Canonical sheaf of total space of vector bundle

Question

Let $X$ be a smooth projective variety and $E$ be a vector bundle on $X$. Let $$Y = Spec(Sym(E^*)).$$ How can I compute the canonical bundle $\omega_Y$ on $Y$?

Answer 1

First, the relative tangent bundle of $Y$ over $X$ is $$ T_{Y/X} \cong p^*E, $$ where $p:Y \to X$ is the projection. Therefore, $$ \omega_{Y/X} \cong p^*\det E^*, $$ and hence $$ \omega_Y \cong p^*\omega_X \otimes \omega_{Y/X} \cong p^*(\omega_X \otimes \det E^*). $$