An arithmetic count of osculating lines

TL;DR

This paper introduces a quadratic enrichment of the classical count of osculating lines to a hypersurface, extending the known complex case to arbitrary perfect fields and incorporating algebraic quadratic form data.

Contribution

It defines a new quadratic count of osculating lines over any perfect field, generalizing the classical complex count and linking it to the Grothendieck--Witt ring.

Findings

Count takes values in Grothendieck--Witt ring

Count depends linearly on the degree of the hypersurface

Generalizes classical complex count to arbitrary perfect fields

Abstract

We say that a line in $\mathbb P^{n+1}_k$ is osculating to a hypersurface $Y$ if they meet with contact order $n+1$. When $k=\mathbb C$, it is known that through a fixed point of $Y$, there are exactly $n!$ of such lines. Under some parity condition on $n$ and $\mathrm{deg}(Y)$, we define a quadratically enriched count of these lines over any perfect field $k$. The count takes values in the Grothendieck--Witt ring of quadratic forms over $k$ and depends linearly on $\mathrm{deg}(Y)$.