Let $X$ be a Toric Variety and let $x\in X$ be a point (not necessarily smooth). Then the blow up $Bl\_{x}X$ of $X$ in $x$ is Toric. 
Let $Y:=WBl\_{x}X$ be the weighted blow up of $X$ in $x$ with weights $a\_{1},...,a\_{n}$.


**Is $Y$ Toric ?**


In particular, **is a weighted blow up of a weighted projective space Toric ?**



Weighted blow up of a Toric Variety
(Essentially reposting Jesus Martinez Garcia and Karl Schwede's comments as an answer)


If you are blowing up a torus-invariant sheaf of ideals, then the Rees algebra (the thing you take Proj of to get the blow-up) has grading by characters of the torus, so the blowup has an action of the torus, so it is toric (since it also contains a dense copy of the torus that acts on it). Thus, if your weighted blow-up is weighted "along the coordinate hyperplanes", it will be toric.


If you blow up a non-invariant sheaf of ideals, you do not get something toric in general. If you blow up a non-invariant point of $\mathbb A^2$, then the blow-up *map* already cannot be toric, even though both varieties are toric. Blowing up two different points of $\mathbb A^2$ gives a total space which has no toric structure at all: the two (-1)-curves must be torus-invariant and blowing them down gives a toric map to $\mathbb A^2$, but any toric blow-up of $\mathbb A^2$ must have the exceptional locus lying over a single point.


