On Seshadri constants and point-curve configurations

Marek Janasz; Piotr Pokora; Marek Janasz; Piotr Pokora

doi:10.3934/era.2020040

Electronic Research Archive

2020, Volume 28, Issue 2: 795-805. doi: 10.3934/era.2020040

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On Seshadri constants and point-curve configurations

Department of Mathematics, Pedagogical University of Cracow, Podchorążych 2, PL-30-084 Kraków, Poland

Received: 01 February 2020 Revised: 01 April 2020
14C20, 14N10, 14N20

In the note we study the multipoint Seshadri constants of $ \mathcal{O}_{\mathbb{P}^{2}_{\mathbb{C}}}(1) $ centered at singular loci of certain curve arrangements in the complex projective plane. Our first aim is to show that the values of Seshadri constants can be approximated with use of a combinatorial invariant which we call the configurational Seshadri constant. We study specific examples of point-curve configurations for which we provide actual values of the associated Seshadri constants. In particular, we provide an example based on Hesse point-conic configuration for which the associated Seshadri constant is computed by a line. This shows that multipoint Seshadri constants are not purely combinatorial.
- Seshadri constants,
- point-curve configurations,
- projective plane,
- plane curves
Citation: Marek Janasz, Piotr Pokora. On Seshadri constants and point-curve configurations[J]. Electronic Research Archive, 2020, 28(2): 795-805. doi: 10.3934/era.2020040

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Abstract

In the note we study the multipoint Seshadri constants of $ \mathcal{O}_{\mathbb{P}^{2}_{\mathbb{C}}}(1) $ centered at singular loci of certain curve arrangements in the complex projective plane. Our first aim is to show that the values of Seshadri constants can be approximated with use of a combinatorial invariant which we call the configurational Seshadri constant. We study specific examples of point-curve configurations for which we provide actual values of the associated Seshadri constants. In particular, we provide an example based on Hesse point-conic configuration for which the associated Seshadri constant is computed by a line. This shows that multipoint Seshadri constants are not purely combinatorial.

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