An arithmetic count of osculating lines
Giosuè Muratore
TL;DR
This work extends the classical count of osculating lines to a smooth hypersurface by lifting the problem to A1-enumerative geometry. The authors construct a quadratically enriched count valued in the Grothendieck–Witt ring GW(k) for lines with maximal contact, under parity conditions n ≡ 2 (mod 4) and d even, yielding a GW(k)-class that specializes to the known n! over the complex numbers. The approach hinges on the relative orientability of a principal-parts–based vector bundle on a flag-variety setup and a careful Euler-class computation that localizes to the osculating data via explicit global sections. The resulting enriched count scales with the degree d and provides a robust invariant that recovers classical counts in characteristic zero while suggesting enriched counts for osculating curves of higher degree in future work. The framework integrates A1-homotopy, relative orientations, and Euler theory to extend enumerative geometry beyond the complex setting, with potential applications to other osculation problems and more general 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 $\mathrm{deg}(Y)$.