Is the blow-up of a （singular）projective variety still projective? The following  is  well-known:
Let$X$ be a  $n$-dimensional projective manifold and $Y$ a closed submanifold
with $\rm{codim}_X Y = s$.   Let $\sigma: \widetilde{X} \rightarrow X$ be the blow-up of $X$ with center $Y$. Then for every positive line bundle $L$ over $X$, the line bundle $\mathcal{O}(-E) \otimes \sigma^{\star}\left(L^{k}\right)$ over $\widetilde{X}$ is positive for $k>0$ large enough.
Then
Question: If $X$ is replaced by a (singular) projective variety (assume irreducible or reduced or some other good property) and $Y$ a subvariety  of $X$, Can one still conclude that $\widetilde{X}$ is still a projective variety? If not, can someone give an counterexample?

Yes. Blowups are projective morphisms, and the composition of projective morphisms is projective, so the blown-up variety is still projective. Once you've established you're working with varieties, no additional properties are needed besides projectivity of your original variety.
(Parentheticals for people interested in maximum generality: one should blow up a finite-type ideal sheaf in order for the blowup to be projective; one should work over a qcqs base to get that the composition of projective morphisms is still projective. Both of these are automatic in the case of varieties.)
