Math 6421
Algebraic Geometry I
Fall 2021


Matthias Paulsen


Class Meetings
Monday/Wednesday, 11:0012:15 pm, MRDC 3403.
Text
There is no text for this course. Some supplementary references are listed below.
Office Hours
Fri 1:302:30 on Teams, and by appointment
Homework
There will be weekly homework assignments, each with 24 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.
Grading
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 
2228 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 
Homework
 Let C be the affine algebraic variety Z(xy) in A^{2}. Describe the intersection of C with the unit sphere x^{2}+y^{2}=1.
 Consider the three ideals I_{1} = (xy+y^{2},xz+yz) and I_{2}=(xy+y^{2},xz+yz+xyz+y^{2}) and I_{3}=(xy^{2}+y^{3},xz+yz). Are any of these equal? Do any of them define the same affine algebraic varieties? Sketch them!
 Consider the affine algebraic variety C=Z(x^{2}yz,xzx) in A^{3}. Decompose C into its irreducible components.
 Consider the semicubical parabola C=Z(y^{2}x^{3}). Show that the map from the complex plane to C given by t maps to (t^{2},t^{3}) 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.)
 Is the coordinate ring for the unit circle Z(x^{2}+y^{2}1)) isomorphic to the coordinate ring for the parabola Z(yx^{2})? Hint: Find two factorizations of y^{2} in one of the rings...
 Prove that subvarieties of an affine algebraic variety X are in bijection with reduced ksubalgebra quotients of k[X].
 Prove that the product of two affine algebraic varieties is an affine algebraic variety.
 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 subalgebra 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 Gorbits 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.
 Apply the previous exercise to the standard action of the symmetric group S_{n} on A^{n}. (a) What are the invariants for the action of S_{n} on A^{n}? (b) Show that the map from A^{n}/S_{n} to A^{n} that takes an unordered set of roots to a polynomial is an isomorphism of varieties. Assume that k is algebraically closed.
 Show that the rational normal curve of degree 3, parametrized by [t_{0}^{3}:t_{0}^{2}t_{1}:t_{0}t_{1}^{2}:t_{1}^{3}] is equal to the intersection of three quadrics, namely, Z(xzy^{2}, xwyz, ywz^{2}). Show that it is not the intersection of two quadrics.
 Consider Y=Z_{p}(xwyz). (a) Show that xwyz is irreducible, and conclude that Y is irreducible. (b) Show that f : P^{2} → Y given by [x:y:z] → [x^{2}: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 → P^{2} 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).
 Consider the projective twisted cubic Y=Z_{p}(xzy^{2},ywz^{2},xwyz) in P^{3}. (a) Show that f : P^{1} → Y given by [u:v] → [u^{3}:u^{2}v:uv^{2}:v^{3}] 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 P^{1}
 We consider the Grassmannian G(2,4) as the space of lines in P^{3}. The Plucker embedding puts this in P^{5}. Consider two skew lines in P^{3}. Show that the set Q of lines in P^{3} meeting both is the intersection of G(2,4) with a 3plane P^{3} in P^{5}, and so it is a quadric surface. Deduce that Q is isomorphic to P^{1} x P^{1}. What happens if the two original lines meet?
 Let X be an affine algebraic variety. Show that an injective, finite map k[y_{1},...,y_{d}] → k[X] corresponds to a finitetoone surjective map X → A^{d}. Recall that a map of kalgebras 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.
 Prove that the dimension of a hypersurface in A^{n} or P^{n} is n1. 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 Krullâ€™s Hauptidealsatz, which you can find in Dolgachev's book, for example.
 For which [a:b] in P^{1} is the curve C_{[a:b]} = Z((a+b)y^{2}(a+b)x^{3}bx) singular? Sketch these curves over the reals. Bonus: Do the same for the projective closure in P^{2}.
 Find all singular points of the zero sets of the following three polynomials: (x^{2}+y^{2})^{3}4x^{2}y^{2} in k[x,y], xy^{2}z^{2} in k[x,y,z], and xy+x^{3}+y^{3} in k[x,y,z]
 Show that Z(xyz^{2}) in P^{2} is nonsingular. Hint: once you do it for the first affine chart, it suffices to check the points not in that chart.
 Recall that the blowup of an affine variety X along an ideal I=(f_{1},...,f_{m}) is the graph of the map X → P^{m1} sending x to [f_{1}(x):...:f_{m}(x)]. Show that the blowup of any irreducible affine variety along any principal ideal is an isomorphism.
 Show that the blowup of A^{2} along (x^{2},y^{3}) is not smooth (so blowup can make a variety more singular!).
 Show that the blowup of A^{3} along any line is smooth (find the defining equations for the blowup).
 Find the intersection points and multiplicities: (x+z)^{3}+3y^{3}z^{3} and x^{3}+x^{2}(y+z)+xz^{2}+y^{3}
 Find the intersection points and multiplicities: y^{2}zx(x2z)(x+z) and y^{2}+x^{2}2xz
 Find the intersection points and multiplicities: (x^{2}+y^{2})z+x^{3}+y^{3} and x^{3}+y^{3}2xyz
 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.
 Find explicit formulas for the group law on Z(y^{2}xz^{3}x^{3}). Use [0:1:0] as the identity.
 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.
 Use the local ring definition of intersection multiplicity to show that the multiplicity of the intersection of Z(y) and Z(yx^{3}) is 3.
Resources
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
CayleyBacharach Geogebra