Let $Y$ be a smooth projective scheme and $X$ a projective subscheme of $Y$. Let us consider the blowup morphism of $Y$ along $X$, denotated by $\pi_{X} : \widetilde{Y} \longrightarrow Y$.
Let $\mathcal{F}$ be a coherent sheaf of rank $r$ in $Y$ and $\widetilde{\mathcal{F}} = \pi_{X}^{*}(\mathcal{F})$.
What is the definition of $\text{deg}(\widetilde{\mathcal{F}})$? Does the exceptional divisor contribute to the $\text{deg}(\widetilde{\mathcal{F}}) $ "calculation"? Is there any relationship between $\text{deg}(\widetilde{\mathcal{F}})$ and $\text{deg}(\mathcal{F})$?
References and suggestions on this subject will be welcome.
Thanks in advance.