The lifting question

Goal

For a given $f$ and $p$, improve the inequality $\rho(X_{\mathbf{Q}^{\mathrm{al}}}) \leq \rho(X_p^{\mathrm{al}})$.

Which classes in $\operatorname{Pic}(X_p^{\mathrm{al}})$ actually come from $X$?

Every excess counted yesterday is a class that exists in the special fibre and doesn't lift.
Reduction ranks cannot see the difference: they count the special fibre and nothing else.

We will do this by considering the thickenings

$$Z(f \bmod p^i) \subset \mathbf{P}^3_{\mathbf{Z}/(p)^i} \quad i = 1, 2, \ldots$$

Not "which prime" but "which classes": lifting, not counting.