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Toric blowup of a line
Question: Let $X$ be a toric variety corresponding to a complete fan $\Delta$. Is it true that a toric blowup of a line in $X$ is a line?
The case of smooth blowup is well understood. Any line in a smooth variety is the zero set of a non-zero regular function on the variety. Therefore in a smooth blowup, if the zero set of the regular function is blown up, it is a curve.
But I am not sure about non-smooth blowup. This is because, I am not sure that the regular function has a zero set which is the blown up version of the zero set of the regular function. For example, when the fan $\Delta$ in the above example is the fan of $Z(x_1^2 – x_2^2 – x_3^2 + x_4^2)$, the line $(1,0,0,0)$ has a regular function $x_1^2 – x_2^2$. My understanding is that it is not a zero set in the smooth blowup of the variety, so it is not the zero set of a regular function. Therefore the line itself is not blown up in the smooth blowup.
Does anyone know if a line can be blown up by a blowup with non-smooth center? Thanks.
A:
So, the answer is yes. In fact, this is true for any Gorenstein toric Fano variety. And this is Proposition 7.2 of the following paper,
At the beginning of Section 7, I have used this Lemma for an explicit toric computation of the minimal volume of F

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