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Relative cone of curves and contractions of a successive blowup

  • Yuto Masamura
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Abstract

Let X be a normal variety in arbitrary characteristic, and let X̃ be the successive blowup along subvarieties Z1,dotsc,Zn⊆ X of codimension at least two that have simple normal crossings and satisfy Zhnot⊇ Zi whenever h<i. We prove that the relative cone of curves NE̅(X̃/X) is generated by the classes of finitely many elementary curves, and that every face admits a contraction over X. Moreover, we prove that every small face contraction admits a D-flip for every ℝ-Cartier divisor D that is relatively anti-ample. We also compute the relative Cox sheaf and prove its finite generation.

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Cite

BibTeXRIS
BibTeX
@misc{masamura2026relative,
  title = {{Relative cone of curves and contractions of a successive blowup}},
  author = {Yuto Masamura},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.18936},
  eprint = {2609.18936},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.18936},
}
RIS
TY  - UNPB
TI  - Relative cone of curves and contractions of a successive blowup
AU  - Yuto Masamura
PY  - 2026
DA  - 2026/10/12
PB  - arXiv
DO  - 10.48550/arxiv.2609.18936
UR  - https://arxiv.org/abs/2609.18936
AB  - Let X be a normal variety in arbitrary characteristic, and let X̃ be the successive blowup along subvarieties Z_1,dotsc,Z_n⊆ X of codimension at least two that have simple normal crossings and satisfy Z_hnot⊇ Z_i whenever h<i. We prove that the relative cone of curves NE̅(X̃/X) is generated by the classes of finitely many elementary curves, and that every face admits a contraction over X. Moreover, we prove that every small face contraction admits a D-flip for every ℝ-Cartier divisor D that is relatively anti-ample. We also compute the relative Cox sheaf and prove its finite generation.
ER  -