Let $C$ be a generic curve (over algebraically closed field of characteristic $0$) of genus $g\geq10$ and $\eta$ a non-trivial torsion line bundle of level $l\geq3$ i.e. $\eta^{\otimes l}\cong\mathcal{O}_C$. Is it true that $\omega_C\otimes\eta$ is very ample? I don't know whether this question is silly or whether I missed something, but I did not find any references giving determined answer. It seems that many authors working on paracanonical curves presumed this. I found some references talking about the Prym-canonical curves, for example Lange and Sernesi, but the condition $\eta^{\otimes2}\cong\mathcal{O}_C$ is important there.
To show the very ampleness, it is equivalent to check $h^0(\eta^\vee+A+B)=0$ for all $A,B\in C$. In other words, it suffices to show $\eta\notin C_2-C_2$. If $g$ is odd, it is solved by Chiodo, Farkas, Eisenbud and Schreyer by the arguments of difference varieties. So in particular, I'm curious about the even genus case.
Some further related questions:
1.Is the natural map $\phi_{\omega_C\otimes\eta}:C\to\mathbb{P}^{g-2}$ projectively normal?
2.(For people familiar with Koszul cohomology)The Prym-Green Conjecture A said $K_{\frac{g}{2}-3,2}(C,\omega_C\otimes\eta)=0$ for even genus. Is it easy/trivial to prove a weaker result: $K_{\frac{g}{2}-4,2}(C,\omega_C\otimes\eta)=0$?
3.In particular, is the paracanonical model of a generic curve of genus $g\geq10$ cut out by quadrics?
