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<p>In this note, we continue the study of Seshadri constants on blow-ups of Hirzebruch surfaces initiated in (Hanumanthu, K. et al.: Math. Nachr. <b>298</b>(2), 437–455 2025). 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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\((-1)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-curves conjecture in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathbb {P}^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">P</mi> </mrow> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>. 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.</p>

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Rationality of Seshadri constants on blow-ups of ruled surfaces

  • 期刊论文

  • 发表日期：

    2026年4月2日

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

摘要

In this note, we continue the study of Seshadri constants on blow-ups of Hirzebruch surfaces initiated in (Hanumanthu, K. et al.: Math. Nachr. 298(2), 437–455 2025). 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 \((-1)\) ( - 1 ) -curves conjecture in \(\mathbb {P}^2\) 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.