In $R^5$ we further consider the constant level set algebraic hypersurfaces of the multi lener polynomial
$$c =x_1x_4 + x_2x_5 + x_2x_3x_4 - x_1x_2x_3x_4 - x_1x_2x_4x_5 - x_2x_3x_4x_5 + x_1x_2x_3x_4x_5.$$
How many straight lines are included in each hypersurface ?
As an example, the constant level set zero hypersurface contains the linear varieties $Ox_3x_4x_5 :x_1 = 0;x_2 = 0; Ox_1x_3x_5 : x_2 = 0;x_4 = 0; Ox_2x_3 : x_1 = 0;x_4 = 0;x_5 = 0$. Indeed, we have
$$c =x_1(x_4 - x_2x_3x_4 - x_2x_4x_5 + x_2x_3x_4x_5)+ x_2(x_5 + x_3x_4- x_3x_4x_5) .$$
Can anyone help me to study this case ?
Mentioning some sources or lecture.
If $c = 0$, the $3$-plane given by $x_1 = 0, x_2 = 0$ is contained in your surface. So if $c = 0$ you have infinitely many lines contained in the hypersurface.
Similarly, for $ c \neq 0$, again the $2$-plane $x_2 = 0, x_1 = 1, x_4 = c$ is contained in your hypersurface, so again it contains infinitely many lines.
In general, determine how many lines an hypersurface contains can be tricky, for example a cubic surface in $\mathbb P^3$ contains $27$ lines (I am working with complex coefficients) but it's not easy to prove.