Suppose $f: X \rightarrow Y$ is a finite, flat (hence locally free) morphism of curves (i.e. schemes of dimension 1, not smooth or even reduced). Suppose $L$ is a reflexive sheaf on $X$, locally free of rank 1 at each generic point of $X$.


Is the direct image $f\_\* L$ still reflexive on $Y$? (Better its top exterior power). What if $Y$ is smooth?



Direct image of reflexive sheaf via finite, flat map
Here is a (probably non-optimal) statement that may apply in your situation. In your situation with curves, the hypothesis says that you need $X$ and $Y$ to be Gorenstein.


**Claim.** *Let $X$ and $Y$ be noetherian schemes satisfying $G\_1$ and $S\_2$. If $f\colon X \to Y$ is a finite surjective morphism and $\mathscr{F}$ is a coherent reflexive sheaf on $X$, then $f\_\*\mathscr{F}$ is a coherent reflexive sheaf on $Y$.*


*Proof.* On noetherian schemes satisfying $G\_1$ and $S\_2$, reflexivity is equivalent to being $S\_2$ (in Hartshorne's sense) [[Hartshorne 1994](https://doi.org/10.1007/BF00960866), Thm. 1.9]. The claim then follows since the $S\_r$ property is preserved under pushforward by finite surjective morphisms by [[EGAIV$\_2$](https://doi.org/10.1007/bf02684322), Prop. 5.7.9]. $\blacksquare$


I wanted to prove a statement for non-finite morphisms as well, and for integral schemes, you can say a bit more:


**Claim.** *Let $X$ and $Y$ be integral noetherian schemes satisfying $G\_1$ and $S\_2$. If $f\colon X \to Y$ is a proper dominant morphism with all fibers of the same dimension. If $\mathscr{F}$ is a coherent reflexive sheaf on $X$, then $f\_\*\mathscr{F}$ is a coherent reflexive sheaf on $Y$.*


*Proof.* The fact that $f\_\*\mathscr{F}$ is coherent and normal follows from the proof of [[Hartshorne 1980](https://doi.org/10.1007/BF01467074), Cor. 1.7]. By [[Hartshorne 1994](https://doi.org/10.1007/BF00960866), Rem. 1.11], to show $f\_\*\mathscr{F}$ is reflexive, it therefore suffices to show that it satisfies $S\_1$. But being $S\_1$ is equivalent to torsion-freeness for integral noetherian schemes [[Hartshorne 1994](https://doi.org/10.1007/BF00960866), Lem. 1.5], hence the claim follows by the fact that torsion-freeness is preserved under pushforwards by dominant morphisms. $\blacksquare$


**Edit.** Added the hypothesis that $f$ is surjective in the first claim.


