/sci/ - Science & Math

>>11653283
>>11653315
>>11653321
The need for such a relation is my attempt to answer this question.

My idea is something like: all quadrics in [math]\mathbb P^3[/math] are projectively equivalent, and the Segre variety there is a quadric, so it is enough to show that the union of the lines forms a quadric. The lines [math]L,M,N[/math] are given as an intersection of two planes, that is, two linear relations. After some thought, one can try the particularly convenient setup with:

[eqn]L: \begin{cases}X=0\\ Y=0\end{cases} ;\;\;M:\begin{cases}Z=0\\ T=0\end{cases};\;\; N:\begin{cases}X+Z=0\\ Y+T=0\end{cases}[/eqn]Any line through these three lines can be written as [eqn]l:\begin{cases}aX+bY+cZ+dT=0\\eX+fY+gZ+hT=0\end{cases}[/eqn]Subject to the relations [math]af=be,dg=hc,cf+ah=de+bg[/math], as well as them not being proportional and not all coefficients being [math]0[/math] (we can write the former as not all minors of the determinant of the planes of [math]l[/math] are [math]0[/math]).

Now from here I don't know how to turn it into a variety.

An attempt is by considering a point on the union of lines [math]U[/math]; every such line [math]l[/math] can be written as a point of [math]\mathbb P^7[/math] as [math][a:b:c:d:e:f:g:h][/math], and further, it is inside the variety [math]V=V(af-be,dg-hc,cf+ah-de-bg)[/math]. Every point in [math]U[/math] defines a line, so this could define a map [math]U\to V\subseteq\mathbb P^7[/math]. My attempt at describing the minor relations algebraically is to 'refine' this image into perhaps something that is isomorphic to [math]U[/math].

The fact that not both planes are proportional could be taken to be that [math]V[/math] lies inside at least one of the open sets defined by a minor.