\`A propos des vari\'et\'es de Poncelet

TL;DR

This paper explores Poncelet varieties, showing that quadrics and smooth cubics in three-dimensional projective space are Poncelet surfaces, while most higher-degree surfaces are not, thus extending classical geometric concepts.

Contribution

It characterizes which surfaces in three-dimensional projective space are Poncelet, specifically identifying quadrics and smooth cubics as Poncelet surfaces and excluding higher-degree surfaces.

Findings

Quadrics are Poncelet surfaces.

Smooth cubics are Poncelet surfaces.

Most surfaces of degree greater than four are not Poncelet.

Abstract

We recall the definition of Poncelet varieties that generalize the celebrated Poncelet curves introduced by Darboux. We show that any quadric and any smooth cubic in the projective space of dimension three is a Poncelet surface but that a general surface of degree greater than four is not.