Let $f: X \to S$ be a smooth morphism of schemes of relative dimension $1$. Then $K=\Omega^1_{X/S}$ is a line bundle on $X$. I am interested in the following question:
When does $\Omega_{X/S}$ have a square root $K^{1/2}$ on $X$?
Example: start with a rank $2$ vector bundle $E$ over $S$, we could consider the projective bundle $\mathbb P(E) \to S$. Then we could compute $K$ via the relative Euler sequence (see Canonical sheaf projective bundle).
Motivation: Assume $f: X \to S$ is a family of smooth projective curves of genus $g>0$ over $\mathbb C$ (in particular, the fiber of $f$ at any closed point is a smooth projective curve over $\mathbb C$). Then fiberwise on $S$ the answer is yes.
For example, we could consider the universal curve over $M_g$ the moduli space of curves and consider good locus of $M_g$ where $K^{1/2}$ exists.
The case I am most interested in is when $S$ is also a smooth projective curve.