Can anyone give an example of a projective, regular, geometrically reduced but non-smooth curve ?
Of course, the base field should be imperfect.
In Exercise 4.3.22 of Qing Liu's book *Algebraic Geometry and Arithmetic Curves*, a regular but non-smooth curve is given. But that curve is not geometrically reduced.



A regular, geometrically reduced but non-smooth curve
I believe a classic example is the curve define in $\mathbb P^2\_{\mathbb F\_p(t)}$, with coordinates $(x:y:z)$, by the equation


$$ t x^p + z^{p-1} y + y^p=0$$


for $p>2$.


Differentiating with respect to $y$, one can see that the curve is smooth wherever $z \neq 0$, and substituting in $z=0$, one can see the curve is smooth away from a single geometric point $(1:t^{1/p}:0)$. Since the derivatives of the defining equation with respect to $x,y,$ and $z$ all vanish at this point, using $p>2$ for the $z$ derivative, this curve is indeed non-smooth at that point. But since it has only one non-smooth point, it is geometrically reduced.


However, the curve is regular at this nonsmooth point, because the derivative with respect to $t$ of the defining equation is nonzero there.


