5.A line l in P" is a one-dimensional projective variety of the form l =V,(f1,..., fr), where each f; is (homogeneous) of degree 1.a) Show that there are two disjoint lines in F" if and only if n > 3.b) Show that if ly and l2 are two disjoint lines in P3 and a e P a pointnot belonging to lı U l2, then there exists a unique line containing a andintersecting l1 and l2.c) What further conditions need to be imposed for b) to be true in the affinesetting, with l1 and l2 two ordinary lines and a E A³ a point not belongingto either?

From the given information. The line l in is a one-dimensional projective variety of form, .

A line l in P" is a one-dimensional projective variety of the form { = 5. V,(f1,..., fr), where each f; is (homogeneous) of degree 1. a) Show that there are two disjoint lines in P" if and only if n > 3. b) Show that if lị and l2 are two disjoint lines in P³ and a € P³ a point not belonging to l1 U l2, then there exists a unique line containing a and intersecting (1 and l2.

A line l in P" is a one-dimensional projective variety of the form { = 5. V,(f1,..., fr), where each f; is (homogeneous) of degree 1. a) Show that there are two disjoint lines in P" if and only if n > 3. b) Show that if lị and l2 are two disjoint lines in P³ and a € P³ a point not belonging to l1 U l2, then there exists a unique line containing a and intersecting (1 and l2.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Concept explainers

Maximization

Have you ever seen a bridge in the shape of a parabola? Where is its maximum on the bridge? The maximum of that bridge is the highest point on that bridge. The process of finding the maximum of the bridge here is called the maximization.

Question

5.
A line l in P" is a one-dimensional projective variety of the form l =
V,(f1,..., fr), where each f; is (homogeneous) of degree 1.
a) Show that there are two disjoint lines in F" if and only if n > 3.
b) Show that if ly and l2 are two disjoint lines in P3 and a e P a point
not belonging to lı U l2, then there exists a unique line containing a and
intersecting l1 and l2.
c) What further conditions need to be imposed for b) to be true in the affine
setting, with l1 and l2 two ordinary lines and a E A³ a point not belonging
to either?

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