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 $V$?



Extension of the trivial bundle by the canonical bundle on a curve
If $X$ is a smooth curve, the canonical bundle $\omega\_X$ is nothing but the bundle of differentials $\Omega\_X^1$, and the corresponding extension defines the **jet bundle**
$$
0 \to \Omega\_X^1 \to J\_X \to \mathcal{O}\_X \to 0.
$$


