On the Sylvester-Gallai theorem for conics

TL;DR

This paper provides a new algebraic geometry-based proof of an extension of the Sylvester-Gallai theorem to conics, using Cremona transformations and Hirzebruch inequality.

Contribution

It offers a novel proof of the Sylvester-Gallai theorem analogue for degree-two curves, expanding the theorem's applicability with algebraic geometry tools.

Findings

Established the Sylvester-Gallai theorem for conics.

Introduced a proof using Cremona transformations.

Applied Hirzebruch inequality in the proof.

Abstract

In the present note we give a new proof of a result due to Wiseman and Wilson which establishes an analogue of the Sylvester-Gallai theorem valid for curves of degree two. The main ingredients of the proof come from algebraic geometry. Specifically, we use Cremona transformation of the projective plane and Hirzebruch inequality.