Math 6421 Algebraic Geometry I Fall 2019 -

1. Let k be an infinite field. Show that f,g in k[x₁,...,x_n] are equal if and only if f(a)=g(a) for all a in Aⁿ.
2. Let k = Z/pZ where p is prime. Is the affine space Aⁿ Hausdorff under the Zariski topology?
3. Find a compact set in the Zariski topology that is not closed.
4. Use the Nullstullensatzes to prove the correspondence between radical/prime/maximal ideals of k[X] and general/irreducible/one-point varieties of X.
5. Show that f_* : k[Y] → k[X] is injective if and only if f is dominant, that is, theimage f(X) is dense in Y.
6. Show that f_* : k[Y] → k[X] is surjective if and only f f defines an isomorphism between X and a subvariety of Y.
7. Prove the uniqueness statement: If f : X → Y and g : X → Y are maps with f_* = g_* then f=g.
8. Let X be an affine algebraic variety in Aⁿ and let k[X] be its coordinate ring, where k is algebraically closed. For each f in k[X], let U_f be the set of points p in X so that f(p) is nonzero. (a) Show that {U_f} forms a basis for the Zariski topology on X. (These are called the basic, or principal, open sets.) (b) Prove that X is compact, meaning that every open cover has a finite subcover (in algebraic geometry this condition is usually called quasi-compactness, with the word compact reserved for Hausdorff spaces). (c) Describe all irreducible Hausdorff affine algebraic varieties in X.
9. Let V be an affine subspace of Aⁿ (that is, a translate of a linear subspace). Give a description of k[V]. Show that the algebro-geometric dimension is equal to the linear-algebraic dimension.
10. The twisted cubic is the image of the map P¹ → P³ given by [x₀:x₁] → [x₀³:x₀²x₁:x₀x₁²:x₁³]. Verify that the twisted cubic is a projective variety. Hint: use the polynomials z₀z₂-z₁², z₀z₃-z₁z₂, z₁z₃-z₂² in k[z₀,z₀,z₁,z₂,z₃].
11. Show that (i) every element of Λ¹V is totally decomposable, (ii) every element of Λ²V is totally decomposable if dim V = 3, and (iii) if v₁, v₂, v₃, v₄ are linearly independent then v₁ ∧ v₂ + v₃ ∧ v₄ is not totally decomposable.
12. Find a set of homogeneous polynomials S so that the image of G_2,4 under the Plucker embedding is Z(S). What is the smallest such S? Here is how to do it: Choose a basis e_i for k⁴. This gives you 6 basis elements e_I for Λ²V. Let x be an arbitrary point Λ²V; this is given by the 6 a_I-coordinates. As in class, there is a matrix M_x corresponding to φ_x; this should be a 4 x 4 matrix. To say that x is in the image of the Plucker embedding is to say that x is totally decomposable, which is to say (by the Lemma in class) that all 3 x 3 minors vanish. If you write down the 16 3 x 3 minors, you will get 16 polynomials in the a_I. You will see that you can throw most of them away.
13. Show that the intersection, product, or sum of homogeneous ideals is homogeneous.
14. Consider the twisted cubic X = Z(y-x²,z-xy). Show that the sub variety of P³ given by the homogenizations of the defining polynomials for X has two components: the projective closure of X and one additional line.
15. Show that the projective completion of an affine variety is the Zariski closure of that variety, thought of as a subset of Aⁿ, hence as a subset of Pⁿ.
16. Show that the projective completion of an irreducible variety is irreducible.
17. Show that the twisted cubic curve C in A³ can be defined by the polynomials y-x² and z-x³, and that these polynomials generate the full radical ideal of polynomials vanishing on C. Show that the two homogeneous polynomials obtained by homogenizing these polynomials define the projective closure of C in P³, but that the ideal they generate is not radical.
18. Show that the homogenization of a radical ideal is radical.
19. Prove that the homogeneous ideal of the set {[1:0:0],[0:1:0]} in P² is (x₂,x₀x₁). Interpret this fact geometrically and then find the homogeneous ideal of the set {[2:3:-1],[1:-2:2]}. Try to generalize the exercise to sets of 3 and 4 points in P² (you will need to think about the different possible embeddings of points).
20. Show that the Segre embedding φ_m,n maps {linear subspace} x {pt} and {pt} x {linear subspace} to linear subspaces of P^(m+1)(n+1).
21. Define the product of two affine varieties to be the image under the appropriate Segre map. Show that the product of two affine varieties is an affine variety.
22. Regard the twisted cubic as the image of the map P¹ → P³ given by [x:y] → [x³:x²y:y²:y³]. Show that the twisted cubic lies in the Segre variety V_1,1. Find bihomogeneous polynomials describing the twisted cubic as a subset of P¹ x P¹.
23. Assuming that the Segre map is an isomorphism of Pⁿ x Pⁿ onto V_m,n, show that the diagonal in Pⁿ x Pⁿ is a closed set.
24. Define the topology on a product of varieties by declaring the Segre map to be an isomorphism of Pⁿ x Pⁿ onto V_m,n. Show that the induced topology is not the product topology, assuming that neither of the varieties in the product is a finite collection of points.
25. Consider the Veronese map ν₂ : P² → P⁵. Its image is called the Veronese surface. Describe the images of the lines in P² in the Veronese surface.
26. Verify the defining equations for the images of the Veronese maps.
27. Prove the classification of conics in P².
28. Consider the curve in A² given by y²=x³+ax+b, where a and b lie in k. Find conditions on a and b that determine whether or not the given curve is smooth.
29. Assume that f and g are coprime elements of k[x₁,...,x_n] and let V=Z(fg). Show that a point of V is singular if and only if it is a singular point of Z(f), Z(g), or their intersection.
30. Show that a morphism of affine algebraic varieties induces a linear map on each tangent space.
31. Describe Spec R[x].
32. Let R = C[x,y]. (a) Show that (x-y) is a prime ideal in R, and hence is a point in Spec R. (b) For two fixed complex numbers a and b, show that (x-a,y-b) is a maximal ideal in R and hence is also a point in Spec R. (c) For every complex number a, the ideal (x-a,y-a) contains the ideal (x-y). (d) Comment on this.
33. Determine Spec Q[x] as a set.
34. Show that the only non-principal prime ideals in C[x,y] are of the form (x-a,y-b). Hint: suppose p is not principal. Show you can find f,g in p with no common factor. Using the Euclidean algorithm in the Euclidean domain C(x)[y], show that you can find a nonzero h in (f,g) < p. Using primality, show that one of the linear factors of h, say x-a, is in p. Similarly show there is some y-b in p.
35. Let R be a ring and S a multiplicative subset. Describe an order-preserving bijection between the primes of S^-1R and the primes of R that don't meet S.

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