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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 (short) way to prove 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.

2 added 33 characters in body

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 want plan to use it for teaching and really want to know the answer.

Question 1. Is there some slick (short) way to prove that there are exactly $27$ lines on Femat cubic in $\mathbb CP^3$ using 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.

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# Lines on degree 2n-3 Fermat hypersufaces

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 want to use it for teaching.

Question 1. Is there some slick (short) way to prove that there are exactly $27$ lines on Femat cubic in $\mathbb CP^3$ using 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.