Very ampleness of $\omega_{C}^n$ Let $C$ be a genus $g$ curve over complex numbers. How can I prove that $\omega_{C}^n$ is very ample for $n\ge2$ if $g=2$ and $n\ge 3$ if $g\ge 3$? Also, I wonder if this still true for other fields other than complex numbers or not. 
Thank you for your help.

Let $C$ be an irreducible,  complete smooth curve over an algebraically closed field $k$ (not necessarily equal to $\mathbb C$).
According to Hartshorne Chapter IV, Corollary 3.2 (b),  a line bundle $L$ on $C$ is very ample as soon as the degree of $L$ satisfies  $$deg(L)\geq 2g+1 \quad (\bigstar)$$
Since $deg(\omega_C)=2g-2$ , we have $deg (\omega_C^{\otimes n})=n(2g-2)$ and because of $(\bigstar)$, the line bundle  $\omega_C^{\otimes n}$ will be very ample as soon as  $n(2g-2)\geq 2g+1$.
So for $g=2$ this translates as $n\geq 5/2$, i.e. $n\geq 3$ and for $g\geq 3$ it translates as $n\geq \frac {2g+1}{2g-2}=1+\frac {3}{2g-2}$ i.e. $n\geq 2$.
The inequalities you mentioned that guarantee very ampleness  are thus justified.
