June  2020, 28(2): 795-805. doi: 10.3934/era.2020040

On Seshadri constants and point-curve configurations

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

Received  February 2020 Revised  April 2020 Published  May 2020

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.

Citation: Marek Janasz, Piotr Pokora. On Seshadri constants and point-curve configurations. Electronic Research Archive, 2020, 28 (2) : 795-805. doi: 10.3934/era.2020040
References:
[1]

Th. Bauer, Ł. Farnik, K. Hanumanthu and J. Huizenga, Mini-Workshop: Seshadri Constants, Oberwolfach Report No. $53/2019$, 2020. Google Scholar

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M. Cuntz, Simplicial arrangements with up to 27 lines, Discrete Comput. Geom., 48 (2012), 682-701.  doi: 10.1007/s00454-012-9423-7.  Google Scholar

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I. Dolgachev, A. Laface, U. Persson and G. Urzúa, Chilean configuration of conics, lines and points, Preprint. Google Scholar

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D. Kohel, X. Roulleau and A. Sarti, A special configuration of 12 conics and generalized Kummer surfaces, Preprint. Google Scholar

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K. Oguiso, Seshadri constants in a family of surfaces, Math. Ann., 323 (2002), 625-631.  doi: 10.1007/s002080200317.  Google Scholar

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P. Pokora, Seshadri constants and special point configurations in the projective plane, Rocky Mountain J. Math., 49 (2019), 963-978.  doi: 10.1216/RMJ-2019-49-3-963.  Google Scholar

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P. PokoraX. Roulleau and T. Szemberg, Bounded negativity, Harbourne constants and transversal arrangements of curves, Ann. Inst. Fourier (Grenoble), 67 (2017), 2719-2735.  doi: 10.5802/aif.3149.  Google Scholar

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P. Pokora and T. Szemberg, Conic-line arrangements in the complex projective plane, arXiv: 2002.01760. Google Scholar

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B. Strycharz-Szemberg and T. Szemberg, Remarks on the Nagata conjecture, Serdica Math. J., 30 (2004), 405-430.   Google Scholar

show all references

References:
[1]

Th. Bauer, Ł. Farnik, K. Hanumanthu and J. Huizenga, Mini-Workshop: Seshadri Constants, Oberwolfach Report No. $53/2019$, 2020. Google Scholar

[2]

M. Cuntz, Simplicial arrangements with up to 27 lines, Discrete Comput. Geom., 48 (2012), 682-701.  doi: 10.1007/s00454-012-9423-7.  Google Scholar

[3]

I. Dolgachev, A. Laface, U. Persson and G. Urzúa, Chilean configuration of conics, lines and points, Preprint. Google Scholar

[4]

F. Hirzebruch, Arrangements of lines and algebraic surfaces, Arithmetic and Geometry, Vol. II, Progr. Math., Birkhäuser, Boston, Mass., 36 (1983), 113–140.  Google Scholar

[5]

D. Kohel, X. Roulleau and A. Sarti, A special configuration of 12 conics and generalized Kummer surfaces, Preprint. Google Scholar

[6]

K. Oguiso, Seshadri constants in a family of surfaces, Math. Ann., 323 (2002), 625-631.  doi: 10.1007/s002080200317.  Google Scholar

[7]

P. Pokora, Seshadri constants and special point configurations in the projective plane, Rocky Mountain J. Math., 49 (2019), 963-978.  doi: 10.1216/RMJ-2019-49-3-963.  Google Scholar

[8]

P. PokoraX. Roulleau and T. Szemberg, Bounded negativity, Harbourne constants and transversal arrangements of curves, Ann. Inst. Fourier (Grenoble), 67 (2017), 2719-2735.  doi: 10.5802/aif.3149.  Google Scholar

[9]

P. Pokora and T. Szemberg, Conic-line arrangements in the complex projective plane, arXiv: 2002.01760. Google Scholar

[10]

B. Strycharz-Szemberg and T. Szemberg, Remarks on the Nagata conjecture, Serdica Math. J., 30 (2004), 405-430.   Google Scholar

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