Let $X$ be a smooth projective variety over a finite field.
In [Poonen - Bertini theorems over finite fields] it is shown that one can find a smooth geometrically integral hypersurface $S$ of degree $d$ such that the intersection $X \cap S$ is smooth, if one allows the degree $d$ to be high enough. Applying this theorem again to $X \cap S$, one finds another smooth geometrically integral hypersurface $T$ of degree $e$ such that $C \cap S \cap T$ is smooth.
I am looking for a way to ensure that we can find such $S$ and $T$ of the same degree. This is less trivial than it looks: Once we have fixed an $S$ there seems a priori no way to ensure that the degree of $T$ can be chosen to be the same as that of $S$. Is this somewhere in the literature? If not, does someone know a trick to adapt Poonens argument to show what I want?
In [Bucur, Kedlaya - The probability that a complete intersection is smooth], theorem 1.2 seems to be what I am looking for, since it allows to choose multiple hypersurfaces at the same time. However, the hypersurfaces obtained in that way are not guaranteed to be smooth, nor geometrically integral, which is what I really need for my application.
To finish, let me give you the application I had in mind. I need to find a smooth projective variety $Y$ that is birational to $X$ and that admits a proper surjective map to a smooth curve, with generic fibre smooth and geometrically integral. Once I have two hypersurfaces $S, T$ of the same degree as descibed, this follows easily. If someone knows another way to do this, this would also be greatly appreciated.