Let $X$ be a singular normal quartic surface in $\mathbb{P}^3$ over an algebraically closed field $k$ of even characteristic. Is there an quasielliptic fibration on $X$?
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$\begingroup$ What do you mean by "quasielliptic fibration"? A cone over a plane quartic curve is singular and normal, yet it admits no nonconstant morphism to $\mathbb{P}^1$ with connected fibers. $\endgroup$ – Jason Starr Aug 5 '16 at 13:41

$\begingroup$ I mean by "quasielliptic fibration" a nonconstant morphism to $\mathbb{P}^1$ such that the generic fiber is a singular irreducible curve of arithmetic genus 1. $\endgroup$ – Dima Koshelev Aug 5 '16 at 13:53

$\begingroup$ I am interested in the case when X has one elliptic singularity. $\endgroup$ – Dima Koshelev Aug 5 '16 at 13:59

$\begingroup$ @SándorKovács. "If the general fiber is singular, then $X$ is not normal." That is not correct. If the function field of the base $\mathbb{P}^1$ is $k(t)$, and if the generic fiber is affine locally the zero scheme in $\mathbb{A}^2_{k(t)}$ of the polynomial $y^2x^3t$, then the fiber is normal, and even regular, even though it is not smooth over $k(t)$. $\endgroup$ – Jason Starr Aug 6 '16 at 12:33

$\begingroup$ Jason, of course you are right. I got stuck thinking in char 0 even though the OP clearly said otherwise... At the same time you probably didn't write exactly what you meant.... :) $\endgroup$ – Sándor Kovács Aug 6 '16 at 14:11