Does there exists a cubic or quartic $3$fold $X\subset\mathbb{P}^4$ such that $Sing(X)$ is a smooth curve $C$ of genus $g(C)\geq 2$ and $X$ has $A_1$singularities along $C$?
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3$\begingroup$ No. If $\operatorname{Sing}(X) =C$ every bisecant line to $C$ must be contained in $X$. This implies easily that $C$ is rational. $\endgroup$ – abx Mar 4 at 18:05

$\begingroup$ You are right. Thank you. Do you know if there exists a quartic 3fold with this property? $\endgroup$ – japin Mar 4 at 18:55

1$\begingroup$ I think so. Let $C$ be a general complete intersection of 3 quadrics in $\mathbb{P}^4$, given by $P=Q=R=0$. The quartic threefold $X$ defined by $P^2+Q^2+R^2=0$ has $\operatorname{Sing}(X)=C $. Now I don't know what you call "$A_1$singularities along $C$", but the singularities of $X$ along $C$ are as nice as they can be. $\endgroup$ – abx Mar 4 at 19:56