Math 6421
Algebraic Geometry I
Fall 2019




Class Meetings
Monday, Wednesday, Friday, 11:1512:05 pm, Skiles 308.
Text
Algebraic geometry, Andreas Gathmann
Office Hours
In Skiles 234, Monday 34, Wednesday 23, and by appointment
Homework
There will be daily homework assignments, each with 13 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.
Grading
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 #13 
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
 Let k be an infinite field. Show that f,g in k[x_{1},...,x_{n}] are equal if and only if f(a)=g(a) for all a in A^{n}.
 Let k = Z/pZ where p is prime. Is the affine space A^{n} Hausdorff under the Zariski topology?
 Find a compact set in the Zariski topology that is not closed.
 Use the Nullstullensatzes to prove the correspondence between radical/prime/maximal ideals of k[X] and general/irreducible/onepoint varieties of X.
 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.
 Show that f_{*} : k[Y] → k[X] is surjective if and only f f defines an isomorphism between X and a subvariety of Y.
 Prove the uniqueness statement: If f : X → Y and g : X → Y are maps with f_{*} = g_{*} then f=g.
 Let X be an affine algebraic variety in A^{n} 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 quasicompactness, with the word compact reserved for Hausdorff spaces). (c) Describe all irreducible Hausdorff affine algebraic varieties in X.
 Let V be an affine subspace of A^{n} (that is, a translate of a linear subspace). Give a description of k[V]. Show that the algebrogeometric dimension is equal to the linearalgebraic dimension.
 The twisted cubic is the image of the map P^{1} → P^{3} given by [x_{0}:x_{1}] → [x_{0}^{3}:x_{0}^{2}x_{1}:x_{0}x_{1}^{2}:x_{1}^{3}]. Verify that the twisted cubic is a projective variety. Hint: use the polynomials z_{0}z_{2}z_{1}^{2}, z_{0}z_{3}z_{1}z_{2}, z_{1}z_{3}z_{2}^{2} in k[z_{0},z_{0},z_{1},z_{2},z_{3}].
 Show that (i) every element of Λ^{1}V is totally decomposable, (ii) every element of Λ^{2}V is totally decomposable if dim V = 3, and (iii) if v_{1}, v_{2}, v_{3}, v_{4} are linearly independent then v_{1} ∧ v_{2} + v_{3} ∧ v_{4} is not totally decomposable.
 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 k^{4}. This gives you 6 basis elements e_{I} for Λ^{2}V. Let x be an arbitrary point Λ^{2}V; 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.
 Show that the intersection, product, or sum of homogeneous ideals is homogeneous.
 Consider the twisted cubic X = Z(yx^{2},zxy). Show that the sub variety of P^{3} given by the homogenizations of the defining polynomials for X has two components: the projective closure of X and one additional line.
 Show that the projective completion of an affine variety is the Zariski closure of that variety, thought of as a subset of A^{n}, hence as a subset of P^{n}.
 Show that the projective completion of an irreducible variety is irreducible.
 Show that the twisted cubic curve C in A^{3} can be defined by the polynomials yx^{2} and zx^{3}, 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 P^{3}, but that the ideal they generate is not radical.
 Show that the homogenization of a radical ideal is radical.
 Prove that the homogeneous ideal of the set {[1:0:0],[0:1:0]} in P^{2} is (x_{2},x_{0}x_{1}). 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 P^{2} (you will need to think about the different possible embeddings of points).
 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)}.
 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.
 Regard the twisted cubic as the image of the map P^{1} → P^{3} given by [x:y] → [x^{3}:x^{2}y:y^{2}:y^{3}]. Show that the twisted cubic lies in the Segre variety V_{1,1}. Find bihomogeneous polynomials describing the twisted cubic as a subset of P^{1} x P^{1}.
 Assuming that the Segre map is an isomorphism of P^{n} x P^{n} onto V_{m,n}, show that the diagonal in P^{n} x P^{n} is a closed set.
 Define the topology on a product of varieties by declaring the Segre map to be an isomorphism of P^{n} x P^{n} 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.
 Consider the Veronese map ν_{2} : P^{2} → P^{5}. Its image is called the Veronese surface. Describe the images of the lines in P^{2} in the Veronese surface.
 Verify the defining equations for the images of the Veronese maps.
 Prove the classification of conics in P^{2}.
 Consider the curve in A^{2} given by y^{2}=x^{3}+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.
 Assume that f and g are coprime elements of k[x_{1},...,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.
 Show that a morphism of affine algebraic varieties induces a linear map on each tangent space.
 Describe Spec R[x].
 Let R = C[x,y]. (a) Show that (xy) 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 (xa,yb) is a maximal ideal in R and hence is also a point in Spec R. (c) For every complex number a, the ideal (xa,ya) contains the ideal (xy). (d) Comment on this.
 Determine Spec Q[x] as a set.
 Show that the only nonprincipal prime ideals in C[x,y] are of the form (xa,yb). 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 xa, is in p. Similarly show there is some yb in p.
 Let R be a ring and S a multiplicative subset. Describe an orderpreserving bijection between the primes of S^{1}R and the primes of R that don't meet S.
Resources
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, Mathpages.com
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