# What is the subdivision corresponding to the blowup of a toric divisor of a singular toric variety?

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?

• The blow up of a divisor yields an isomorphic map Commented Dec 21, 2022 at 16:20
• @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). Commented Dec 21, 2022 at 16:25
• The blowup is an identity if and only if the center is the Cartier divisor (not necessarily smooth). Commented Dec 21, 2022 at 20:02
• @Sergey Yeah, I just wanted to communicate that there are examples when it is not an isomorphism but you're right of course. Commented Dec 21, 2022 at 22:47

1. 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 $$k[P]$$.
2. 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 $$\min\left(\langle \cdot, e_{k_1} \rangle, \dots, \langle \cdot, e_{k_m} \rangle\right)$$.
• 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 $f≥\epsilon$. Commented Dec 24, 2022 at 15:06
• 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. Commented Dec 24, 2022 at 15:15