Jake Pennel
100 Point Extra Credit Connect the Dots
1.
a.
b.
c.
d.
e.
f.
g.
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a.
Social and Personal topics in the film
The different parts of the brain and how they are used.
Evolution of the human
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 prope
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
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
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 fol
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 ringe
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
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 abel
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. S
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
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 homoge
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 surje
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
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 inver
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
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,
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 ni
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 , . . . , f
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
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
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
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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 irreduc
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
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 a
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) Sh
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, i
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