I followed a course in projective geometry and I'm not sure about 2 things:
- If I have 6 lines in projective space (IP³) with commun secant, why are the 6 corresponding tensors linearly dependent?
- Why do 4 lines in complex projective space always have a line which intersects all 4 lines?
Thanks in advance
If I am correctly interpreting your first question, here's one way to look at it. Every line $\ell$ in $\mathbb P^3$ is given by its Plücker coordinates. That is, if the two points $P=\sum_{j=0}^3 s_je_j$ and $Q=\sum_{j=0}^3 t_je_j$ determine our line $\ell$, then it is given by the coordinates of $[P\wedge Q]\in\mathbb P(\Lambda^2\mathbb C^4)$. We can think of homogeneous coordinates $[x_{01},x_{02},x_{03},x_{12},x_{13},x_{23}]\in\mathbb P^5$.
Suppose $\ell$ intersects the line $x_0=x_1=0$. Thus, we can choose $P$ in this line and so $P= s_2e_2+s_3e_3$ for some $s_2,s_3$, not both $0$. This means that in $P\wedge Q$ the coefficient of $e_0\wedge e_1$ must equal $0$, and so the $01$-coordinate of $\ell$ in our $\mathbb P^5$ must be $0$. If all six of your lines have this property, that means that they all lie in the hyperplane $\{x_{01}=0\}\subset\mathbb P^5$.