Exercise 1 (Due Friday, October 17)
Let X be the curve with a triple point which was
considered in the second example in the notes, and let
:
A1
- {i, -i} -> X be the parametrization
constructed there.
Show that
extends to a parametrization
:
P1 -> Y, where
Y is the projective closure of X. Find the points
in P1 which are mapped
to the origin in A2
(homogeneous coordinates [x,y,z] = [0,0,1] in
P2 ) and find the points
in P1 which are mapped
by to points at infinity on the
projective closure of X.
The solution to Exercise 1 is linked here.
Exercise 2 (Due Friday, October 17)
Let X be the plane quintic curve whose defining
equation is x5 +
y5 + xy(x - y) = 0.
Recall that this curve has a triple point at (0,0). Let
L be
the line x + y = 2, and let
:
X - {(0,0)} -> L be the (central) projection
from (0,0).
A sketch of this curve, showing a similar projection, is linked here. (If you are printing this exercise, you also may want to print a copy of the sketch.)
(a) Show that
is a generically two-to-one mapping from X - {(0,0)} to
L. Determine which points of L are omitted
from the image.
(b) We can identify L with the affine line A1 by means of the parametrization t -> (1 - t , 1 + t ).
Use this identification to solve locally for (x,y) as a
function of t , thereby obtaining locally defined parametrizations
+:
U ->X and
-:
U -> X, which are defined in terms of square roots of
rational functions of t . Here, U is a suitable
open subset of A1. Show
that there are finitely many values of t where the
parametrization cannot be defined (regardless of which square root
is chosen), and find these values. Also, determine which values of
t are mapped to (0,0).
SUGGESTION: One can make the calculations look reasonably nice by working with the auxiliary variable z , defined by the relations x = (1 - t )z and x = (1 + t )z .
(c) Determine the image of the set of real points under
the projection
:
X - {(0,0)} -> L.
(d) Consider the restrictions of the parametrizations
+
and -
to the set of real points of
A1. Find the maximal
intervals on which each of these maps is defined.
The solution to Exercise 2 is linked here.
Exercise 3
(Due Monday, November 10)
= Exercise 2.17 in the text.
Exercise 4
(Due Monday, November 10)
= Exercise 2.18 in the text.
NOTE: In the definition of rational quartic curves
which is referenced in this exercise, one should assume that
1.
This is because if = 1, then the map is undefined
at the point [, 1].
Exercise 5
(Due Monday, November 10)
= Exercise 2.22 in the text.
Exercise 6 (due Monday, November 24)
Let X be an affine variety in An, i.e. a Zariski closed subset of An, and let I(X) be its ideal in K[x1,..., xn], i.e. the ideal whose elements are all polynomials f such that f(p) = 0 for every p in X. Prove that if X is irreducible, then I(X) is a prime ideal.
A HINT for Exercise 6 is linked here.
Exercise 7 (due Monday, November 24)
- (a) Let
f(x1,...,
xn) and
g(x1,...,
xn) be elements of
K[x1,...,
xn] with GCD = 1. Prove that there
exist
polynomials a(x1,...,
xn) and
b(x1,...,
xn) such than af + bg is
an element of
K[x1,...,
xn-1]. Thus:
a(x1,..., xn) f(x1,..., xn) + b(x1,..., xn)g (x1,..., xn) = h(x1,..., xn-1) for some h in K[x1,..., xn-1].
HINT: Consider f(x1,..., xn) and g(x1,..., xn) as elements of K(x1,..., xn-1) [xn],
where K(x1,..., xn-1) is the fraction field of K[x1,..., xn-1]. - (b) Let f(x,y) and g(x,y) be elements
of
K[x,y], and let X and Y be the zero sets of
f and g respectively in
A2. Show that if f
and
g have GCD = 1, then X
Y is a finite set of points.
HINT: Show first that there are only finitely many possibilities for the x-coordinate of a point of XY.
- (c) Show that if f(x,y) is an irreducible polynomial, then the zeroset of f is an irreducible variety in A2.
A HINT for part (c) of Exercise 7 is linked here.
Exercise 8 (due Wednesday, December 3)
- Let f =
f(x1,...,
xn) be a
polynomial of degree d
in n variables over an algebraically closed field K,
and write f
as a sum of homogeneous polynomials:
f(x1,..., xn) = F0 (x1,..., xn) + F1 (x1,..., xn) + ... + Fd (x1,..., xn), where Fd is nonzero. Show that there is a (homogeneous) linear change of variables in the polynomial ring K[x1,..., xn] such that f is monic relative to the last variable in the new set of generators. Explicitly, show that there exist elements t1,..., tn which are homogeneous of degree 1 in x1,..., xn such that:- K[x1,..., xn] = K[t1,..., tn], and:
- If f is expanded as a polynomial in the new set of generators, then tnd occurs with coefficient = 1.
- Let X be a hypersurface in
An with
defining equation
f(x1,...,
xn) = 0. Assume that
f is monic
with respect to xn .
[By the result of part
(a), this is always possible after a linear change of variables.] Let
:
An ->
An
-1
be the usual parallel projection. Thus:
- The condition K[x1,..., xn] = K[t1,..., tn] is equivalent to saying that the homogeneous linear expressions t1,..., tn are linearly independent.
- If f is written as a sum of homogeneous polynomials,
specifically:
f(x1,..., xn) = F0 (x1,..., xn) + F1 (x1,..., xn) + ... + Fd (x1,..., xn), where Fd is nonzero, then the property of f being monic relative to xn is really just a property of the highest degree form Fd .
- Let X = V(f) and Y =
V(g) be hypersurfaces in
An. Assume that
f
and g have GCD = 1, and let Z = X
Y. Show that
(Z)
An
-1, or more
specifically, that there is a hypersurface in
An
-1 that
contains (Z).
HINT: Apply one of the results from Exercise 7. - Let f(x1,...,
xn) be an irreducible
polynomial in
n variables. Show that the hypersurface V(f)
An is irreducible.
A HINT for part (d) of Exercise 8 is linked here.
SOME COMMENTS about the solution of Exercise 8 are linked here.
Exercise 9
(due Wednesday, December 3)
= Exercise 3.9 in the text.
Exercise 10
(optional: please submit by 10 a.m. on
Thursday, December 11)
= Exercise 2.19 in the text.
Exercise 11
(optional: please submit by 10 a.m. on
Thursday, December 11)
Let Pk
be identified with a linear subspace
Pk
Pn, and
let 
Pn be a
linear subspace of dimension n-k-1 such that
Pk is empty.
Let :
Pn -
->
Pk
be the projection from .
Suppose that is
defined by the equations:
Pn, then:
(HINT: If we write in the usual way:
Li =
j
cijXj ,
then there is a set of k+1 columns such that the corresponding
submatrix of the matrix of coefficients
cij is invertible. Try to show that
one can arrange for this submatrix to be the identity matrix, and choose the
linear subspace
Pk
Pn to be
suitably related to these column indices.
Comments and inquiries to: roberts@math.umn.edu
Back to my homepage: http://www.math.umn.edu/~roberts
Last updated December 9, 1997.
