Let $ X $ and $ Y $ be integral curves over some perfect field $ k $ and suppose that $ X $ is smooth. Moreover, let $ \pi\_1 : Y \to X $ and $ \pi\_2 : Y \to X $ be finite morphisms such that the function field $ k( Y ) $ of $ Y $ is the composite field $ \pi\_1^\*( k( X ) ) \cdot \pi\_2^\*( k( X ) ) $ where $ \pi\_i^\* : k( X ) \to k( Y ) $ denotes the pullback of $ \pi\_i $ for all $ i = 1, 2 $.


Finally, let $ q $ be a closed point on $ Y $ and let $ Q $ be a place in $ k( Y ) $ which dominates $ q $ (not sure if this is the right terminology; I mean that $ Q $ contains the maximal ideal in the local ring of $ q $).


Is it possible that $ Q $ is unramified in $ k( Y ) / \pi\_i^\*( k( X ) ) $ for all $ i = 1, 2 $ but $ q $ is a singular point on $ Y $?


Moreover, if this can happen:
Can this also happen if $ Y $ is even a projective curve which is contained in $ X \times\_k X $ and $ \pi\_1 $ and $ \pi\_2 $ are the restrictions of the canonical projections $ X \times X \to X $?


I don't believe that it matters. But just to be sure.


Thank you



Ramification and singular points
Sure. Take $X = \mathbb P^1$, $Y$ a nodal curve in $\mathbb P^1 \times \mathbb P^1$ given by some equation like $x^2 - y^2 + x y^2$. The point $(0,0)$ is singular but splits into two places in the function field.


Each of these two places is unramified since it is a degree $2$ extension in of both $k(x)$ and $k(y)$, so every pair of places lying over the same place of $k(x)$ or $k(y)$ is a pair of unramified places.


