Some remarks on osculating self-dual varieties

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 $({\mathbb{P}}^3{)}^{\ast }$ 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 $({\mathbb{P}}^{2k+1}{)}^{\ast }$ 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 ${\mathbb{P}}^{2k+1}$.

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