Some remarks on osculating self-dual varieties

TL;DR

This paper investigates osculating self-dual varieties in projective spaces, establishing the existence of many such subvarieties for each dimension k in the space a72k+1, expanding understanding of their geometric properties.

Contribution

It demonstrates the existence of numerous osculating self-dual subvarieties in a72k+1 for all ka01, providing new examples and insights into their structure.

Findings

Existence of many osculating self-dual k-dimensional subvarieties for each ka01.

Construction methods for osculating self-dual varieties in projective spaces.

Extension of known results to higher-dimensional cases.

Abstract

Let us say that a curve $C\subset\mathbb P^3$ is osculating self-dual if it is projectively equivalent to the curve in the dual space $(\mathbb P^3)^*$ whose points are osculating planes to~$C$. Similarly, we say that a $k$-dimensional subvariety $X\subset\mathbb P^{2k+1}$ is osculating self-dual if its second osculating space at the general point is a hyperplane and $X$ is projectively equivalent to the variety in $(\mathbb P^{2k+1})^*$ whose points are second osculating spaces to $X$. In this note we show that for each $k\ge 1$ there exist many osculating self-dual $k$-dimensional subvarieties in $\mathbb P^{2k+1}$.