Projective automorphisms of a plane cubic curves

Question

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.

Answer 1

You said you are particularly interested in the group of translations. Building on what dhy said, this group has a normal subgroupthe subgroup inducing translations is always $E[3]$, since translation by $\sigma \in E$ acts by adding $n\sigma$ on line bundles of degree $n$, and thus preserves a degree $n$ line bundle if and only if $E$ is $n$-torsion. So the translation group has order $9$ if the characteristic is not $3$ or less if the characteristic is $3$.

Note that every automorphism of the curve can be composed with a translation to preserve the identity element, and any automorphism preserving the identity element automatically preserves the fixed line bundle of degree $3$ (its divisor being thrice the identity element). It follows that the group of automorphisms preserving the identity is a subgroup of the projective automorphism group, whose projection onto the quotient by translations is an isomorphism. In other words, the projective automorphism group is a semidirect product of the translation group with the group of automorphisms preserving the identity.

The group of automorphisms preserving the identity is classified: In characteristic greater than $3$ it has the usual three possibilities, cyclic of order $2$, $4$, or $6$, whereas in characteristics $2$ or $3$ there are nonabelian groups arising from supersingular elliptic curves.