Is it true that a general curve of genus 3 is a plane section of an appropriate Kummer surface in $\mathbb P^3$? By Kummer surface I mean image of a principally polarized Abelian surface w.r.t. the map defined by the complete linear system $|2\Theta|$, where $\Theta$ is a theta divisor of polarization.


This assertion agrees with the naive dimension count, but I have no idea whether it is true or false.


Thanks in advance,  

Serge



General curves of genus 3 as plane sections of Kummer surfaces
Let $C$ be a non hyperelliptic curve of genus 3, let $f\colon C'\to C$ be an e'tale double cover and let $A$ be the Prym variety of $f$. Then $A$ is a principally polarized surface and the Abel-Prym map embeds $C'$ in $A$ as an element of $|2\Theta|$. Multiplication by $-1$ on $A$ restricts on $C'$ to the involution induced by $f$, hence the image of $C'$ via the map given by $|2\Theta|$ is a plane section isomorphic to $C$.


A reference for these facts is chapter 12 of the book "Complex abelian varieties'' by Birkenhake-Lange. 


