Let $E$ be a smooth cubic curve in $\mathbb{P}^2$ over an algebraically closed field $k$.

What is the group of the projective transformations preserving $E$ ?

In characteristic $0$ the answer is known (see e.g. https://arxiv.org/abs/1603.09018), that group is a sub-group of the Hesse group, for the generic cubic it has order $18$ and is generated by the translation by order $3$ torsion points and the multiplication by $-1$ (after the choice of a flex point as a neutral element). Moreover, for non-generic curve (which means if the $j$-invariant is $0$ or $1728$) there are other automorphisms (and a group of order $54$ and $64$ respectively).

But what is known about positive characteristic ? Is it also true that the same order 18 group acts (when the characteristic is not $3$) on the generic case ? What about the non-generic case ? and characteristic $3$ ?

Let $E$ be a smooth cubic curve in $\mathbb{P}^2$ over an algebraically closed field $k$.

What is the group of the projective transformations preserving $E$ ?

In characteristic $0$ the answer is known (see e.g. https://arxiv.org/abs/1603.09018), that group is a sub-group of the Hesse group, for the generic cubic it has order $18$ and is generated by the translation by order $3$ torsion points and the multiplication by $-1$ (after the choice of a flex point as a neutral element). Moreover, for non-generic curve (which means if the $j$-invariant is $0$ or $1728$) there are other automorphisms (and a group of order $54$ and $64$ respectively).

But what is known about positive characteristic ? Is it also true that the same order 18 group acts (when the characteristic is not $3$) on the generic case ? What about the non-generic case ? and characteristic $3$ ?

I am particularly interested to know the sub-group inducing translations on the elliptic curve.

Let $E$ be a smooth cubic curve in $\mathbb{P}^2$ over an algebraically closed field $k$.

What is the group of the projective transformations preserving $E$ ?

In characteristic $0$ the answer is known, that group is a sub-group of the Hesse group, for the generic element it has order $18$ and is generated by the translation by order $3$ torsion points and the multiplication by $-1$ (after the choice of a flex point as a neutral element). Moreover, for non-generic curve (which means if the $j$-invariant is $0$ or $1728$) there are other automorphisms (and a group of order $54$ and $64$ respectively).

But what is known about positive characteristic ? Is it also true that the same order 18 group acts (when the characteristic is not $2$ nor $3$) on the generic case ? What about the non-generic case ? and characteristics $2$ and $3$ ?

Let $E$ be a smooth cubic curve in $\mathbb{P}^2$ over an algebraically closed field $k$.

What is the group of the projective transformations preserving $E$ ?

In characteristic $0$ the answer is known (see e.g. https://arxiv.org/abs/1603.09018), that group is a sub-group of the Hesse group, for the generic cubic it has order $18$ and is generated by the translation by order $3$ torsion points and the multiplication by $-1$ (after the choice of a flex point as a neutral element). Moreover, for non-generic curve (which means if the $j$-invariant is $0$ or $1728$) there are other automorphisms (and a group of order $54$ and $64$ respectively).

But what is known about positive characteristic ? Is it also true that the same order 18 group acts (when the characteristic is not $3$) on the generic case ? What about the non-generic case ? and characteristic $3$ ?