Le degre de la variete des courbes de Poncelet

TL;DR

This paper calculates the degree of the projective variety of Poncelet curves of degree n, linking geometric properties with moduli spaces and providing explicit formulas for n ≥ 4, including numerical results for n ≤ 6.

Contribution

It introduces a method to compute the degree of Poncelet curve varieties using moduli spaces and provides explicit formulas for degrees for n ≥ 4.

Findings

Degree of Poncelet curve variety for n=4 is 54.

Explicit formulas for degrees of Poncelet curves for n ≥ 4.

Numerical results for degrees when n ≤ 6.

Abstract

We compute the degree of the projective variety of Poncelet curves of degree $n$. This variety is irreducible of dimension $2 n + 5$, and lies inside the projective space of degree $n$ plane curves. It is classically defined as the closure on this projective space of the locally closed subset of curves passing through the vertices of some nondegenerate $n$ sided polygone inscribed in some smooth conic (the polygone and the conic being variable). It is related to a specific class of semi-stable sheaves on the projective (dual) plane, named Poncelet sheaves. Using moduli spaces birational to the variety of Poncelet curves, we compute the requested degree. It involves quite cumbersome computations, and we obtain general formulas for $n \geq 4$. We do numerical applications for $n \leq 6$. For $n=4$ we find back the well known Donaldson number of the projective plane, 54, which is the degree of the hypersurface of Luroth quartics.