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Functionality for isomorphism classes of curves and hypersurfaces

Thomas Bouchet, Reynald Lercier, Jeroen Sijsling, Christophe Ritzenthaler

Abstract

We describe algorithms based on invariant theory to solve problems on the geometry of curves, mainly those of genus 2, 3 and 4. New theoretical results building on the first author's PhD thesis are also included.

Functionality for isomorphism classes of curves and hypersurfaces

Abstract

We describe algorithms based on invariant theory to solve problems on the geometry of curves, mainly those of genus 2, 3 and 4. New theoretical results building on the first author's PhD thesis are also included.

Paper Structure

This paper contains 14 sections, 13 theorems, 27 equations, 1 table.

Table of Contents

  1. Invariants and reconstruction
  2. Invariants: hyperelliptic curves of small genus
  3. Invariants: non-hyperelliptic curves of genus 3 and 4
  4. Reconstruction: general philosophy
  5. Reconstruction: hyperelliptic curves of genus 2, 3 and 4
  6. Reconstruction: non-hyperelliptic curves of genus 3 and 4
  7. Isomorphisms, automorphisms, and twists
  8. A generic strategy for hypersurfaces
  9. The hyperelliptic case
  10. The case of plane quartics
  11. The case of non-hyperelliptic curves of genus 4
  12. Dixmier--Ohno invariants in positive characteristic
  13. Good filtrations
  14. Application to the invariant ring of ternary quartics

Key Result

Lemma 2.1.1

If $M . f = g$, then there exists a scalar $\lambda\in k^\times$ such that $\hat{M} . C_f = \lambda C_g$. In particular, $\hat{M} = 1/\lambda \cdot {\ }^t(C_fC_g^{-1})$.

Theorems & Definitions (33)

  • Remark 1.2.1
  • Example 1.3.1
  • Example 1.4.1
  • Lemma 2.1.1
  • Example 2.1.2
  • Proposition 2.1.3
  • proof
  • Remark 2.1.4
  • Remark 2.1.5
  • Example 2.3.1
  • ...and 23 more