This is a problem of Hartshorne 1.3.7(b). I have solved in Hartshorne's hint way and I am trying something else.
If $Y\subset P^n$ is projective of dimension $\geq 1$, for any hypersurface $H\subset P^n$, $H\cap Y\neq\emptyset$.
My plan is lifting the variety through cone map to $A^{n+1}$. I want to show $\dim(H\cap Y)\geq 1$. This is done in the similarly way to show any two hypersurface of $P^n$ has non-trivial intersection for $n\geq 2$. So I will denote cone over $Y$ as $Y$ and cone over $H$ as $H$.
Denote coordinate ring of $A^{n+1}$ as $R$. Since $I(H)\subset R$ is generated by single irreducible element, I will consider $(I(H)+I(Y))/I(H)\subset R/I(H)$.
It suffices to show its height is $<n$. I know I can reduce to the case $1\leq \dim(Y)<n-1$ which in turn says $2<\mathrm{ht}(I(Y))\leq n-1$. I wish to show $\mathrm{ht}(P)<n-1$, $P$ minimal over $I(Y)+I(H)$. However, it is not clear I can adjust the chain of $I(Y)$ easily as I do not know whether $I(H)$ is contained in some maximal ideal containing $I(Y)$.
Any hint will suffice.