## STNB 2016 (30è any)

### On loci of smooth plane curves with non-trivial automorphism groups

Eslam Farag

#### Resum

In the STNB2005 non-singular plane curves of degree 4 were extensively studied, the aim in the talk is to obtain certain generalizations for any degree $d\geq 4$.

Let $M_g$ be the moduli space of smooth, genus $g$ curves over an
algebraically closed field $K$ of zero characteristic. Consider the locus $M_g^{Pl}$ where
$$M_g^{Pl}:=\{\delta\in M_g|\,\,\exists\, a\,\,smooth,\,\,\, plane\,\, model\,\, for\,\, \delta \}.$$
Now, if $G$ is a finite non-trivial group then define the loci $M_g^{Pl}(G)$ and $\widetilde{M_g^{Pl}(G)}$ as
\begin{eqnarray*}
M_g^{Pl}(G)&:=&\{\delta\in M_g^{Pl}|\,G\cong\, a\, subgroup\, of\, Aut(\delta)\},\\
\widetilde{M_g^{Pl}(G)}&:=&\{\delta\in M_g|\,G\cong Aut(\delta)\}.
\end{eqnarray*}
In particular, we have $\widetilde{M_g^{Pl}(G)}\subseteq M_g^{Pl}(G)\subseteq M_g^{Pl}\subseteq M_g$.

This talk is devoted to present the results, which have been obtained on these loci. For instance, some aspects on the irreducibility of $\widetilde{M_g^{Pl}(G)}$ and its interrelation with the existence of "normal forms", the analogy of Henn's and Komiya-Kuribayashi's results on quartic curves, but for degree $5$ curves (jointly with Francesc Bars), and also an account on the set of twists of such loci (jointly with Francesc Bars and Elisa Lorenzo).

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