A question on effective divisors Let $X$ be a projective variety with two morphisms $f:X\rightarrow Y$ and $g:X\rightarrow Z$ with irreducible fibers of positive dimension. Assume that $Pic(X) = f^{*}Pic(Y)\oplus g^{*}Pic(Z)$. Then if $D$ is a divisor on $X$ we can write $D = f^{*}D_Y + g^{*}D_Z$, where $D_Y,D_Z$ are divisors on $Y$ and $Z$ respectively.
If $D$ is effective are then $D_Y$ and $D_Z$ effective as well?
This holds for instance when $X = \mathbb{P}^n \times \mathbb{P}^m$ is a product and $f,g$ are the projections onto the factors.
 A: This is false.  The original post, included below, had some mistakes.
The example of @Pop is better, and in fact that example is where I started.  It is straightforward to modify that example into an example satisfying the constraints.  If @Pop wants to add an answer, then I am happy to delete this answer.
Let $Y$ be the projective plane.  Fix a point $p$ in $Y$, and denote by $Z'$ the $1$-dimensional projective space parameterizing lines $L$ in $Y$ that contain $p$.  Let $Z$ be the product of $Z'$ with a projective space $W$ of dimension $\geq 1$.  Let $X'$ be the parameter of ordered pairs $(q,L)$ of a points of $Y$ and a line $L$ in $Y$ that contains both $p$ and $q$.  Let $X$ be the product of $X'$ with $W$.  Then, as a subvariety of the product $Y\times Z$ considered as a projective space bundle (of relative dimension $2$) over $Z$, the variety $X$ is a projective space subbundle over $Z$ (of relative dimension $1$).  Thus, the pullback homomorphism from $\text{Pic}(Y)\oplus \text{Pic}(Z)$ to $\text{Pic}(X)$ is an isomorphism.
Now let $D'$ be the divisor in $X'$ parameterizing pairs $(q,L)$ such that $q$ equals $p$, and let $D$ be $D'\times W$.  The normal bundle of $D'$ in $X'$ is anti-ample.  Thus, the normal bundle of $D$ in $X$ is not nef.  For the reason mentioned above, the divisors $D_Y$ and $D_Z$ are not both effective.
Original post.
Here are the details.  Fix a vector space $V$ of dimension $4$ together with a linear subspace $U$ of dimension $1$.  Denote by $\text{Flag}(1,3;V)$ the partial flag variety parameterizing ordered pairs $(A,B)$ of a $1$-dimensional linear subspace $A$ contained in a $3$-dimensional linear subspace $B$ contained in $V$.  Denote by $X$ the closed subvariety of $X$ parameterizing ordered pairs such that $U$ is contained in $B$.
Let $Y$ be $\mathbb{P}V$, the parameter space of $1$-dimensional linear subspaces $A$ of $V$.  Let $Z$ be the linear $2$-plane in $\text{Grass}(3,V)$ that parameterizes $3$-dimensional subspaces $B$ of $V$ that contain $U$, i.e., the dual projective space of $V/U$.  Denote by $f$ and $g$ the forgetful morphisms from $X$ that remember only $A$, respectively $B$.
The fiber of $f$ over every point of $Y$ other than $[U]$ is a projective space $\mathbb{P}^1$.  The fiber of $f$ over $[U]$ is the full projective space $Z = \mathbb{P}(V/U)^\vee \cong \mathbb{P}^2$.  In all cases, the fiber is irreducible of positive dimension.  The fiber of $g$ over every point of $Z$ is a P^1 $\mathbb{P}^2$.
In fact, the embedding of $X$ in $Y\times Z$ realizes $X$ as a projective space subbundle (of relative dimension 1 $2$)over $Z$ inside the (constant) projective space bundle $Y\times Z$ over $Z$ (of relative dimension $2$).  From this and the formula for the Picard group of a projective space bundle, it is straightforward to see that the Picard group of $X$ is the isomorphic image under pullback of $\text{Pic}(Y)\oplus \text{Pic}(Z)$.
Now consider the divisor $D$ in $X$ parameterizing pairs $(A,B)$ such that $A$ equals $U$.  The divisor $D_Y$ is effective, linearly equivalent to a hyperplane class in the projective space $Y$.  However, the divisor $D_Z$ is not effective.  In fact, it is linearly equivalent to the negative of the hyperplane class in the projective space $Z$.
