| Math 431 Algebraic Geometry Homework Page |
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ALL HOMEWORK SHOULD BE WRITTEN ON STANDARD A4 WHITE PAPER
AND SHOULD
BE IN FINAL FORM.
LaTeX OUTPUT WILL BE GRADED WITH MORE TOLERANCE!
HW 1 (due 16 Sept 1998):
Show that if R^3 is given the standard topology, then the quotient
topology on P^2 is Hausdorff.
HW 2 (due 16 Sept 1998):
Assume that we define P^2 as the set of equivalence classes on R^3
where (a,b,c) is related to (e,f,g) in R^3 when (ka,kb,kc)=(e,f,g)
for some k in R. Show that in this case P^2 is not Hausdorff.
HW 3 (due 16 Sept 1998):
Show that P^1 and the unit circle in R^2 are the `same'. i.e. you
can find an invertible 1-1 polynomial mapping from one onto the other.
Try this also with P^2 and the unit sphere in R^3...!
HW 4 (due 16 Sept 1998):
Draw the cone in R^3 given by x^2=yz. Show how we obtain y=x^2 as a
plane section of this cone.
HW 5,6 (due 23 Sept 1998 Wed):
Show that any two plane curves intersect. (We work in P^2 over
an algebraically closed field. You will probably need to derive some
basic properties of resultants and apply them to the two polynomials
in three variables defining the curves. Some library browsing might
prove useful! Start for instance from Cox, Little & O'Shea's books
"Ideals, Varieties and Algorithms" (QA564.C688 (next time you find
the access code yourself...)) and "Using Algebraic Geometry" (I have
a copy) and Abhyankar's "Algebraic Geometry for Scientists and
Engineers." [WARNING: One pitfall in solving complicated looking
homework problems is to give complicated looking but otherwise
meaningless solutions with the hope that the instructor will make
something out of it. He never does!!]
HW 7 (due 23 Sept 1998 Wed):
Solve Problem 2.11 on page 41 of the text book. In particular
explain what Reid means by saying "..., and the point of the
question is that the usual construction gives a group law on the
complement of the singular point."
HW 8 (due 23 Sept 1998 Wed):
Find out what Noether's Fundamental Theorem, sometimes known as
Noether AF+BG Theorem, is and how it provides an immediate proof of
Corollary 2.7 on page 33 of your book. (Explain but don't prove
Noether's theorem. Explain how it applies to Cor 2.7.)
HW 9 (due 2 November 1998 Monday):
Let k be an algebraically closed field, X and Y affine algebraic
sets, F:X--->Y a morphism of varieties with F^*:k[Y]--->k[X] the map
induced by F on the coordinate rings. Prove or disprove the
following statements.
i) If F is injective then F^* is surjective.
ii) If F^* is surjective the F is injective.
iii) If F is surjective then F^* is injective.
iv) If F^* is injective then F is surjective.
Where did you use the assumption that k is algebraically closed?
Now solve Exercise 4.4 on page 77 of Reid's book.
HW 10 (due 2 November 1998 Monday):
Let X=D(f) for some f in k[x_1,...,x_n], and O(X) the ring of
regular functions on X. Show that O(X) is isomorphic to the
localization of the ring k[x_1,...,x_n] at the multiplicative set
{1,f,f^2,f^3,...}. (Notation: k[x_1,...,x_n]_f). Show that this
localization is isomorphic to the quotient ring k[x_1,...,x_{n+1}]/I
where I=(x_{n+1}f(x_1,....,x_n)-1). We now showed that O(X) is
isomorphic to k[Y] where Y=V(I). Can we conclude that X is
isomorphic to Y?
HW 11 (due 2 November 1998 Monday):
Let k be an algebraically closed field and let A^n be the affine
n-space. Show that A^n-{(0,...,0)}, the complement of the origin, is
not isomorphic to an affine variety. Where did you use the algebraic
closure of k?
HW 12 (due 2 November 1998 Monday):
Let k be algebraically closed, X=V(y-x^2), Y=V(xy-1) in A^2. For any
irreducible quadratic polynomial f in k[x,y], show that V(f) is
isomorphic either to X or to Y. Where did you use k algebraically
closed?
(Hartshorne, Algebraic Geometry, page 5, Ex 1.1)
HW 13 (due 2 November 1998 Monday):
Show that a k-algebra B is isomorphic to the affine coordinate ring
of some algebraic set in A^n, for some n, if and only if B is a
finitely generated k-algebra with no nilpotent elements. Where does
nilpotent condition come into play?
(Hartshorne, Algebraic Geometry, page 6, Ex 1.5)
HW 14 (due 11 November 1998 Wednesday):
Exercise 5.2, on page 90-91.
HW 15 (due 11 November 1998 Wednesday):
Exercise 5.10, on page 92.
HW 16 (due 11 November 1998 Wednesday):
The last sentence on page 89 claims that it is easy to see that the
image is given as the vanishing of certain minors. It is clear that
these minors vanish on the image. Prove that any polynomial
vanishing on the image is in the ideal generated by these minors.
HW 17 (due 11 November 1998 Wednesday):
Exercise 7.2 on page 110.
HW 18 (due 9 December 1998 Wednesday):
Exercise 7.3 on page 111.
HW 19 (due 9 December 1998 Wednesday):
Exercise 7.4 on page 111. Note here that the reference to (7.3,(b))
should read (7.4, (b)).
HW 20 (due 9 December 1998 Wednesday):
Draw a diagram which shows all the 27 lines of a cubic and indicate
their intersection properties.
That is all for this term...