Math 203, Problem Set 6. Due Monday, November 19.
For this problem set, you may assume that the ground eld is k = C. Solve the
following problems, and hand in solutions to three of them.
1. (Resolving curve singularities.) Resolve the following Ak plane c
Math 203, Problem Set 3. Due Wednesday October 24.
Hand in (at least) 3 problems from the list below.
For this problem set, you may assume that the ground eld is algebraically closed.
1. (Cubic curves are not rational.) We have seen in the last problem se
Math 203A, Solution Set 4.
Problem 1. (Quadrics are rational.)
(i) Show that a non-singular irreducible quadric Q in P3 can be written in the form
xy = zw
after a suitable change of homogeneous coordinates. Combining this result with
the Segre embedding,
Math 203A, Solution Set 3.
Problem 1. Let k \ cfw_0, 1. Consider the cubic curve E A2 given by the equation
y 2 x(x 1)(x ) = 0.
Show that E is not birational to A1 . In fact, show that there are no non-constant
rational maps
/ E .
F : A1
(i) Write
F (t) =
Math 203, Problem Set 2. Due Monday October 15.
Hand in (at least) 3 problems from the list below.
For this problem set, you may assume that the ground eld is algebraically closed.
1. Find the irreducible components of the ane algebraic set xz y 2 = z 3 x
Math 203, Problem Set 1. Due Friday October 5.
Hand in (at least) 3 problems from the list below.
For this problem set, you may assume that the ground eld is algebraically closed. .
1. Show that the Zariski topology on A2 is not the product of the Zariski
Math 203A, Solution Set 2.
Problem 1. Find the irreducible components of the ane algebraic set xz y 2 = z 3
x5 = 0 in A3 .
Answer: If x = 0, then the two equations imply that y = z = 0. Similarly, if y = 0, then
x = z = 0. Let us assume that x = 0, y = 0
Math 203, Problem Set 4. Due Friday, November 2.
For this problem set, you may assume that the ground eld is algebraically closed.
Solve all problems below. Hand in at least 3 problems from the list below subject to the
rules:
(i) you need to hand in eith
Math 203A, Solution Set 5.
Problem 1. Let a1 , . . . , a2g+1 be pairwise distinct constants. Find the singularities of
the projective hyperelliptic curve of genus g :
y 2 z 2g1 = (x a1 z ) . . . (x a2g+1 z ).
Answer: Let
f (x, y, z ) = y 2 z 2g1 (x a1 z )
Math 203A, Solution Set 8.
Problem 1. Show that the Segre embedding
Pn Pm P(n+1)(m+1)1
has degree
n+m
n
.
Answer: Let m,n be the image of the Segre embedding. Degree homogeneous polynomials on P(n+1)(m+1)1 restrict to m,n as polynomials in the variables x
Math 203, Problem Set 8. Due Friday, December 7.
Solve the following problems, and hand in solutions to three of them.
1. (Degree of the Segre embedding.) Show that the Segre embedding
Pn Pm P(n+1)(m+1)1
has degree
n+m
n
.
2. (Arithmetic genus.) Let X Pn
Math 203, Problem Set 7. Due Wednesday, November 28.
Solve the following problems, and hand in solutions to three of them.
1. (Distinguished open sets.) Let X = Spec(A) be an ane scheme. For each f A,
recall the basic open sets
Xf = cfw_p X : f p.
Show th
Math 203A, Solution Set 7.
Problem 1. Let X = Spec(A) be an ane scheme. For each f A, recall the basic
open sets
Xf = cfw_p X : f p.
Show that
(i) Xf Xg i
(f )
(g ). In particular,
Xf = f is nilpotent,
and
Xf = X f is a unit.
(ii) Show that X is quasi-co
Math 203, Problem Set 5. Due Friday, November 9.
For this problem set, you may assume that the ground eld is algebraically closed.
Solve all problems below. Hand in at least 3 problems from the list. Please hand in at
least one of the last three problems:
Math 203, Solution Set 6.
Problem 1. Resolve the singularities of the following curve by subsequent blow-ups
y 2 xk+1 = 0.
Answer: Assume rst k = 2n is even. Let Cn be the curve y 2 x2n+1 = 0 and let be the
blowup of A2 at the origin. The curve 1 (Cn ) is
Math 203A, Solution Set 1.
Problem 1. Show that the Zariski topology on A2 is not the product of the Zariski
topologies on A1 A1 .
Answer: Clearly, the diagonal
Z = cfw_(x, y ) : x y = 0 A2
is closed in the Zariski topology of A2 . We claim this Z is not