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PreprintCC BY 4.0arXiv

Seshadri slope K-semistability for the blow-up of projective space along a linear subspace

  • Nathan Grieve
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Abstract

We study a particular instance of the frame work from \cite{Grieve:CM:Line:Slope:Stab}, building on earlier work of Arezzo-et-al \cite{Arezzo:DellaVedova:LaNave}, Ross and Thomas \cite{Ross:Thomas:2007}, \cite{Ross:Thomas:2006} and others, which applies the theory of the Chow-Mumford line bundle to determine the Donaldson-Futaki invariant of deformation to the normal cone test configurations with respect to big and nef line bundles. In particular, here we consider the case of projective n-space blown-up along a linear subspace. Our main result arises as an application of \cite[Theorem 1.1]{Grieve:CM:Line:Slope:Stab}. It establishes K-semistability of the deformation to the normal cone test configuration for the blow-up PP(OshPPs⊕ r ⊕ OshPPs(1)) ≃ BlPPr−1(PPs) along the exceptional divisor and with respect to the tautological line bundle OshPP(OshPPs⊕ r ⊕ OshPPs(1))(1).

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Cite

BibTeXRIS
BibTeX
@misc{grieve2026seshadri,
  title = {{Seshadri slope $\K$-semistability for the blow-up of projective space along a linear subspace}},
  author = {Nathan Grieve},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.12319},
  eprint = {2609.12319},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.12319},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Seshadri slope K-semistability for the blow-up of projective space along a linear subspace
AU  - Nathan Grieve
PY  - 2026
DA  - 2026/09/11
PB  - arXiv
DO  - 10.48550/arxiv.2609.12319
UR  - https://arxiv.org/abs/2609.12319
AB  - We study a particular instance of the frame work from \cite{Grieve:CM:Line:Slope:Stab}, building on earlier work of Arezzo-et-al \cite{Arezzo:DellaVedova:LaNave}, Ross and Thomas \cite{Ross:Thomas:2007}, \cite{Ross:Thomas:2006} and others, which applies the theory of the Chow-Mumford line bundle to determine the Donaldson-Futaki invariant of deformation to the normal cone test configurations with respect to big and nef line bundles. In particular, here we consider the case of projective n-space blown-up along a linear subspace. Our main result arises as an application of \cite[Theorem 1.1]{Grieve:CM:Line:Slope:Stab}. It establishes K-semistability of the deformation to the normal cone test configuration for the blow-up PP(Osh_PP^s^⊕ r ⊕ Osh_PP^s(1)) ≃ Bl_PP^r−1(PP^s) along the exceptional divisor and with respect to the tautological line bundle Osh_PP(Osh_PP^s^⊕ r ⊕ Osh_PP^s(1))(1).
ER  -