Let $X$ be a very general surface of degree $\ge 5$ in $\mathbb{P}^3$ and $ Y$ is arbitrary irreducible cubic hypersurface. Is $X \cap Y$ reduced ? 



Reducedness of complete intersection
Yes (of course, $Y$ should be reduced). Since $X$ is very general we may assume $\operatorname{Pic}(X)=\mathbb{Z}\cdot [\mathscr{O}\_X(1)]$. If the divisor $Y\_{|X}$ on $X$ is not reduced, it is of the form $2H+H'$, where $H$ and $H'$ are hyperplane sections of $X$ (possibly equal). But since the restriction map $H^0(\mathbb{P}^3,\mathscr{O}\_{\mathbb{P}^3}(3))\rightarrow H^0(X,\mathscr{O}\_X(3))$ is injective, this implies that $Y$ is non reduced. 


