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Tango Structures on Curves in Characteristic 2

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Polynomial Rings and Affine Algebraic Geometry (PRAAG 2018)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 319))

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Abstract

The (pre-)Tango structure is a certain ample invertible sheaf of exact differential 1-forms on a projective algebraic variety and it implies some typical pathological phenomena in positive characteristic. Moreover, by using the notion of (pre-)Tango structure, we can construct another variety accompanied by similar pathological phenomena. In this article, we explicitly show several interesting and mysterious phenomena on the induced uniruled surfaces from (pre-)Tango structures on curves in characteristic 2.

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Notes

  1. 1.

    Here, note the divisibility of the Picard variety.

  2. 2.

    Recall the extension \(0\rightarrow \mathcal {O}_C\rightarrow \mathcal {E}\rightarrow \mathcal {N}^n\rightarrow \ 0\), where \(\mathcal E\) is generated locally by 1 and \(q_i\)’s subjected to \(q_i=d^n_{ij}q_j +b_{ij}\) (see Sect. 3).

  3. 3.

    Here ‘involving’ means ‘inducing’ or ‘inducing a normal uniruled surface whose desingularization is’.

  4. 4.

    The induced uniruled surface is a quasi-elliptic surface of \(\kappa =1\).

  5. 5.

    The induced uniruled surface is also a quasi-elliptic surface of \(\kappa =1\).

  6. 6.

    For details, see the author’s forthcoming paper.

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Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Wakayama Medical University, Wakayama, 641-8509, Japan

    Yoshifumi Takeda

Authors
  1. Yoshifumi Takeda
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Correspondence to Yoshifumi Takeda .

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Editors and Affiliations

  1. Department of Mathematical Sciences, Tokyo Metropolitan University, Hachioji, Tokyo, Japan

    Shigeru Kuroda

  2. Faculty of Applied Physics, University of Fukui, Fukui, Japan

    Nobuharu Onoda

  3. Department of Mathematics, Western Michigan University, Kalamazoo, MI, USA

    Gene Freudenburg

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Takeda, Y. (2020). Tango Structures on Curves in Characteristic 2. In: Kuroda, S., Onoda, N., Freudenburg, G. (eds) Polynomial Rings and Affine Algebraic Geometry. PRAAG 2018. Springer Proceedings in Mathematics & Statistics, vol 319. Springer, Cham. https://doi.org/10.1007/978-3-030-42136-6_12

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