Does there exist a smooth projective surface over a given field such that every curve on it is smooth？
2026-03-29 16:54:19.1774803259
Smooth surface without singular curves
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No such surfaces exist. Let $O(1)$ be an ample line bundle on a smooth projective surface $X$ and if $M$ is the maximal ideal sheaf of a point $p\in X$, $M^2(n)$ is globally generated for $n\gg 0$ and by Bertini, a general section will give an irreducible curve which is singular at $p$.