Let $S$ be a smooth projective surface over $\mathbb{C}$ and $L$ be an ample line bundle on $S$. Let $d$ be a positive integer such that $dL$ is very ample and $D$ a very general member of the linear system $|dL|$. Let $T_D \rightarrow S$ be the degree $d$ cyclic cover branched along $D$. Assume that the Picard number $ \rho(S) =1 $.

Q.** (Edited)** Is the picard number $\rho(T_D)$ $1$ for a

**sufficiently large**integer $d$?