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Effective methods for plane quartics, their theta characteristics and the Scorza map

Giorgio Ottaviani

Abstract

This is a revised version of the lecture notes prepared for the workshop on "Plane quartics, Scorza map and related topics", held in Catania, January 19-21, 2016. The last section contains eight Macaulay2 scripts on theta characteristics and the Scorza map, with a tutorial. The first sections give an introduction to these scripts. The tutorial contains a list of the 36 Scorza preimages of the Edge quartic.

Effective methods for plane quartics, their theta characteristics and the Scorza map

Abstract

This is a revised version of the lecture notes prepared for the workshop on "Plane quartics, Scorza map and related topics", held in Catania, January 19-21, 2016. The last section contains eight Macaulay2 scripts on theta characteristics and the Scorza map, with a tutorial. The first sections give an introduction to these scripts. The tutorial contains a list of the 36 Scorza preimages of the Edge quartic.
Paper Structure (18 sections, 12 theorems, 46 equations, 1 figure)

This paper contains 18 sections, 12 theorems, 46 equations, 1 figure.

Table of Contents

  1. Introduction
  2. How to write down plane quartics and their theta characteristics
  3. Clebsch and Lüroth quartics
  4. The Scorza map
  5. Description of the content
  6. Summary of the eight M2 scripts presented in § \ref{['sec:alg']}
  7. Acknowledgements
  8. Apolarity, Waring decompositions
  9. The Aronhold invariant of plane cubics
  10. Three descriptions of an even theta characteristic
  11. The symmetric determinantal description
  12. The sextic model
  13. The $(3,3)$-correspondence
  14. The Aronhold covariant of plane quartics and the Scorza map.
  15. Contact cubics and contact triangles
  16. ...and 3 more sections

Key Result

Theorem 3.1

LO, Ot Let $f(x,y,z)\in\mathrm{ Sym}^3{\mathbb C}^3$ be a homogeneous cubic polynomial. Let $C(f_x)$, $C(f_y)$, $C(f_z)$ be the three $3\times 3$ symmetric matrices which are the Hessian of the three partial derivatives of $f$. All the $8$-pfaffians of the $9\times 9$ skew-symmetric matrix coincide (up to scalar) with the Aronhold invariant. Note the beautiful analogy with (eq:paff6).

Figures (1)

  • Figure 1: One of the $36$ Scorza preimages of the Edge quartic

Theorems & Definitions (20)

  • Definition 1.1
  • Definition 2.1
  • Theorem 3.1
  • Theorem 3.2: Nonabelian apolarity for plane cubics
  • proof
  • Remark 3.3
  • Proposition 3.4
  • Proposition 3.5
  • Proposition 3.6
  • Theorem 4.1: Dixon
  • ...and 10 more