It is well known that a generic hypersurface of degree $2n-3$ in $\mathbb CP^n$ has finite number of lines. I would like to ask a couple of questions about lines on Fermat hypersurfaces and their symmetries: 


$$\sum\_{i=1}^{n+1}x\_i^{2n-3}=0.$$


Fermat hypersurfaces have a group of automorphisms of order $(2n-3)^n(n+1)!$. In the case $n=3$ (the case of cubic) this group is acting transitively on the collection of $27$ lines and this rases some questions.


The first question is pedagogical, I plan to use it for teaching and really want to know the answer. 


*Question 1.* Is there some slick way to give a high-school proof of the fact that there are exactly $27$ lines on Femat cubic in $\mathbb CP^3$ using (or not) the symmetries of the cubic but without using any theory at all?


Further questions are not for teaching, I am just curious about them.


*Question 2.* Is it known that a Fermat hypersurface of degree $2n-3$ has finite number of lines for any 
$n$? Is it known that these lines are never multiple?


*Question 3.* Can one say something about the number of orbits of the action of symmetries on lines on 
a Fermat hypersurface of degree $2n-3$? For example, what happen in the case of quintic, $n=5$? According
to wiki a generic quintic has $2875=125\cdot 23$ lines, so if Fermat quintic is generic, there should be more than one orbit in the action on lines on it. What is the number of orbits?


I would be happy to know the answer on any of these questions.



Lines on degree 2n-3 Fermat hypersufaces
Regarding question 1, any line in the Fermat cubic $C = \{X\_0^3 + X\_1^3 + X\_2^3 + X\_3^3 = 0\}$ must meet the coordinate hyperplane $H\_0 = \{X\_0 = 0\}$. So which points $x \in (C \cap H\_0)$ can lie on lines? If $Y, Z$ are homogenous coordinates on $T\_x(C \cap H\_0) \cong \mathbb{P}^1$, then the restriction of $X\_0^3 + X\_1^3 + X\_2^3 + X\_3^3$ to $T\_x C$ is of the form $X\_0^3 + F(Y,Z)$ for a homogeneous cubic $F$. For $x$ to lie on a line, $X\_0^3 + F$ must factorise, so $F$ is a cube. This means that $x$ is an inflection point of the plane cubic curve $C \cap H\_0 = \{X\_1^3 + X\_2^3 + X\_3^3 = 0\}$. The inflection points are given by intersection with the zero set of the Hessian determinant $216X\_1X\_2X\_3$. Hence the intersection of any line in $C$ with any coordinate hyperplane must actually have two corrdinates equal to 0, and it follows that the lines consist of $\{X\_0^3 + X\_1^3 = X\_2^3 + X\_3^3 = 0\}$ and its two images under permutating the coordinates (9 lines in each).


P.S. Here is a related exercise I like. Once one has identified the 27 lines in the Fermat cubic $C$, one can use the symmetries of $C$ to guess how to arrange 6 points in $\mathbb{P}^2$ so that the blow-up is isomorphic to $C$, and then write down an explicit rational map $\mathbb{P}^2 \dashrightarrow \mathbb{P}^3$ that maps birationally onto $C$.


