$D$-affinity of Quadrics Revisited

TL;DR

This paper investigates the $D$-affinity of smooth quadric hypersurfaces over algebraically closed fields of characteristic $p extgreater{}=3$, showing non $D$-affinity for even dimensions and providing examples of non $D$-affine flag varieties.

Contribution

It establishes that even-dimensional quadrics are not $D$-affine in characteristic $p extgreater{}=3$, complementing previous results on odd-dimensional quadrics.

Findings

Even-dimensional quadrics are not $D$-affine for $n=2m extgreater{}=4$.

The Grassmannian ${Gr}(2,4)$ is not $D$-affine.

Provides examples of minimal dimension non $D$-affine flag varieties.

Abstract

Let $K$ be aa algebraically closed field of characteristic $p\geq3$ and let $Q_{n}\subset\mathbb{P}^{n+1}_{K}$ be a smooth quadric hypersurface. We show that if $n=2m\geq4$ then $Q_{n}$ is not $D$-affine. In particular, we show the grassmannian ${Gr}(2,4)$ is not $D$-affine, which gives an example of a non $D$-affine flag variety of minimal possible dimension in characteristic $p\geq3$. Our result complements previous work of A. Langer, who showed that if $p\geq n=2m+1$ then $Q_{n}$ is $D$-affine.