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Lines in supersingular quartics

Alex Degtyarev

Abstract

We show that the number of lines contained in a supersingular quartic surface is 40 or at most 32, if the characteristic of the field equals 2, and it is 112, 58, or at most 52, if the characteristic equals 3. If the quartic is not supersingular, the number of lines is at most 60 in both cases. We also give a complete classification of large configurations of lines.

Lines in supersingular quartics

Abstract

We show that the number of lines contained in a supersingular quartic surface is 40 or at most 32, if the characteristic of the field equals 2, and it is 112, 58, or at most 52, if the characteristic equals 3. If the quartic is not supersingular, the number of lines is at most 60 in both cases. We also give a complete classification of large configurations of lines.

Paper Structure

This paper contains 20 sections, 26 theorems, 41 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Motivation
  3. Principal results
  4. Contents of the paper
  5. Acknowledgements
  6. Lattices
  7. Finite quadratic forms (see Miranda.Morrison:bookNikulin:forms)
  8. Integral lattices (see Nikulin:forms)
  9. Lattice extensions (see Nikulin:forms)
  10. Lemmas on discriminant forms
  11. Generalized quadrangles (see Payne.Thas)
  12. $K3$-surfaces
  13. $K3$-surfaces
  14. Quartics
  15. (Quasi-)elliptic pencils
  16. ...and 5 more sections

Key Result

Theorem 1.1

Assume that $\mathop{\operator@font char}\nolimits\Bbbk=2$ and $X$ is supersingular. Then either $\mathopen|\operatorname{Fn} X\mathclose|=40$, and there are at most five configurations: or $\mathopen|\operatorname{Fn} X\mathclose|\leqslant32$.

Figures (3)

  • Figure : The configuration in \ref{['lem.4-4']} (the graph $K_{4,4}$)
  • Figure : The configuration in \ref{['lem.10-2']} (the graph $K_{10,2}$)
  • Figure : The configuration in \ref{['lem.6-2']}

Theorems & Definitions (30)

  • Theorem 1.1: see \ref{['proof.char=2']}
  • Theorem 1.2: see \ref{['proof.char=3']}
  • Theorem 1.3: see \ref{['proof.ordinary']}
  • Conjecture 1.1: see \ref{['rem.char=2.counts']}
  • Conjecture 1.2: see \ref{['rem.char=3.counts']}
  • Theorem 2.1: see Nikulin:forms
  • Theorem 2.2: see Nikulin:forms
  • Proposition 2.1: see Nikulin:forms
  • Lemma 2.1
  • Lemma 2.2
  • ...and 20 more