Example of an ample line bundle $L$ with $L^{\otimes m}$ very ample and $L^{\otimes (m+1)}$ not very ample

Let $X$ be a smooth projective variety over $\mathbb{C}$. Is there an ample line bundle $L$ such that $L^{\otimes m}$ is very ample, but $L^{\otimes(m+1)}$ is not very ample?

I expect such an $L$ to exist, though I have not been able to construct one. Of course, there is a threshold $M > 0$ such that $L^{\otimes m}$ is very ample for all $m \geq M$ (this is Matsusaka's theorem), but my interest is in the interval of $m$'s before this threshold is reached.

If this is true or false under different hypotheses (say, over a field of positive characteristic or by removing/weakening smoothness), I'd be interested to hear it!

Solution 1:

Take a smooth quartic curve $C$ in the projective plane with a point $P\in C$ such that the tangent line at $P$ meets it only in $P$ (four times). For example, you may take $C$ to be defined by $x^4+y^4+yz^3=0$ and $P=(0,0,1)$. Then $\mathcal{O}_C(4P)=\mathcal{O}_C(1)=K_C$ and thus very ample. But $\mathcal{O}_C(5P)=K_C+P$ always has $P$ as a base point and thus not very ample.