Castelnuovo bound for curves in projective 3-folds
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866914891585028096 |
|---|---|
| author | Liu, Zhiyu |
| author_facet | Liu, Zhiyu |
| contents | The Castelnuovo bound conjecture, which is proposed by physicists, predicts an effective vanishing result for Gopakumar-Vafa invariants of Calabi-Yau 3-folds of Picard number one. Previously, it is only known for a few cases and all the proofs rely on the Bogomolov-Gieseker conjecture of Bayer-Macrì-Toda.
In this paper, we prove the Castelnuovo bound conjecture for any Calabi-Yau 3-folds of Picard number one, up to a linear term and finitely many degree, without assuming the conjecture of Bayer-Macrì-Toda. Furthermore, we prove an effective vanishing theorem for surface-counting invariants of Calabi-Yau 4-folds of Picard number one. We also apply our techniques to study low-degree curves on some explicit Calabi-Yau 3-folds.
Our approach is based on a general iterative method to obtain upper bounds for the genus of one-dimensional closed subschemes in a fixed 3-fold, which is a combination of classical techniques and the wall-crossing of weak stability conditions on derived categories, and works for any projective 3-fold with at worst isolated singularities over any algebraically closed field. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_20161 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Castelnuovo bound for curves in projective 3-folds Liu, Zhiyu Algebraic Geometry High Energy Physics - Theory Mathematical Physics Symplectic Geometry The Castelnuovo bound conjecture, which is proposed by physicists, predicts an effective vanishing result for Gopakumar-Vafa invariants of Calabi-Yau 3-folds of Picard number one. Previously, it is only known for a few cases and all the proofs rely on the Bogomolov-Gieseker conjecture of Bayer-Macrì-Toda. In this paper, we prove the Castelnuovo bound conjecture for any Calabi-Yau 3-folds of Picard number one, up to a linear term and finitely many degree, without assuming the conjecture of Bayer-Macrì-Toda. Furthermore, we prove an effective vanishing theorem for surface-counting invariants of Calabi-Yau 4-folds of Picard number one. We also apply our techniques to study low-degree curves on some explicit Calabi-Yau 3-folds. Our approach is based on a general iterative method to obtain upper bounds for the genus of one-dimensional closed subschemes in a fixed 3-fold, which is a combination of classical techniques and the wall-crossing of weak stability conditions on derived categories, and works for any projective 3-fold with at worst isolated singularities over any algebraically closed field. |
| title | Castelnuovo bound for curves in projective 3-folds |
| topic | Algebraic Geometry High Energy Physics - Theory Mathematical Physics Symplectic Geometry |
| url | https://arxiv.org/abs/2407.20161 |