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.

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