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Normalized Nash Blowups of Toric Varieties

  • Author(s): Bird, Jasper
  • Advisor(s): McKernan, James
  • et al.
Abstract

We study the problem of whether repeated normalized Nash blowups resolve toric singularities. We first review results of Gonzalez-Sprinberg, describing the normalized Nash blowup combinatorially and proving resolution for toric surfaces. We then investigate the problem in higher dimensions, letting $X$ be an affine toric variety of dimension at least $3$ and considering each invariant divisor $Y$ of its normalized Nash blowup $\text{NNB}(X)$. For the case where $Y$ is the strict transform of an invariant divisor $Y_0$ of $X$, we show that the birational map from $Y$ to $\text{NNB}(Y_0)$ is a morphism. For the case where $X$ is $3$-dimensional and $Y$ is any invariant divisor of $\text{NNB}(X)$, we prove the following result. Let $\tau$ be the ray corresponding to $Y$ in the fan of $\text{NNB}(X)$, and let $Z$ be the surface corresponding to $\tau$ in the stellar subdivision (also known as the star subdivision) of the fan of $X$ along $\tau$. Let $\text{MR\textsuperscript{+}}(Z)$ be the surface obtained from the minimal resolution of $Z$ by blowing up all invariant points. Then the birational map from $\text{MR\textsuperscript{+}}(Z)$ to $Y$ is a morphism. For the case where $X$ is a $3$-dimensional quotient singularity and $Y$ is the strict transform of an invariant divisor $Y_0$ of $X$, we deduce that the singularities of $Y$ are milder than those of $Y_0$.

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