On the Hilbert polynomial of the linked projective space

Abstract

Linked projective spaces are quiver Grassmannians of subspaces of dimension 1 of certain quiver representations. Degenerations of linear series produce these representations, with the limit divisors parameterized by the associated linked projective spaces. It is not known whether all linked projective spaces arise this way. If they do, they are degenerations of the (small) diagonal in a product of projective spaces. In any case, we prove here that they have the (multivariate) Hilbert polynomial of the diagonal. To achieve this, we first extend a Hilbert-polynomial formula for multiplicity-free varieties to (simple) normal-crossings schemes with multiplicity-free strata in products of projective spaces. Then we prove that a linked projective space is normal-crossings, by describing it locally in terms of Mustafin varieties. Finally, we use a relation between intersections of components of the linked projective space and certain polytopes in the tiling of a simplex associated to the linked net to prove that our formula for the Hilbert polynomial applies.

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BibTeX RIS

BibTeX

@misc{leon2026hilbert,
  title = {{On the Hilbert polynomial of the linked projective space}},
  author = {Felipe De León and Eduardo Esteves and Eduardo Vital},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.37634},
  eprint = {2609.37634},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.37634},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - On the Hilbert polynomial of the linked projective space
AU  - Felipe De León
AU  - Eduardo Esteves
AU  - Eduardo Vital
PY  - 2026
DA  - 2026/10/06
PB  - arXiv
DO  - 10.48550/arxiv.2609.37634
UR  - https://arxiv.org/abs/2609.37634
AB  - Linked projective spaces are quiver Grassmannians of subspaces of dimension 1 of certain quiver representations. Degenerations of linear series produce these representations, with the limit divisors parameterized by the associated linked projective spaces. It is not known whether all linked projective spaces arise this way. If they do, they are degenerations of the (small) diagonal in a product of projective spaces. In any case, we prove here that they have the (multivariate) Hilbert polynomial of the diagonal. To achieve this, we first extend a Hilbert-polynomial formula for multiplicity-free varieties to (simple) normal-crossings schemes with multiplicity-free strata in products of projective spaces. Then we prove that a linked projective space is normal-crossings, by describing it locally in terms of Mustafin varieties. Finally, we use a relation between intersections of components of the linked projective space and certain polytopes in the tiling of a simplex associated to the linked net to prove that our formula for the Hilbert polynomial applies.
ER  -

Abstract

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