Math 6421

Algebraic Geometry I

Fall 2021


----------------------------------------Matthias Paulsen

Class Meetings

Monday/Wednesday, 11:00-12:15 pm, MRDC 3403.


There is no text for this course. Some supplementary references are listed below.

Office Hours

Fri 1:30-2:30 on Teams, and by appointment


There will be weekly homework assignments, each with 2-4 problems. Homework is due at the start of class on Monday, and will be turned in on Gradescope. Homework will be posted on this web site below the calendar. Homework may be due during the final instructional days.


Course grades will be determined by homework scores. The lowest three scores will be dropped.

Weekly Schedule

Week Dates Topics Homework Notes Comments
1 Aug 23/25 Overview none Aug 23
Aug 25
First day
2 Aug 30/1 Affine algebraic varieties 1,2 Aug 30
Sep 1
Schedule change deadline
3 Sep 8 Affine algebraic varieties 3,4 Sep 8 Labor day
4 Sep 13 Morphisms
Coordinate rings
5,6,7 Sep 13
Sep 15
5 Sep 20 Projective varieties 8,9,10 Sep 20
Sep 22
6 Sep 27 Birational equivalence 11,12 Sep 27
Sep 29
7 Oct 4 Veronese & Segre none Oct 4
Oct 6
8 Oct 13 Grassmannians 13 Oct 13 Fall break
9 Oct 18/20 Incidence varieties makeups Oct 18
Oct 20
10 Oct 25/27 Dimension 14 (problem 15 canceled) Oct 25
Oct 27
Drop date
11 Nov 1/3 Smoothness 16, 17, 18 Nov 1
Nov 3
12 Nov 8/10 Blowups / Degree 19, 20, 21 Nov 8
Nov 10
13 Nov 15/17 Curves 22-28 or project Nov 15
Nov 17
14 Nov 22 Curves Nov 22 Thanksgiving
15 Nov 29/1 27 lines Nov 29
Dec 1
16 Dec 6 Farewell Dec 6 Last day


  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 Sn on An. (a) What are the invariants for the action of Sn on An? (b) Show that the map from An/Sn 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=Zp(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=Zp(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,...,yd] → k[X] corresponds to a finite-to-one surjective map X → Ad. 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,...,fm) is the graph of the map X → Pm-1 sending x to [f1(x):...:fm(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 Ip(C,E) = Ip(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.


---Elementary Algebraic Geometry, Klaus Hulek

---Ideals, Varieties, and Algorithms, Cox, Little, and o'Shea

---Algebraic Geometry, Andreas Gathmann

---Undergraduate Algebraic Geometry, Joe Harris (notes by Aaron Landesman)

---Algebraic Geometry III/IV, Matt Kerr

---Introduction to Projective Varieties, Enrique Arrondo

---An Invitation to Algebraic Geometry, Karen Smith et al.

---Algebraic Geometry I, Karen Smith (notes by David Bruce)

---Algebraic Geometry, J.S. Milne

---Algebraic Geometry, Jonathan Wise

---Undergraduate Algebraic Geometry, Miles Reid

---Hilbert's Nullstullensatz, Daniel Allcock

---Cayley-Bacharach Geogebra