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On the order sequence of an embedding of the Ree curve

  • Dane C. Skabelund EMAIL logo
Published/Copyright: June 22, 2019
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Advances in Geometry
From the journal Advances in Geometry Volume 20 Issue 2

Abstract

In this paper we compute the Weierstrass order-sequence associated with a certain linear series on the Deligne–Lusztig curve of Ree type. As a result, we show that the set of Weierstrass points of this linear series consists entirely of 𝔽q-rational points.

Keywords: Ree curve; Weierstrass points
MSC 2010: 11G20

  1. Communicated by: G. KorchmΓ‘ros

Acknowledgements

I would like to thank the anonymous referee for his/her detailed comments which improved the exposition of this paper.

7 Appendix

In the following tables, an asterisk in row i and column f indicates that i is in the set Sf described in Section 3. In particular, Di f = 0 wherever there is a blank entry in the table. The first table involves functions of type 1 and the second table involves functions of type 2.

i x w1 w2 w3 w6 w8
0 βˆ— βˆ— βˆ— βˆ— βˆ— βˆ—
1 βˆ— βˆ— βˆ— βˆ— βˆ— βˆ—
3q0 βˆ— βˆ— βˆ— βˆ— βˆ—
3q0 + 1 βˆ— βˆ— βˆ— βˆ— βˆ—
q βˆ— βˆ— βˆ— βˆ— βˆ—
q + 1 βˆ— βˆ— βˆ— βˆ—
q + 3q0 βˆ— βˆ— βˆ— βˆ— βˆ—
q + 3q0 + 1 βˆ— βˆ— βˆ— βˆ—
2q βˆ— βˆ— βˆ— βˆ—
2q + 1 βˆ— βˆ— βˆ—
2q + 3q0 βˆ— βˆ— βˆ— βˆ—
2q + 3q0 + 1 βˆ— βˆ— βˆ—
3q βˆ— βˆ— βˆ—
3q + 3q0 βˆ— βˆ— βˆ—
3qq0 βˆ— βˆ— βˆ— βˆ— βˆ—
3qq0 + 1 βˆ— βˆ— βˆ— βˆ—
3qq0 + 3q0 βˆ— βˆ—
3qq0 + 3q0 + 1 βˆ— βˆ—
3qq0 + q βˆ— βˆ— βˆ— βˆ— βˆ—
3qq0 + q+1 βˆ— βˆ— βˆ— βˆ—
3qq0 + q + 3q0 βˆ— βˆ—
3qq0 + q + 3q0 + 1 βˆ— βˆ—
3qq0 + 2q βˆ— βˆ— βˆ— βˆ—
3qq0 + 2q + 1 βˆ— βˆ— βˆ—
3qq0 + 2q + 3q0 βˆ— βˆ—
3qq0 + 2q + 3q0 + 1 βˆ— βˆ—
3qq0 + 3q βˆ— βˆ— βˆ—
3qq0 + 3q + 3q0 βˆ— βˆ—
6qq0 βˆ— βˆ—
6qq0 + 1 βˆ— βˆ—
6qq0 + q βˆ— βˆ—
6qq0 + q + 1 βˆ— βˆ—
6qq0 + 2q βˆ— βˆ—
6qq0 + 2q + 1 βˆ— βˆ—
6qq0 + 3q βˆ— βˆ—
q2 βˆ— βˆ— βˆ— βˆ— βˆ—

i y z w4 w7 w5 w9 w10
0 βˆ— βˆ— βˆ— βˆ— βˆ— βˆ— βˆ—
1 βˆ— βˆ— βˆ— βˆ— βˆ— βˆ— βˆ—
q0 βˆ— βˆ— βˆ— βˆ— βˆ— βˆ— βˆ—
q0 + 1 βˆ— βˆ— βˆ— βˆ— βˆ— βˆ— βˆ—
2q0 βˆ— βˆ— βˆ— βˆ— βˆ— βˆ—
2q0 + 1 βˆ— βˆ— βˆ— βˆ— βˆ— βˆ—
3q0 βˆ— βˆ— βˆ—
3q0 + 1 βˆ— βˆ— βˆ—
q βˆ— βˆ— βˆ— βˆ— βˆ— βˆ— βˆ—
q + 1 βˆ— βˆ— βˆ— βˆ— βˆ—
q + q0 βˆ— βˆ— βˆ— βˆ— βˆ— βˆ— βˆ—
q + q0 + 1 βˆ— βˆ— βˆ— βˆ— βˆ—
q + 2q0 βˆ— βˆ— βˆ— βˆ— βˆ— βˆ—
q + 2q0 + 1 βˆ— βˆ— βˆ— βˆ— βˆ—
q + 3q0 βˆ— βˆ— βˆ—
q + 3q0 + 1 βˆ— βˆ— βˆ—
2q βˆ— βˆ— βˆ— βˆ— βˆ—
2q + 1 βˆ— βˆ—
2q + q0 βˆ— βˆ— βˆ— βˆ— βˆ—
2q + 2q0 βˆ— βˆ— βˆ— βˆ— βˆ—
2q + 3q0 βˆ— βˆ— βˆ—
2q + 3q0 + 1 βˆ— βˆ—
3q βˆ— βˆ—
3q + 3q0 βˆ— βˆ—
qq0 βˆ— βˆ— βˆ— βˆ— βˆ— βˆ— βˆ—
qq0 + 1 βˆ— βˆ— βˆ— βˆ— βˆ—
qq0 + q0 βˆ— βˆ— βˆ— βˆ— βˆ—
qq0 + q0 + 1 βˆ— βˆ— βˆ— βˆ— βˆ—
qq0 + 2q0 βˆ— βˆ— βˆ— βˆ—
qq0 + 2q0 + 1 βˆ— βˆ— βˆ— βˆ—
qq0 + 3q0 βˆ— βˆ— βˆ—
qq0 + 3q0 + 1 βˆ— βˆ— βˆ—
qq0 + q βˆ— βˆ— βˆ— βˆ— βˆ— βˆ— βˆ—
qq0 + q + 1 βˆ— βˆ— βˆ— βˆ— βˆ—
qq0 + q+q0 βˆ— βˆ— βˆ— βˆ— βˆ—
qq0 + q+q0 + 1 βˆ— βˆ— βˆ— βˆ— βˆ—
qq0 + q + 2q0 βˆ— βˆ— βˆ— βˆ—
qq0 + q + 2q0 + 1 βˆ— βˆ— βˆ— βˆ—
qq0 + q + 3q0 βˆ— βˆ— βˆ—
qq0 + q + 3q0 + 1 βˆ— βˆ— βˆ—
qq0 + 2q βˆ— βˆ— βˆ— βˆ— βˆ—
qq0 + 2q + 1 βˆ— βˆ—
qq0 + 2q + q0 βˆ— βˆ— βˆ— βˆ— βˆ—
qq0 + 2q + 2q0 βˆ— βˆ— βˆ— βˆ—
qq0 + 2q + 3q0 βˆ— βˆ— βˆ—
qq0 + 2q + 3q0 + 1 βˆ— βˆ—
qq0 + 3q βˆ— βˆ—
qq0 + 3q + 3q0 βˆ— βˆ—
2qq0 βˆ— βˆ— βˆ— βˆ— βˆ— βˆ—
2qq0 + 1 βˆ— βˆ— βˆ— βˆ— βˆ—
2qq0 + q0 βˆ— βˆ— βˆ— βˆ—
2qq0 + q0 + 1 βˆ— βˆ— βˆ— βˆ—
2qq0 + 2q0 βˆ— βˆ— βˆ—
2qq0 + 2q0 + 1 βˆ— βˆ— βˆ—
2qq0 + 3q0 βˆ—
2qq0 + 3q0 + 1 βˆ—

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Received: 2017-10-09
Revised: 2018-02-18
Published Online: 2019-06-22
Published in Print: 2020-04-28

Β© 2020 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. On the order sequence of an embedding of the Ree curve
  3. 𝔇βŠ₯-parallel normal Jacobi operators for Hopf hypersurfaces in complex two-plane Grassmannians with generalized Tanaka–Webster connection
  4. A Krasnosel’skii-type theorem for certain orthogonal polytopes starshaped via k-staircase paths
  5. Higher algebraic structures in Hamiltonian Floer theory
  6. Additive structures on f-vector sets of polytopes
  7. The degree of the tangent and secant variety to a projective surface
  8. Building lattices and zeta functions
  9. Quaternionic equiangular lines
  10. Non-reduced moduli spaces of sheaves on multiple curves
https://doi.org/10.1515/advgeom-2019-0007
Keywords for this article
Ree curve; Weierstrass points

Articles in the same Issue

  1. Frontmatter
  2. On the order sequence of an embedding of the Ree curve
  3. 𝔇βŠ₯-parallel normal Jacobi operators for Hopf hypersurfaces in complex two-plane Grassmannians with generalized Tanaka–Webster connection
  4. A Krasnosel’skii-type theorem for certain orthogonal polytopes starshaped via k-staircase paths
  5. Higher algebraic structures in Hamiltonian Floer theory
  6. Additive structures on f-vector sets of polytopes
  7. The degree of the tangent and secant variety to a projective surface
  8. Building lattices and zeta functions
  9. Quaternionic equiangular lines
  10. Non-reduced moduli spaces of sheaves on multiple curves
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