5.A line l in P" is a one-dimensional projective variety of the form { =V,(f1,... f-), 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 lį and l2 are two disjoint lines in P3 and a e P3 a pointnot belonging to l1 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 € A3 a point not belongingto either?

Projective line word is associated with word real projective plane. In a real plane parallel lines…

5. A line l in P" is a one-dimensional projective variety of the form l = Vp(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 e P3 a point not belonging to l1 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 (1 and l2 two ordinary lines and a € A³ a point not belonging to either?

5. A line l in P" is a one-dimensional projective variety of the form l = Vp(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 e P3 a point not belonging to l1 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 (1 and l2 two ordinary lines and a € A³ a point not belonging to either?

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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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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