remember that the product in this category does not have the product topology.)
15. Find the (three) irreducible components in A³ of <yz=x², x=xz>.
16. Find the irreducible components of Z(x² = y⁴, x³ = y⁴ -x²y²+xy²) in A³.

    Exercises on Rational Functions

17. Let f_n : A¹ ® A² be the map given by f_n(x) = (x², xⁿ). If n is odd, show that the image of f_n is isomorphic to A¹ and f_n is 2-to-1 outside 0. If n is odd, show that f_n is bijective, and provide a rational inverse.
18. Show that the map f : (x:y:z) |® (yz:xz:xy) is a rational map from P² to itself. Find the largest open set on which f is defined. Show that f is an isomorphism from this set to its image, and find a rational inverse.
19. Let f = Y/X be a rational function on the affine plane curve Y² = X³ + X². Find the poles of f and of f².

    Exercises on Birationality

20. Let k be an algebraically closed field of characteristic different from 2. Let l Î k with l ¹ 0, 1. Show that the affine plane curve defined by

    y² = x(x-1)(x-l)

    is not rational.
21. Prove that the affine plane curve y³ = x⁴ + x³ is rational.
22. Show that the affine plane curve Y defined by y=x² is isomorphic to A¹. Show that the affine plane curve Z defined by xy=1 is birationally isomorphic, but not isomorphic, to A¹. Finally show that every irreducible affine plane curve of degree two is isomorphic either to Y or to Z.

    Exercises on Projective Space

23. Prove that the algebraic sets are the closed subsets of a topology on Pⁿ.
24. Show that the radical of a homogeneous ideal is homogeneous.
25. Prove the "homogeneous Nullstellensatz", which says that if J Ì S = k[x₀, ¼, x_n] is a homogeneous ideal and if f Î S is a hommogeneous polynomial of positive degree such that f Î I(Z(J)), then fⁿ Î Jfor some positive integer n.
26. Let k be an algebraically closed field. Classify the projective plane curves of degree two (conic curves) over k up to isomorphism.
27. A hypersurface in Pⁿ defined be a linear polynomial is called a hyperplane. A subvariety of Pⁿ whose ideal is generated by linear polynomials is called a linear variety. Show that a subvariety is linear if and only if it is an intersection of hyperplanes.
28. Prove that any two curves in the projective plane have nonempty intersection. In particular, any two distinct lines meet in eactly one point.
29. d-uple (iso to image)
30. Segre
31. quadrics

    Exercises on Sheaves

32. Describe the sheaf associated to the constant presheaf which assigns to all open sets U the same abelian group A.
33. Prove that the exactness of a sequence of morphisms of sheaves can be tested on the stalks.
34. Let Y be the topological space with one point. Describe all sheaves of abelian groups on Y. Give an example of a ring R for which Y is homeomorphic to Spec(R).
35. Let Y be the set with two points, made into a topological space in which all four subsets are open. Describe all sheaves of abelian groups on Y. Give an example of a ring R for which Y is homeomorphic to Spec(R).
36. Let Y be the set with two points {0, 1}, made into a topological space in which only the three subsets Æ, {0}, and {0, 1} are open. Describe all sheaves of abelian groups on Y. Give an example of a ring R for which Y is homeomorphic to Spec(R).
37. Find the stalks of the sheaves in the previous three problems.
38. Prove that a sheaf on a topological space X is completely determined by its sections on a basis for the topology on X.
39. Suppose {U_i} is an open cover of X, that F_i are sheaves on U_i, and that f_ij : F_i½_UiÇUj ® F_j½_UiÇUj are isomorphisms satisfying the compatibility condition f_jkf_ij = f_ik on triple intersections. Prove that there is a unique sheaf F on X whose restriction to U_i is isomorphic to F_i.

    Exercises on Spectra

40. Let R be the coordinate ring of an affine algebraic variety X over an algebraically closed field k, and let P be the maximal ideal corresponding to a point x Î X. Prove that for all f Î R, the value f(P) Î R/P is just the usual value f(x) Î k.
41. Prove that the Zariski topology on Spec(R) really is a topology.
42. Describe the topological space Spec(C[X]).
43. Describe the topological space Spec(R[X]).
44. Describe the topological space Spec(k[X]_(x)).
45. Describe the topological space Spec(k[[X]]).

    Exercises on Affine Schemes

46. Let X=Spec(R) and let P Î X be a point. Show that there is a one-to-one correspondence between the set of prime ideals in O_X,P and the set of irreducible closed subsets of X that contain P.

    Exercises on General Schemes

47. Show that every scheme is a T₀ topological space.

    Exercises on the Functor of Points

    Exercises on Dimension

48. Show that every noetherian topological space is quasicompact. Show that every Hausdorff noetherian topological space is a finite set with the discrete topology.
49. If k is a field, prove that Pⁿ_k is a noetherian topological space. Find its dimension.
50. Show that every PID has dimension one.
51. Let R be a UFD and let P=<t> be a principal, proper, prime ideal. Show that there are no prime ideals properly conatined between P and 0.
52. Let Y be a linear subvariety of Pⁿ_k over a field k. Show that if dim(Y) = r, then I(Y) is minimally generated by n - r linear polynomials.
53. Let Y and Z be linear subvarieties of Pⁿ_k of dimensions r and s. Show that if r + s - n ³ 0, then Y Ç Z ¹ Æ. Moreover, if Y Ç Z ¹ Æ, then Y Ç Z is a linear subvariety of dimension r + s - n.
54. Let A be a noetherian ring of dimension zero. Prove that A is artinian. If, an addition, A is finitely generated as an algebra over a field k, then show that A is a finite-dimensional k-vector space.
55. Let Y Ì A³ be the set { (t³, t⁴, t⁵) : t Î k}. Prove that I(Y) is a prime ideal of height 2 that cannot be generated by two elements.

    Exercises on Local Properties

56. Locate the singular points on the affine plane curves

    y² = x⁴ + y⁴

    xy = x⁶ + y⁶

    x³=y² + x⁴ + y⁴

    (x² + y²)² = y³ - 3x²y

    (x²+y²)³ = 4x²y²

57. Let X be an affine plane curve of degree n containing a multiple point P of multiplicity n. Show that X consists of a collection of n lines through P.
58. Describe the singular points on the surfaces in A³ defined by:

    xy=z²

    xy²=z²

    x² + y² = z²

    xy + x³ +y³ = 0

59. Suppose R is an algebra over a field k that is finitely generated as a k-vector space. Prove that R is isomorphic to a direct product of artinian local rings.
60. Let R be a discrete valuation ring containing a field k that is isomorphic to its residue field R/M. Show that v(z) = dim_k(R/<zgt for all z Î R.

    Exercises on Curves

61. Find (birationally isomorphic) nonsingular projective models of each of the curves in exercise (1) of the section on Local Properties.
62. Every conic in the projective plane is normal.
63. The quadric surfaces xy=zw and xy=z² in P³ are normal.
64. Show that the cuspidal cubic curve y²=x³ is not normal.
65. Show that an irreducible cubic curve has at most one singular point.

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