Math 6421

Algebraic Geometry I

Fall 2019


Class Meetings

Monday, Wednesday, Friday, 11:15-12:05 pm, Skiles 308.


Algebraic geometry, Andreas Gathmann

Office Hours

In Skiles 234, Monday 3-4, Wednesday 2-3, and by appointment


There will be daily homework assignments, each with 1-3 problems. Homework is due at the start of the next class. The three lowest scores will be dropped. Homework will be posted on this web site. If there is no homework, the web site will explicitly say so. 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 Text Sections Homework Notes Comments
1 Aug 19 Intro / Varieties none 0.2.4(i) A #1-3 Intro
Affine varieties
2 Aug 26 Varieties 1.4.1, 1.4.2, 1.4.3 none 1.4.4
3 Sep 2 Morphisms A.4 A.4, A.5, A.6 Morphisms Labor day
3 Sep 9 Dimension A.8 A.9 none
5 Sep 16 Projective varieties A.10 none none
6 Sep 23 Grassmannian A.11, A.12 A.13, A.14, A.15 A.16, A.17, A.18 ProjectiveVarieties
7 Sep 30 Segre maps A.19, A.20, A.21 A.22, A.23, A.24 Rosh Hashanah
8 Oct 7 Veronese maps 3.5.1, A.25, A.26 none none Yom Kippur
9 Oct 14 Bezout's theorem no class A.27 none Fall break
10 Oct 21 Smoothness none A.28, A.29, A.30 none Smoothness Withdrawal deadline
11 Oct 28 Sheaf of regular functions none none 3.23, 3.24 (new notes) Sheaf of Reg Fns
12 Nov 4 Morphisms 4.14 (new notes) none 4.19,5.8,5.11 Varieties
13 Nov 11 Varieties 5.13 (new notes) none 5.22,5.23,5.24
14 Nov 18 Schemes A.31, A.32 A.33, A.34, A.35 Vakil 3.2.Q Schemes
15 Nov 25 27 lines none no class no class 27 Lines Thanksgiving
16 Dec 2 none no class no class Class notes

Additional homework

  1. Let k be an infinite field. Show that f,g in k[x1,...,xn] 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 Uf be the set of points p in X so that f(p) is nonzero. (a) Show that {Uf} 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 G2,4 under the Plucker embedding is Z(S). What is the smallest such S? Here is how to do it: Choose a basis ei for k4. This gives you 6 basis elements eI for Λ2V. Let x be an arbitrary point Λ2V; this is given by the 6 aI-coordinates. As in class, there is a matrix Mx corresponding to φx; 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 aI. 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 φm,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 V1,1. 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 Vm,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 Vm,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,...,xn] 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.


---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

---Hilbert's Nullstullensatz, Terence Tao

---Grassmannian notes, Evan Bullock

---Some naive enumerative geometry, James McKernan

---Foundations of Algebraic Geometry, Ravi Vakil

---Solving the cubic and quartic, Aaron Landesman

---Euclid meets Bezout: Intersecting algebraic plane curves with the Euclidean algorithm, Jan Hilmar and Chris Smyth

---The resultant and Bezout's theorem,

---Bezout's theorem and its applications, Geunho Gim

---Klein's quartic equation, Greg Egan

---Klein's quartic curve, John Baez

---The Eightfold Way: The Beauty of Klein's Quartic Curve, Silvio Levy, ed.

---The Eightfold Way: A Mathematical Sculpture by Helaman Ferguson, William Thurston

---The configuration of bitangents of the Klein curve, R.H. Jeurissen, C.H. van Os, J.H.M. Steenbrink

---List of Algebraic Geometry Lecture Notes, Y.-P. Lee