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Assume $C\subset \mathbb{P}^2$ is a smooth cubic curve. Then there is a cyclic triple cover $\pi: S\rightarrow \mathbb{P}^2$ ramified in $C$. Let $\sigma$ be the covering automorphism (sheet interchange automorphism) associated to $\pi$, i.e. $S/<\sigma>\cong\mathbb{P}^2$, here $\sigma^3=id$. If $H:=\pi^{*}l$ denotes the pullback of a line, then the canonical divisor of $S$ is $K_S=-H$ and $K_S^2=3$. So $S$ can also be seen as the blow up of $\mathbb{P}^2$ in 6 points and is therefore a cubic surface with the famous 27 lines on it.

Looking at the triple cover contruction the lines can be found the following way: $C$ has 9 points of inflection. The preimage of tangent line at such a point decomposes as $\pi^{-1}(l)=E\cup\sigma(E)\cup\sigma^2(E)$, where these are 3 (-1)-curves on $S$, so we get 9$\times$3=27 lines on $S$.

If $\pi: S\rightarrow \mathbb{P}^2$ is the triple cover and we pick 6 mutually skew lines $E_1,\cdots,E_6$ in the preimages of inflection lines, then there is a map $\phi: S \rightarrow \mathbb{P}^2$ such that $S$ is the blow up of $\mathbb{P}^2$ in 6 points $P_1,\cdots,P_6$ and the $E_i$ are the exceptional curves. The strict transforms of the lines in $\mathbb{P}^2$ containing two different points $P_i$ and $P_j$, $1\leq i < j \leq 6$ give 15 (-1)-curves $F_{i,j}$ on $S$. Finally there are six strict transforms of the conics $G_i$ in $\mathbb{P}^2$ containing the $P_j$ for $j\neq i$, $1\leq i \leq 6$.

What can we say about the images of the 27 lines under the automorphism $\sigma$? For example if we pick $E_1$ can we say which lines $\sigma(E_1)$ and $\sigma^2(E_1)$ are in terms of the $F_{i,j}$ and $G_j$, e.g something like $\sigma(E_1)=G_1$? Or is there any other description which tells us exactly which 3 lines are in a preimage of an inflection line?

Background: I recently learned about the "Geiser involution": if one has a double cover $Y$ of $\mathbb{P}^2$ ramified in a smooth quartic $Q$, then $Y$ is the blow up of $\mathbb{P}^2$ in 7 points. $Y$ has 56 (-1)-curves, which arise in the preimages of the 28 bitangents to $Q$. The covering automorphism associated to $Y$ is called the "Geiser involution" and one can describe the images of the 56 (-1)-curves under this involution, see for example arxiv.org/pdf/math/0403245.pdf, Remark 3.3. So i was wondering if there is such a description for a cubic surface and its 27 lines.

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# How to find the action of an automorphism on the 27 lines on a cubic surface?

Assume $C\subset \mathbb{P}^2$ is a smooth cubic curve. Then there is a cyclic triple cover $\pi: S\rightarrow \mathbb{P}^2$ ramified in $C$. Let $\sigma$ be the covering automorphism (sheet interchange automorphism) associated to $\pi$, i.e. $S/<\sigma>\cong\mathbb{P}^2$, here $\sigma^3=id$. If $H:=\pi^{*}l$ denotes the pullback of a line, then the canonical divisor of $S$ is $K_S=-H$ and $K_S^2=3$. So $S$ can also be seen as the blow up of $\mathbb{P}^2$ in 6 points and is therefore a cubic surface with the famous 27 lines on it.

Looking at the triple cover contruction the lines can be found the following way: $C$ has 9 points of inflection. The preimage of tangent line at such a point decomposes as $\pi^{-1}(l)=E\cup\sigma(E)\cup\sigma^2(E)$, where these are 3 (-1)-curves on $S$, so we get 9$\times$3=27 lines on $S$.

If $\pi: S\rightarrow \mathbb{P}^2$ is the triple cover and we pick 6 mutually skew lines $E_1,\cdots,E_6$ in the preimages of inflection lines, then there is a map $\phi: S \rightarrow \mathbb{P}^2$ such that $S$ is the blow up of $\mathbb{P}^2$ in 6 points $P_1,\cdots,P_6$ and the $E_i$ are the exceptional curves. The strict transforms of the lines in $\mathbb{P}^2$ containing two different points $P_i$ and $P_j$, $1\leq i < j \leq 6$ give 15 (-1)-curves $F_{i,j}$ on $S$. Finally there are six strict transforms of the conics $G_i$ in $\mathbb{P}^2$ containing the $P_j$ for $j\neq i$, $1\leq i \leq 6$.

What can we say about the images of the 27 lines under the automorphism $\sigma$? For example if we pick $E_1$ can we say which lines $\sigma(E_1)$ and $\sigma^2(E_1)$ are in terms of the $F_{i,j}$ and $G_j$, e.g something like $\sigma(E_1)=G_1$? Or is there any other description which tells us exactly which 3 lines are in a preimage of an inflection line?