# Is the blowup of a toric variety corresponding to a subdivision normal?

Toroidal Embeddings 1 by KKMS say subdividing the fan of a toric variety yields the fan of a normalized blowup. How do I avoid normalization? Do I need to choose my subdivision carefully, or is it just that they consider all fractional ideals that aren't necessarily "complete," or integrally closed in the fraction field? That is, if I take the canonical fractional ideal corresponding to a subdivision instead of a bad one, can I avoid normalization?

I would appreciate a reference if possible. I have a vague argument using the fact that the integral closure of the Rees algebra has homogeneous pieces given by the integral closure of the powers of the ideals but it seems unwieldy and unnecessary. I want to make sure I can use a delicate operation that works for blowups but perhaps not normalization.

In many cases, normalization is blowup of a conductor ideal anyway. This might change my anticipated blowup center.

In general, what natural conditions on an ideal or fractional ideal $$I$$ ensure the blowup $$Bl_I X$$ will be normal? Does integral closure of $$I \subseteq \mathcal{O}_X$$, resp. $$I$$ in the fraction field suffice?

Likewise, Niziol's "Toric singularities" paper notes log blowups of log regular schemes are normalizations of blowups, I can't tell if this is because the ideal may not be "saturated" in her terminology or if the normalization is inevitable.

I combed Cox, Little, Schenck but didn't find such a statement.

• Whenever you construct a toric variety using a fan, it is automatically normal. You can subdivide your fan in any way you want. The toric variety corresponding to the new fan will be normal by construction. Commented Feb 19, 2021 at 5:04
• Yes, that's right. But it doesn't address the question of whether normalization took place. Normalization may destroy things about the variety I hold dear. I would like to know when I can avoid it. For example, blowing up the plane at (x, y) or at (x^2, y^2) both correspond to subdivision along the diagonal, but one of them is normal and the other isn't. I don't know if integral closure of the ideal is enough to ensure the blowup is normal already, and that normalization is unnecessary. I apologize if I was unclear. Commented Feb 21, 2021 at 18:00
• It was not clear to me what description you have for your original variety and how you want the the blow up described. What I tried to point out is that "subdividing a fan" would always give you normalized version of whatever construction you are trying to get at. On the other extreme, say you have an $\text{affine}$ toric variety described by a set of monomials, and the exponents for a monomial ideal that you want to blow up, then it is easy to write down the monomials corresponding to the (possibly non-normal) toric variety which is the blow up of that ideal. Commented Feb 22, 2021 at 16:34
• Hi Leo, I don't think asking the ideal to be integrally closed in the structure sheaf is enough to make the blowup normal. For example, see mathoverflow.net/questions/37749/… for a blowup of a normal surface at a maximal ideal yielding something non-normal. Maximal ideals are radical hence integrally closed. To your broader question, I don't yet really understand what you are asking. To me, if I'm using the language of cones it means I must be in the fs cat. There is a non-normal version of toric geom, but IIRC there are no cones. Commented Mar 22, 2021 at 14:17
• Karl Schwede kindly told me if I is equal to the normal closure of I^n for a large enough n, it would result in a normal blowup. Apparently it's in Huneke Swanson, sort of here en.wikipedia.org/wiki/…. Valuative ideals should satisfy that property. I keep forgetting to write this up as an answer to this question. Commented Mar 22, 2021 at 15:15

Huneke-Swanson Proposition 5.2.4: if $$\overline R$$ is the integral closure of $$R$$ in its total ring of fractions and $$I$$ an ideal with integral closure $$\overline I$$.
The integral closure of $$\overline R[It]$$ in its total ring of fractions is given by the Rees-like algebra $$\overline R \oplus \overline{\overline R I} t \oplus \overline{\overline R I^2} t \oplus \overline{\overline R I^3} t \oplus \cdots.$$