Download Algebraic Geometry I - Problem Set 5

Algebraic Geometry I - Problem Set 5
Please pick eight of the following problems. Please write up solutions as legibly and clearly
as you can, preferably in LaTeX.
Morphisms of projective varieties:
1. Consider the n-Veronese embedding of P1 :
v : P1 → Pn ,
[u, v] 7→ [un , un−1 v, . . . , v n ]
(where n is positive integer). Let V be its image.
(a) Let I be the homogeneous ideal of the graded ring R = k[x0 , . . . , xn ] generated by the
2 × 2 minors of the matrix:
x0 x1 . . . xn−1
x1 x2 . . . x n
Prove that V is a projective variety with homogeneous coordinate ring R/I.
(b) Prove that the map v is an isomorphism onto its image. The image is called a rational
normal curve. The case n = 2 is a conic (we did this in class). The case n = 3 is a
twisted cubic (projective version).
2. Prove that any automorphism P1 → P1 is of the form [x, y] 7→ [ax + by, cx + dy], for some
a, b, c, d ∈ k such that ad − bc 6= 0. This is often written in dehomogenized form
az + b
z 7→
cz + d
(i.e., write z for [1, z]). This is also called the group of Möbius transformations.
Problems on quadrics/conics. Assume that the characteristic of k is not 2.
3. Prove that the quadrics X : wx = yz and Y : w2 + x2 + y 2 = z 2 in P3 are isomorphic.
Describe both one-parameter families of lines in Y .
4. Prove that all irreducible conics in P2 (i.e., projective varieties in P2 defined by an irreducible homogeneous polynomial of degree 2 in k[x0 , x1 , x2 ]) are isomorphic to P1 . (Hint: use
the classification
of quadratic forms over k: there is a change of basis such that the quadratic
P
form is 0≤i≤j yi2 , for some j ∈ {0, 1, 2}.) Give an example of two non-isomorphic quadrics
in P3 - you don’t have to prove they are not isomorphic. (Your example should not include
quadratic forms that are squares of linear forms.)
5. Recall from class that the image of the Segre map P1 ×P1 → P3 is the quadric X : wx = yz.
Prove that the only lines in P3 contained in X are the images of the fibers of the two projection
maps πi : P1 × P1 → P1 , i = 1, 2.
1
2
Linear Algebra.
A set of points {p1 , . . . , pl } in Pn is said to be in general position if the corresponding l
vectors in An+1 = k n+1 have the following property: for any 1 ≤ j ≤ n + 1, any j of them are
linearly independent (i.e., they span an affine space of dimension j). Two ordered sets of l
points {p1 , . . . , pl }, {q1 , . . . , ql } in Pn are projectively equivalent if there is an automorphism
φ of Pn such that φ(pi ) = qi , for all i.
5. Prove that any two ordered sets of n + 2 points in general position in Pn are projectively
equivalent. For l ≤ n, prove that l + 1 points in general position in Pn are contained in a
unique linear subspace of dimension l, i.e., a projective variety which is the vanishing locus
of n − l linearly independent linear forms (homogeneous polynomials of degree 1).
Some comments/hints/suggestions:
• Try to prove that a linear subspace of dimension l is ismorphic to Pl . We call lines
lines linear subspaces of dimension 1, planes linear subspaces of dimension 2, etc.
• You are asked to prove here in the projective setting what you know from linear
algebra: through 2 distinct points there passes a unique line, through 3 non-collinear
points, there passes a unique plane, etc.
• Hint: Try to prove that any n + 2 points in general position in Pn are projectively
equivalent to the following points:
p1 = [1, 0, . . . , 0],
p2 = [0, 1, . . . , 0],
...
pn+1 = [0, 0, . . . , 1],
pn+2 = [1, 1, . . . , 1].
6. Prove that through five points in general position in P2 there passes a unique irreducible
conic.
Have a guess: how many lines in P3 intersect four general lines? Take the word “general”
to loosely mean that “nothing changes” if we move any of the lines in any direction. Here is
one way to come up with the answer.
7. Let L and M be two disjoint lines in P3 and let P be a point in P3 \ (L ∪ M ).
a. Prove that there is a unique line through P intersecting L and M .
b. Prove that if N is another line that is disjoint from L and M , then there is a unique
quadric surface in P3 (which up to a change of coordinates is xw = yz!) that contains
the three lines L, M, N .
Now note that if a fourth line K is not contained in the above quadric, it must (intuitively
at least) intersect the quadric at two points! (Why?) Any line that intersects all four lines
must be therefore contained in the quadric, passing through the two points of intersection of
K with the quadric and be in the “other ruling” than the one to which L, M, N belong. The
answer is: 2 lines!
3
Here is a concrete question (you don’t have to write this up!): how many lines intersect
the following four lines in P3 (x, y, z, w are the coordinates on P3 ):
L1 : x = y = 0,
L2 : z = w = 0,
L3 : x = y, z = w,
L4 : x + 2y = z + w, x + 2w = y + z.
Miscellanea:
8. Prove that if X is a topological space obtained by gluing irreducible topological spaces
X1 and X2 along open sets U ⊆ X1 , V ⊆ X2 via a homeomorphism f : U → V , then X is
irreducible. Explain how you would generalize this in the case when you glue more than two
spaces.
9. Prove that the product X × Y of two varieties X, Y is a variety. (Only argue why the
product is separated - you don’t need to argue why it is a prevariety.) Prove that X × Y
is irreducible if X, Y are both irreducible. (We proved this when X and Y are affine. Only
argue how to use the gluing construction to prove this in general).
10. Let X ⊆ An be a closed subvariety. Identify An with D+ (x0 ) ⊆ Pn and let X be
the closure of X in Pn . Prove that I(X) = I(X)∗ ⊂ k[x0 , . . . , xn ], where I(X)∗ is the
ideal generated by the homogeneous polynomials f ∗ obtained from elements f ∈ I(X) ⊂
k[x1 , . . . , xn ] by “homogenizing” (i.e., if f = fd + . . . + f0 , with each fi a homogeneous
polynomial of degree i in k[x1 , . . . , xn ], then f ∗ = fd + fd−1 x0 + . . . + f0 xd0 ).
11. Let X be a Hausdorff topological space. Prove that X is compact if and only if for any
topological space Y , the projection X × Y → Y is a closed map (maps closed sets to closed
sets).