Let $X$ be a smooth projective curve over a field $k$ and $K_X$ be its canonical line bundle. By the Serre duality, $\text{H}^1(X,K_X)$ is a one-dimensional $k$-vector space. On the other hand, $\text{H}^1(X,K_X)=\text{Ext}^1(O_X,K_X)$ so this vector space corresponds to the set of extensions of $O_X$ by $K_X$. Let $V$ be such an extension that corresponds to a nonzero vector in $\text{H}^1(X,K_X)$. This is a $2$-dimensional vector bundle on $X$.

**Question.** Is there exists a simple/concrete description of $V$?