Suppose $C$ is a smooth projective curve over, say, $\mathbb{C}$. I'm interested in knowing whether the following is true.
Let $\mathcal{L} \in Pic^d(C)$ be a special line bundle, i.e. its $H^1 \neq 0$. Then is $\mathcal{L}^{\otimes 2} \in Pic^{2d}(C)$ also necessarily special? Are there any degree and genus constraints?
In general, the answer is no.
If $C$ has genus $g \geq 3$, take any effective divisor $D$ of degree $d$ such that $g - 1 < d < 2g-2$ and the support of $D$ is contained in an effective canonical divisor. Then $$h^1(C, \, D) = h^0(C, K_C-D) >0,$$ that is, $D$ is special.
On the other hand, $$\deg 2D = 2d > 2g-2 = \deg K_C,$$ hence $2D$ is non-special.
The simplest example is when $C \subset \mathbb{P}^2$ is a plane quartic (which has genus $3$) and $D$ is an effective divisor of degree $3$ made by three collinear points (recall that the canonical series of $C$ is cut out by lines).
