4 added 148 characters in body

Maybe the following works?

Let's consider the ringed space $(X, \pi_* O_{\widetilde X})$ and also the ringed space $(\widetilde X, O_{\widetilde X})$. Certainly $\pi_{*}$ gives us an isomorphism $$\text{Pic}(\widetilde X, O_{\widetilde X})) \cong \text{Pic}(X, \pi_* O_{\widetilde X})$$ which sends an $O_X$-line bundle $L$ to the $\pi_* O_X$-module $\pi_* L$, right? (I'm using the fact that $X$ is projective here to show that there exist open trivializing sets containing various finite collections of points). Also observe that $(\pi_* O_{\widetilde X})^{*} = \pi_* (O_{\widetilde X}^{*})$. Thus by Hartshorne Chapter III, Exercise 4.5 (which works on arbitrary ringed spaces), we see that $$\text{Pic}(X, \pi_* O_{\widetilde X}) \cong H^1(X, (\pi_* O_{\widetilde X})^*).$$ Combining with the isomorphisms already written, we obtain: $$\text{Pic}(\widetilde X, O_{\widetilde X}) \cong \text{Pic}(X, \pi_* O_{\widetilde X}) \cong H^1(X, (\pi_{*} O_{\widetilde X})^*) \cong H^1(X, \pi_{*} (O_{\widetilde X}^{*}))$$

3 added 293 characters in body

Maybe the following works?

Let's consider the ringed space $(X, \pi_* O_{\widetilde X})$ and also the ringed space $(\widetilde X, O_{\widetilde X})$. Certainly $\pi_{*}$ gives us an isomorphism $\text{Pic}(X,$ \pi_* text{Pic}(\widetilde X, O_{\widetilde X}X})) \cong \text{Pic}(\widetilde X, text{Pic}(X, \pi_* O_{\widetilde X})). X}) $$which sends an O_X-line bundle L to the \pi_* O_X-module \pi_* L, right? Also observe that (\pi_* O_{\widetilde X})^{*} = \pi_* (O_{\widetilde X}^{*}). Thus by Hartshorne Chapter III, Exercise 4.5 (which works on arbitrary ringed spaces), we see that$$ \text{Pic}(X, \pi_* O_{\widetilde X}) \cong H^1(X, (\pi_* O_{\widetilde X})^*). $$Combining with the isomorphisms already writtencompletes the proof., we obtain:$$ \text{Pic}(\widetilde X, O_{\widetilde X}) \cong \text{Pic}(X, \pi_* O_{\widetilde X}) \cong H^1(X, (\pi_{*} O_{\widetilde X})^*) \cong H^1(X, \pi_{*} (O_{\widetilde X}^{*})) $$2 added 61 characters in body Maybe the following works? Let's consider the ringed space (X, \pi_* O_{\widetilde X}) and also the ringed space (\widetilde X, O_{\widetilde X}). Certainly$$ \text{Pic}(X, \pi_* O_{\widetilde X}) \cong \text{Pic}(\widetilde X, O_{\widetilde X})). $$Also observe that (\pi_* O_{\widetilde X})^{*} = \pi_* (O_{\widetilde X}^{*}). Thus by Hartshorne Chapter III, Exercise 4.5 (which works on arbitrary ringed spaces), we see that$$ \text{Pic}(X, \pi_* O_{\widetilde X}) \cong H^1(X, (\pi_* O_{\widetilde X})^*).  Combining with the isomorphisms already written completes the proof.

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