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Line intersecting three lines in $mathbbP^4$ that are not in one hyperplane

0

$begingroup$

Given three lines $L,M,N$ in $mathbbP^4$, not all in one hyperplane, I want to show by example that it is possible that there are multiples lines intersecting $L,M$ and $N$.

What I know:
A projective line through two points $P=(p_0:...p_n), Q=(q_0:...q_n)$ is defined by first moving these points to $mathbbR^n+1$, and then we have

$$
PQ=<P,Q>=lambda p_0 + mu q_0 : ... : lambda x_n + mu q_n mid (lambda,mu)neq (0,0).
$$

So a line PQ in projective space is a plane through the lines OP and OQ in Euclidian space. (I think)

How do I have to think about these lines (planes?) in $mathbbP^4$ and how would I find lines intersecting all these lines?

projective-geometry

share|cite|improve this question

edited Mar 27 at 21:43

The Coding Wombat

asked Mar 27 at 20:54

The Coding WombatThe Coding Wombat

342111

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Would this work? Assume L and M intersect in a point P. By the dimension formula we have that $langle L, M rangle cap N=emptyset$. Choose any point $Q$ on $N$ and draw a line from $P$ to $Q$. You now have multiple lines.
  $endgroup$
  – The Coding Wombat
  Mar 28 at 21:19
  

  
  
  
  

add a comment | 

0

$begingroup$

Given three lines $L,M,N$ in $mathbbP^4$, not all in one hyperplane, I want to show by example that it is possible that there are multiples lines intersecting $L,M$ and $N$.

What I know:
A projective line through two points $P=(p_0:...p_n), Q=(q_0:...q_n)$ is defined by first moving these points to $mathbbR^n+1$, and then we have

$$
PQ=<P,Q>=lambda p_0 + mu q_0 : ... : lambda x_n + mu q_n mid (lambda,mu)neq (0,0).
$$

So a line PQ in projective space is a plane through the lines OP and OQ in Euclidian space. (I think)

How do I have to think about these lines (planes?) in $mathbbP^4$ and how would I find lines intersecting all these lines?

projective-geometry

share|cite|improve this question

edited Mar 27 at 21:43

The Coding Wombat

asked Mar 27 at 20:54

The Coding WombatThe Coding Wombat

342111

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Would this work? Assume L and M intersect in a point P. By the dimension formula we have that $langle L, M rangle cap N=emptyset$. Choose any point $Q$ on $N$ and draw a line from $P$ to $Q$. You now have multiple lines.
  $endgroup$
  – The Coding Wombat
  Mar 28 at 21:19
  

  
  
  
  

add a comment | 

$begingroup$

Given three lines $L,M,N$ in $mathbbP^4$, not all in one hyperplane, I want to show by example that it is possible that there are multiples lines intersecting $L,M$ and $N$.

What I know:
A projective line through two points $P=(p_0:...p_n), Q=(q_0:...q_n)$ is defined by first moving these points to $mathbbR^n+1$, and then we have

$$
PQ=<P,Q>=lambda p_0 + mu q_0 : ... : lambda x_n + mu q_n mid (lambda,mu)neq (0,0).
$$

So a line PQ in projective space is a plane through the lines OP and OQ in Euclidian space. (I think)

How do I have to think about these lines (planes?) in $mathbbP^4$ and how would I find lines intersecting all these lines?

projective-geometry

share|cite|improve this question

edited Mar 27 at 21:43

The Coding Wombat

asked Mar 27 at 20:54

The Coding WombatThe Coding Wombat

342111

$endgroup$

share|cite|improve this question

edited Mar 27 at 21:43

The Coding Wombat

asked Mar 27 at 20:54

The Coding WombatThe Coding Wombat

342111

share|cite|improve this question

edited Mar 27 at 21:43

The Coding Wombat

asked Mar 27 at 20:54

The Coding WombatThe Coding Wombat

342111

asked Mar 27 at 20:54

The Coding WombatThe Coding Wombat

342111

asked Mar 27 at 20:54

The Coding WombatThe Coding Wombat

342111