Lemma 19.7.27`B1ED8431B1B9`

Lemma 19.7.27.B1ED8431B1B9 Given $S\in \mathcal{S}'(C)$, we have $S\sim \operatorname{Reg}(S)$.

Proof. Since $S$ and $\operatorname{Reg}(S)$ have the same Okounkov body, we have $\operatorname{vol}S=\operatorname{vol}\operatorname{Reg}(S)$ by theorem 19.2.5. By corollary 19.2.8 again,

\[ d_{\mathrm{sg}}(\operatorname{Reg}(S),S)=\operatorname{vol}\operatorname{Reg}(S)-\operatorname{vol}S=0. \]

This completes the proof. $\square$

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BibTeX citation

@misc{pome-tag-B1ED8431B1B9,
  author = {{Pome contributors}},
  title = {Lemma 19.7.27},
  howpublished = {Pome},
  url = {https://mingchenxia.github.io/Pome/tag/B1ED8431B1B9/}
}

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Lemma 19.7.27B1ED8431B1B9

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Lemma 19.7.27`B1ED8431B1B9`