Assigned Exercises (Spring Quarter)

Math 8205
Spring 1998

Assigned exercises

Last updated Thursday, June 18, 1998.

Exercise 1 (Due Friday, April 24)
Exercise 2 (Due Friday, April 24)
Exercise 3 (Due Thursday, June 11 )
Exercise 4 (Due Thursday, June 11 )
Exercise 5 (Due Thursday, June 11 )
Exercise 6 (Due Thursday, June 11 )

The first exercise provides an alternate proof of the claim on p. 40 of the text, specifically that if U is a locally closed subset of Aⁿ, then the image of U under the linear projection of Aⁿ to A^n-1 contains a dense open subset of its image. We start by considering the case where U is closed in Aⁿ.

In the second exercise, we will apply this result to prove that the image of any regular map contains a dense open subset of its closure. So, in some sense, this is a review exercise.

Exercise 1 (Due Friday, April 24)

Let X $\subset$ Aⁿ be a closed subvariety, and let $\pi$ : Aⁿ -> A^{n- 1} be the linear projection. Thus, $\pi$ corresponds to the inclusion of K[x₁,...,x_n_-1] in K[x₁,..., x_n]. Let Y be the closure of $\pi$ (X) in A^{n- 1}. Show that Y is the zeroset of the ideal I(X) $\pi$ K[x₁,..., x_n_-1].

A solution

Let g₁, ..., g_r be a set of generators of I(X) $\cap$ K[x₁,...,x_n_-1]. Show that either:

I(X) is generated by g₁, ..., g_r (considered as elements of K[x₁,..., x_n]), or
If f $\in$ I(X) is an element of minimal positive x_n-degree, written in the following form:

f(x₁,..., x_n) = h₀ + h₁ x_n + ... + h_d x_n^d,

₀

h_d

$\in$

₁

x_n

_-1

h_d

$\neq$

I(X)

₁

x_n

h_d

i.e.,

I(X)· K[x₁,..., x_n] [1/h_d],

₁

g_r

A solution

With the same notation as before, show that either:

$\pi$ (X) is closed in A^n-1, and X $\cong$ $\pi$ (X) $\times$ A¹, or
If U = D(h_d) Aⁿ, then U $\cap$ Y $\pi$ (X), and the restriction of $\pi$ to $\pi$ ^-1 (U $\cap$ Y) $\cap$ X is finite-to-one and surjective (onto U $\cap$ Y).

A solution

Let U Aⁿ be a locally closed subset, and let $\pi$ : Aⁿ -> A^{n- 1} be the linear projection. Show that $\pi$ (U) contains a dense open subset of its closure in A^{n- 1}.

A hint

A somewhat different hint

A solution

Exercise 2 (Due Friday, April 24)

Let m $\leq$ n, and let X Aⁿ be a closed subvariety. Show that if $\pi$ : Aⁿ -> A^m is a linear projection, then $\pi$ (X) contains a dense open subset of its closure.

Hint.

n - m

A more detailed hint

A solution

Show that if X is an affine variety, and $\varphi$ : X -> Aⁿ is a regular map, then $\varphi$ (X) contains a dense open subset of its closure.

A solution

Show that if X is a quasi-projective variety, and $\varphi$ : X -> Pⁿ is a regular map, then $\varphi$ (X) contains a dense open subset of its closure.

A solution

Exercise 3 (Due Thursday, June 11 ) Let Q

P⁵ be a smooth quadric hypersurface: more specifically, Q = G(1,3), the Grassmannian of lines in P³. Recall from last quarter that every line on Q is of the form L = $\Sigma$ _p,H, where p is a point in P³ and H is a plane in P³. Let W be the universal hyperplane in P³ $\times$ (P³)*. Thus, W is the set of all pairs (p,L) such that p $\in$ L. Finally, let F₁(Q)

G(1,5) be the Fano variety of lines in Q.

Show that there is a regular map $\varphi$ : W -> G(1,5) whose image is F₁(Q).

A solution

Show that F₁(Q) is irreducible, and determine its dimension.

A solution

Exercise 4 (Due Thursday, June 11 ) This problem is about the secant variety of the Veronese n-fold X = v₂(Pⁿ)

P^N, where n $\geq$ 2, and N = (n+1)(n+2)/2 - 1 = n(n+3)/2. (The exact value of N isn't particularly important here, however.) Recall from the lecture that the secant deficiency satisfies $\delta$ $\geq$ 1, so that dim(Sec X) $\leq$ 2n. Specifically, every general point of Sec X lies on (at least) every line in some 1-parameter family of secant lines of X. We proved that fact by observing that if L is a line in Pⁿ, then C := v₂(L) is a curve of degree 2 in some plane $\Pi$

P^N. Thus, v₂(Pⁿ) $\cap$ $\Pi$ = v₂(L), and each point of $\Pi$ lies on every secant line of C in some family parametrized by P¹, and hence on every line in a 1-parameter family of secant lines of X. This is illustrated in the following figure. (See pp. 144-145 of the text for the case n = 2.) Veronese secant line diagram

In this exercise, our goal will be to show that the inequality $\delta$ $\geq$ 1 is in fact an equality.

Assume that n $\geq$ 3, let p,q,r,s be 4 non-coplanar points of Pⁿ, and consider the lines L₁: = <p,q> and L₂: = <r,s>. Thus, L₁ and L₂ are skew lines. Let $\Pi$ ₁ and $\Pi$ ₂ be the planes that contain the curves C₁ := v₂(L₁) and C₂ := v₂(L₂) respectively. Show that $\Pi$ ₁ $\cap$ $\Pi$ ₂ is empty.
Let L₁ and L₂ be distinct coplanar lines in Pⁿ, and let $\Pi$ ₁ and $\Pi$ ₂ be defined as in part a. Show that $\Pi$ ₁ $\cap$ $\Pi$ ₂ is a single point of X = v₂(Pⁿ).
Let z be a point of Sec(X) - X. Show that the set of secant lines passing through z is irreducible and 1-dimensional.
Show that dim (Sec X ) = 2n.

Exercise 5 (Due Thursday, June 11 ) Let A

B be an integral ring extension.

Assume that B is an integral domain. Show that A is a field if and only if B is a field.
Let P be a prime ideal in B, and let p = P $\cap$ A. Show that p is a maximal ideal in A if and only if P is a maximal ideal in B.
Let S be a subset of A which is closed under multiplication, and let S^-1A be the ring whose elements are all quotients f/s, where f $\in$ A and s $\in$ S. (As usual, two quotients f/s and g/t, are equal if and only if ft - gs is anninilated by some element of S.) We define S^-1B similarly.

^-1

Let p be a prime ideal in A, and let S = A - p. Show that if P is a prime ideal in B which is maximal with respect to the property that P $\cap$ S is empty, then the extended ideal P^e = P· S^-1B is a maximal ideal in S^-1B.
Use the result of part d. to prove the existence part of the Lying Over theorem.

Exercise 6 (Due Thursday, June 11 ) Let X be a smooth projective variety. Assume that dim X = r, and that X

P^{2r + 1}. If L is a line in P^{2r +1}, and L $\cap$ X is empty, then $\pi$ = $\pi$ _L: X -> P^{2r - 1} is the projection whose center is L. Show that if L is a general line in P^{2r + 1}, then the following statements hold:

there exist at most finitely many points p $\in$ X such that (d $\pi$ )_p : T_p(X) -> T_(p)(P^{2r
- 1}) is not injective.

A solution

If D = {p : p $\in$ X and $\pi$ ^-1( $\pi$ (p)) contains 2 or more distinct points of X }, then dim(D) $\leq$ 1.
A hint is linked here.
A solution is linked here.

REMARK: The meanings of "general" will be slightly different in part a. and part b. Thus, the lines which satisfy the each of two conditions correspond to points of two different dense open subsets of the Grassmann variety G(1,2r + 1). Of course, the lines which satisfy both conditions correspond to points in the intersection of the two open subsets.

Comments and inquiries to: roberts@math.umn.edu

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Math 8205 Spring 1998

Assigned exercises

Contents

Math 8205
Spring 1998