Recovering plane curves of low degree from their inflection lines and inflection points

Marco Pacini,Damiano Testa

doi:10.1007/s11856-012-0129-6

Marco Pacini, Damiano Testa

Open Access

https://doi.org/10.1007/s11856-012-0129-6

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Abstract

Abstract In this paper we consider the following problem: is it possible to recover a smooth plane curve of degree d ≥ 3 from its inflection lines? We answer the posed question positively for a general smooth plane quartic curve, making the additional assumption that also one inflection point is given, and for any smooth plane cubic curve.

Highlights

In the last decade, several reconstruction theorems for plane and canonical curves defined over the field of complex numbers appeared in the literature
Theorem 1: The general smooth plane quartic curve defined over the field of complex numbers is uniquely determined by its inflection lines and one inflection point
In Lemma 3.2 we prove that the map associating to a smooth plane quartic its configuration of inflection lines is generically finite onto its image

Summary

Introduction

Several reconstruction theorems for plane and canonical curves defined over the field of complex numbers appeared in the literature. Theorem 1: The general smooth plane quartic curve defined over the field of complex numbers is uniquely determined by its inflection lines and one inflection point. Theorem 2: Let C ⊂ P2 be a smooth plane cubic curve over a field k of characteristic different from three and let TC ⊂ (P2)∨ be the set of inflection lines of C. If the characteristic of the ground field is different from two, we prove that an integral plane sextic with a singularity at one of these points is automatically singular at all the nine points and the singularities are all cuspidal This cuspidal curve is the dual of the initial plane cubic and the reconstruction follows by projective duality. The inflection lines to a plane quartic define cubic roots of pencils with a base point that are contained in the canonical system. If S is a scheme, we denote by Sred the reduced scheme associated to S

Limits of inflection lines

Recovering quartics from inflection lines and inflection points

Inflection lines of plane cubics