Does exist a smooth, complex, projective variety $X$ of dimension $d\geq2$ such that $X$ does not contain smooth, complex, projective curves of wichever genus $g\geq2$?

**Answer by Bertie**: No, it does not exists.

More in general, does previous statement hold for $X$ over any field of characteristic $0$?

**Partial Answer**: If the field is algebraically closed: no, it does not.

**Open question**: Does exist a smooth projective variety $X$ of dimension $d\geq2$ over a field $\mathbb{K}$ not algebraically closed of characteristic $0$, such that $X$ does not contain smooth projective curves of wichever genus $g\geq2$?