Let $X$ be an affine toric variety corresponding to the cone $\sigma$. If $X$ is smooth, blowups of toric strata correspond to star subdivisions of $\sigma$. Suppose that $X$ is singular and let $D \subseteq X$ be a toric divisor corresponding to a ray $\rho \subseteq \sigma$. What is the subdivision of $\sigma$ corresponding to blowing up $D$? What about the toric strata of higher codimension?
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$\begingroup$ The blow up of a divisor yields an isomorphic map $\endgroup$– HenriDec 21, 2022 at 16:20
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$\begingroup$ @Henri Only if the locus of the blowup is smooth, I am asking about the singular case. E.g. if you blow up the divisor $x=t=0$ in $xy-tw=0$ you resolve the singularity (this corresponds to a toric variety that is the cone over a square). $\endgroup$– Evgeny GoncharovDec 21, 2022 at 16:25
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$\begingroup$ The blowup is an identity if and only if the center is the Cartier divisor (not necessarily smooth). $\endgroup$– SergeyDec 21, 2022 at 20:02
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$\begingroup$ @Sergey Yeah, I just wanted to communicate that there are examples when it is not an isomorphism but you're right of course. $\endgroup$– Evgeny GoncharovDec 21, 2022 at 22:47
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1 Answer
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I guess the answer is that, in general, these subdivisions can be rather complicated. In the divisor case, one can obtain the subdivision as follows (the general case is similar):
Let $\check{\sigma}$ be the dual cone and let $P$ be the monoid of integral points of $\check{\sigma}$. Suppose that $P$ is generated (as a monoid) by $e_1, \dots, e_n$ and take all the generators $e_{k_1}, \dots, e_{k_m}$ that are not in $\rho^{\perp} \cap P$. Then these generate the ideal $I_D$ of $D$ in the monoid algebra $k[P]$.
By reversing the logic in section 3 of https://arxiv.org/abs/1612.09206, the subdivision of $\sigma$ by blowing up $I_D$ is the cut locus of the PL-function $\min\left(\langle \cdot, e_{k_1} \rangle, \dots, \langle \cdot, e_{k_m} \rangle\right)$.
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$\begingroup$ And in the dual picture the answer is quite easy: if the divisor D corresponds to the facet F given by inequality $f≥0$, then to obtain the blowup you just move this facet inside, i.e. replace it by inequality $f≥\epsilon$. $\endgroup$– SergeyDec 24, 2022 at 15:06
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$\begingroup$ The blowup $f : Y \to X$with centre $Z\subset X$ is the terminal object in the category of morphisms $f': Y' \to X$ with $(f')^* Z$ being a Cartier divisor on $Y'$ (pullback in the sense of sheaves of ideals). So you want the minimal subdivision, such that the preimage of your divisor D is Cartier. This means that there exists a piecewise-linear function that takes value $1$ on the ray $\rho$ and takes value $0$ on all other rays (old and new). Then your answer 2 and references solves this problem. $\endgroup$– SergeyDec 24, 2022 at 15:15