Let $C$ be some integral projective curve on some field $k$. Can $C$ consist only singular points (except the generic point)? From Is this true that, any algebraic curve has finitely many singularities? I see that $C$ can contain only finitely many singular points. Hence if $C$ consists only singular ponints (except the generic point, which is always regular under the integral assumption), then it has only finitely many points.
Then $k$ should be a finite field (I dont know how to show this, but I think it should be true). And then is such a $C$?
Just wanna to find an (counter)example while studying projective curves. Any algebraic curves with only singular points, or different assumptions are OK too. Thank you in advanced!