Let $f:X\rightarrow Y$ be a flat morphism of factorial projective varieties and denote by $X_{\eta}$ the generic fiber of $f$. Does the inequality $$\rho(X)\leq \rho(X_{\eta}) + \rho(Y)$$ among the Picard numbers hold?
Here is a potential argument: consider the exact sequence $$Pic(Y)\rightarrow Pic(X)\rightarrow Pic(X_{\eta})\rightarrow 0$$ and let $K$ be the kernel of the first map. Then we have an exact sequence $$0\rightarrow K\rightarrow Pic(Y)\rightarrow Pic(X)\rightarrow Pic(X_{\eta})\rightarrow 0$$ So $\rho(X)+\dim(K) = \rho(X_{\eta}) + \rho(Y)$.
Thank you very much.
This is not true, here is a counter example. Elliptic fibrations are flat, this is a consequence of the so called "miracle flatness theorem".
Consider an Enriques surface $X$, (quotient of K3 surface by $\mathbb{Z}_2$), every such surface admits an elliptic fibration $f: X \rightarrow \mathbb{P}^1$. By Bombieri-Mumford every Enriquez surface has picard rank $10$, and since the base has picard rank $1$ and a generic fibre (= smooth elliptic curve) has picard rank $1$, we get a contradiction with the inequality.