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21. Rationality of Seshadri constants on blow-ups of ruled surfaces

Author

Hanumanthu, Krishna, Jacob, Cyril J., N., Suhas B., and Singh, Amit Kumar

Subjects
Mathematics - Algebraic Geometry, 14C20, 14E05, 14J26
Abstract

In this note, we continue the study of Seshadri constants on blow-ups of Hirzebruch surfaces initiated in arXiv:2312.14555. Now we consider blow-ups of ruled surfaces more generally. We propose a conjecture for classifying all the negative self-intersection curves on the blow-up of a ruled surface at very general points, analogous to the $(-1)$-curves conjecture in $\mathbb{P}^2$. Assuming this conjecture is true, we exhibit an ample line bundle with an irrational Seshadri constant at a very general point on such a surface., Comment: 13 Pages

Published
2024

36. Seshadri constants on blow-ups of Hirzebruch surfaces

Author

Hanumanthu, Krishna, Jacob, Cyril J., N., Suhas B., and Singh, Amit Kumar

Subjects
Mathematics - Algebraic Geometry, 14C20, 14E05, 14J26
Abstract

Let $e,r \ge 0$ be integers and let $\mathbb{F}_e : = \mathbb{P}(\mathcal{O}_{\mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1}(-e))$ denote the Hirzebruch surface with invariant $e$. We compute the Seshadri constants of an ample line bundle at an arbitrary point of the $r$-point blow-up of $\mathbb{F}_e$ when $r \leq e-1$ and at a very general point when $r=e$ or $r=e+1$. We also discuss several conjectures on linear systems of curves on the blow-up of $\mathbb{F}_e$ at $r$ very general points., Comment: 20 pages; minor changes since the previous version; some errors corrected

Published
2023

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