Math 6421 Algebraic Geometry I Fall 2021 -	----------------------------------------Matthias Paulsen

1. Let C be the affine algebraic variety Z(xy) in A². Describe the intersection of C with the unit sphere |x|²+|y|²=1.
2. Consider the three ideals I₁ = (xy+y²,xz+yz) and I₂=(xy+y²,xz+yz+xyz+y²) and I₃=(xy²+y³,xz+yz). Are any of these equal? Do any of them define the same affine algebraic varieties? Sketch them!
3. Consider the affine algebraic variety C=Z(x²-yz,xz-x) in A³. Decompose C into its irreducible components.
4. Consider the semi-cubical parabola C=Z(y²-x³). Show that the map from the complex plane to C given by t maps to (t²,t³) is a homeomorphism in the Zariski topology (here we mean for both to have the Zariski topology). Is it a homeomorphism with respect to the Euclidean topology? (We will see that this is not an isomorphism of varieties.)
5. Is the coordinate ring for the unit circle Z(x²+y²-1)) isomorphic to the coordinate ring for the parabola Z(y-x²)? Hint: Find two factorizations of y² in one of the rings...
6. Prove that subvarieties of an affine algebraic variety X are in bijection with reduced k-subalgebra quotients of k[X].
7. Prove that the product of two affine algebraic varieties is an affine algebraic variety.
8. Let X be an affine algebraic variety and suppose that a finite group G acts on X by automorphisms. We would like to define the quotient variety X/G. The quotient X/G does not naturally live in affine space, so it is not immediately realized as a variety. (a) Show that there is an induced action of G on the coordinate ring k[X]. (b) Show that the invariants k[X]^G of this action is a finitely generated reduced sub-algebra of k[X] (the invariants are the elements that are fixed by every element of G). (c) Let X' be the (isomorphism class of) affine algebraic varieties corresponding to k[X]^G. Show that the points of X' are in bijection with X/G (the set of G-orbits in X). (d) Show that the quotient map from X to X/G is a quotient map in the Zariski topology (the preimages of a set U in X' = X/G is open iff U is open). (e) Conclude that X' is a reasonable definition for X/G.
9. Apply the previous exercise to the standard action of the symmetric group S_n on Aⁿ. (a) What are the invariants for the action of S_n on Aⁿ? (b) Show that the map from Aⁿ/S_n to Aⁿ that takes an unordered set of roots to a polynomial is an isomorphism of varieties. Assume that k is algebraically closed.
10. Show that the rational normal curve of degree 3, parametrized by [t₀³:t₀²t₁:t₀t₁²:t₁³] is equal to the intersection of three quadrics, namely, Z(xz-y², xw-yz, yw-z²). Show that it is not the intersection of two quadrics.
11. Consider Y=Z_p(xw-yz). (a) Show that xw-yz is irreducible, and conclude that Y is irreducible. (b) Show that f : P² → Y given by [x:y:z] → [x²:xy:xz:yz] is a rational map. Show that f is dominant. (Hint 1: Show that each point [a:b:c:d] with a not equal to zero is in the image. Hint 2: Use/prove the lemma that if a map contains the complement of any subvariety then it is dominant.) (c) Show that g : Y → P² given by [x:y:z:w] → [x:y:z] is a rational map. Show that it is dominant. (d) Show that f and g are birational. Conclude that Y is rational (meaning that it is birationally equivalent to projective space).
12. Consider the projective twisted cubic Y=Z_p(xz-y²,yw-z²,xw-yz) in P³. (a) Show that f : P¹ → Y given by [u:v] → [u³:u²v:uv²:v³] is a morphism. (b) Show that f is an isomorphism by finding the inverse rational map and computing their compositions. Conclude that Y is isomorphic to P¹
13. We consider the Grassmannian G(2,4) as the space of lines in P³. The Plucker embedding puts this in P⁵. Consider two skew lines in P³. Show that the set Q of lines in P³ meeting both is the intersection of G(2,4) with a 3-plane P³ in P⁵, and so it is a quadric surface. Deduce that Q is isomorphic to P¹ x P¹. What happens if the two original lines meet?
14. Let X be an affine algebraic variety. Show that an injective, finite map k[y₁,...,y_d] → k[X] corresponds to a finite-to-one surjective map X → A^d. Recall that a map of k-algebras f : A → B is finite if B is a finitely generated module over f(A). There is a discussion of this in Hulek's book (and many other places). Explain in your own words.
15. Prove that the dimension of a hypersurface in Aⁿ or Pⁿ is n-1. Recall that a hypersurface is a variety of the form Z(f) for a polynomial f. UPDATE: This problem is canceled, because it is harder than I thought it was. If you want to read about it, it is Geometric Krull’s Hauptidealsatz, which you can find in Dolgachev's book, for example.
16. For which [a:b] in P¹ is the curve C_[a:b] = Z((a+b)y²-(a+b)x³-bx) singular? Sketch these curves over the reals. Bonus: Do the same for the projective closure in P².
17. Find all singular points of the zero sets of the following three polynomials: (x²+y²)³-4x²y² in k[x,y], xy²-z² in k[x,y,z], and xy+x³+y³ in k[x,y,z]
18. Show that Z(xy-z²) in P² is nonsingular. Hint: once you do it for the first affine chart, it suffices to check the points not in that chart.
19. Recall that the blowup of an affine variety X along an ideal I=(f₁,...,f_m) is the graph of the map X → P^m-1 sending x to [f₁(x):...:f_m(x)]. Show that the blowup of any irreducible affine variety along any principal ideal is an isomorphism.
20. Show that the blowup of A² along (x²,y³) is not smooth (so blowup can make a variety more singular!).
21. Show that the blowup of A³ along any line is smooth (find the defining equations for the blowup).
22. Find the intersection points and multiplicities: (x+z)³+3y³-z³ and x³+x²(y+z)+xz²+y³
23. Find the intersection points and multiplicities: y²z-x(x-2z)(x+z) and y²+x²-2xz
24. Find the intersection points and multiplicities: (x²+y²)z+x³+y³ and x³+y³-2xyz
25. Show that R(f,g+af)=R(f,g) if deg a = deg g - deg f. Let C=Z(f), D=Z(g), and E=Z(g+af). Show that I_p(C,E) = I_p(C,D) for all p.
26. Find explicit formulas for the group law on Z(y²x-z³-x³). Use [0:1:0] as the identity.
27. Let f be a polynomial and f' its derivative. The discriminant of f is Res(f,f'). Compute the discriminant in terms of the coefficients of f (what do you get when deg f = 2?). Show that the discriminant is 0 exactly at the multiples roots of f.
28. Use the local ring definition of intersection multiplicity to show that the multiplicity of the intersection of Z(y) and Z(y-x³) is 3.

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