Jake Pennel
100 Point Extra Credit Connect the Dots
1.
a.
b.
c.
d.
e.
f.
g.
2.
a.
Social and Personal topics in the film
The different parts of the brain and how they are used.
Evolution of the human species
Creation of the written language
Life and death
Problem session 11
While we will not mention this explicitly in what follows, all schemes are assumed
to be of nite type over an algebraically closed eld k .
Problem 1. Show that if f : X Y is a proper morphism of schemes, then we have the
following semic
Problem session 10
In the rst problem we consider the Segre embedding to show that the product of
two projective schemes is again projective.
Problem 1. Consider two projective spaces Pm and Pn . Let N = (m + 1)(n + 1) 1,
and we denote the coordinates on
Problem session 9
While we will not mention this explicitly in what follows, all schemes are assumed
to be of nite type over an algebraically closed eld k .
Problem 1. Show that if (X, O) is a reduced scheme and if U X is a dense open
subset, then the res
Problem session 8
As usual, all schemes are assumed to be of nite type over an algebraically closed
eld k .
Problem 1. Blow-ups appear naturally when resolving indeterminacies of rational maps,
as follows. Suppose that L is a line bundle on an integral sc
Problem session 8
While we will not mention this explicitly in what follows, all schemes are assumed
to be of nite type over an algebraically closed eld k .
Problem 1. Let X and Y be two locally ringed spaces over k , and let X = U1 . . . Ur
be an open co
Problem session 7
As usual, all schemes are assumed to be of nite type over an algebraically closed
eld k .
Problem 1. Show that if X is a reduced scheme, then X is ane if and only if each
irreducible component of X is ane.
Problem 2. Prove the following
Problem session 7
Problem 1. Let F and F be two subsheaves of the sheaf G . Show that if Fx = Fx for
every x X , then F = F .
Problem 2. Give an example of a surjective morphism F G of sheaves of abelian
groups on X , such that the induced morphism F (X )
Problem session 6
As usual, all schemes are assumed to be of nite type over an algebraically closed
eld k .
Problem 1. Show that if X is an integral projective scheme, then (X, OX ) = k .
Problem 2. Show that if L is an invertible sheaf on a projective in
Problem session 6
Problem 1. Let f : X Y be an arbitrary morphism of (quasi-ane) varieties. For
every x X , we put
e(x) := maxcfw_dim(Z ) | Z = irreducible component of f 1 (f (x), x Z .
Show that the function x e(x) is upper semicontinuous, that is, for
Problem session 5
Problem 1. Let X = Projm(R), and M = iZ Mi a nitely generated graded R-module.
Show that M = 0 if and only if there is d0 such that Md = 0 for all d d0 .
Problem 2. Let I be a homogeneous ideal in R = A[x0 , . . . , xn ], dening the clos
Problem session 5
Problem 1. Let f : X Y be a morphism of algebraic varieties. Suppose that Y is
irreducible, and that all bers of f are irreducible, of the same dimension d (in particular,
f is surjective).
i) There is a unique irreducible component of X
Problem session 4
As usual, all schemes are assumed to be of nite type over an algebraically closed
eld k .
Problem 1. Let X be an integral scheme, and F a coherent sheaf on X . Show that
there is an open subset U X such that F|U is locally free.
Problem
Problem session 3
Problem 1. Let X be a scheme. A family of transition functions (Ui , i,j ) on X is given
by a (nite) open cover X = iI Ui , and by a family (i,j )i,j I , where i,j O(Ui Uj )
is invertible, satisfying the following cocycle condition:
i,j
Problem session 3
Problem 1. Let X be a quasiane variety, and let X1 , . . . , Xr be its irreducible components. Show that there is a canonical isomorphism
K (X )
K (X1 ) K (Xr ).
Problem 2. Let f : X
Y be a birational map between the quasiane varieties X
Problem session 2
Problem 1. Show that if X and Y are topological spaces, with X irreducible, and
f : X Y is a continuous map, then f (X ) is irreducible. Use this to show that the
closed subset
r
Mm,n (k ) = cfw_A Mm,n (k ) | rank(A) r
of Amn is irreduci
Problem session 1
Problem 1. Let (X, OX ) be a ringed space for which there is a nite open cover X =
U1 . . . Ur , such that each Ui is isomorphic as a ringed space to some Spec(Ai ), where
Ai is a nitely generated k -algebra (k is an algebraically closed
Problem session 1
Problem 1. Describe the closed algebraic subsets of A1 .
Problem 2. Show that every algebraically closed eld is innite.
Problem 3. Let k be an innite eld. Show that if f1 , . . . , fm k [x1 , . . . , xn ], then
there is x = (x1 , . . . ,
Problem session 2
Problem 1. Let F be a coherent sheaf on a scheme X . Show that F is locally free of
rank r if and only if Fx is a free OX,x -module of rank r for every x X .
Probem 2. Let f : X Y be a morphism of schemes. Show that if F1 and F2 are
OY -
An irreducibility criterion
This is a solution of the rst problem on Pb. set 5.
Proposition. Let f : X Y be a morphism of algebraic varieties. Suppose that Y is
irreducible, and that all bers of f are irreducible, of the same dimension d (in particular,
f
M4121, MIDTERM TEST
March 220, 2005.
Instructions: Dont be afraid of the number of problems on the test. You
are supposed to do the rst 3 problems and then choose another 3 from the
rest. If you want to do the other problems, you can, but only 6 problems
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Homework Set 2
Solutions are due Friday, September 28th.
Problem 1. Let X be an algebraic prevariety, and consider a nite open cover
X = U1 . . . Un ,
where each Ui is nonempty. Show that X is irreducible if and only if the following hold:
i) Each Ui is i
Homework Set 1
Solutions are due Friday, September 21st.
Problem 1. Let Y be the closed algebraic subset of A3 dened by the two polynomials
x2 yz and xz x. Show that Y is a union of three irreducible components. Describe
them and nd their prime ideals.
Pr
Math 632. Homework Set 5
Solutions are due Tuesday, April 20.
All our schemes are of nite type over an algebraically closed eld k .
Problem 1. Let f : X Y be a morphism of schemes. Show that if F is an OX -module,
and E is a locally free sheaf on Y , then
Homework Set 5
Solutions are due Monday, December 14th.
Problem 1. Let S = k [x0 , . . . , xn ] be the homogeneous coordinate ring of Pn , and let
m = (x0 , . . . , xn ) be the irrelevant ideal.
i) Show that two homogeneous ideals I and J in S dene the sa
Math 632. Homework Set 4
Solutions are due Tuesday, April 6.
All our schemes are of nite type over an algebraically closed eld k .
Problem 1. Show that every automorphism : Pn Pn is linear, that is, it is induced
by an element of P GLn .
Problem 2. Let L1
Homework Set 4
Solutions are due Monday, November 23rd.
As usual, all our schemes are assumed to be of nite type over an algebraically
closed eld k .
Problem 1.
i) Let f : Y X be a closed immersion of schemes. Show that for every scheme Z ,
the induced ma