Take $f \in \Bbb{Z}[x,y,z,w]$ and $X := Z(f) \subset \mathbf{P}^3_{\Bbb{Q}}$.
We may consider the surface $X_p := Z(f \bmod{p}) \subset \mathbf{P}^3_{\Bbb{F}_p}$.
If $X$ and $X_p$ are smooth then the specialization map is injective
$$\operatorname{Pic}(X^{\mathrm{al}}) \hookrightarrow \operatorname{Pic}(X_p^{\mathrm{al}}) \quad \text{and} \quad \rho(X^{\mathrm{al}}) \leq \rho(X_p^{\mathrm{al}}).$$
Goal
For a given $f$ and $p$, improve the inequality $\rho(X^{\mathrm{al}}) \leq \rho(X_p^{\mathrm{al}})$.
Parity reasons might already force the inequality to not be sharp.
Endomorphisms of the transcendental lattice can complicate things even further.
APPROVED (2026-09-14): [s07-m01] Keep the even-rank fact on slide 6 and the parity use on slide 7. Contact-sheet verdict: APPROVED; no note supplied.
NEEDS APPROVAL (2026-09-14): [s07-m02] Recommendation: Add "homogeneous of degree 4" after "$f\in\Bbb{Z}[x,y,z,w]$". The projective-space correction to $\mathbf{P}^3_{\Bbb{F}_p}$ is applied under s07-c01; only the degree condition remains open. Both setup paragraphs, the theorem, Goal and closing paragraphs retain the source structure. Checked O:L1144-1165 and Stacks 01NF.