Gauss map

《incompressible-fluid-equations》 · 第 188 页 · Example 15.2

The description of the projective tangent space $T_p(X)$ of a variety $X\subseteq\mathbb P^n$ at a smooth point $p$ as a linear subspace of $\mathbb P^n$ allows us to define the Gauss map associated to a smooth variety. If $X$ is smooth of pure dimension $k$, this is just the map $\gamma_X:X\to G(k,n)$ sending a point $p\in X$ to its tangent plane $T_p(X)$. Note that if $X$ is singular then $\gamma_X$ is still defined and regular on the open set of smooth points of $X$ and so gives a rational map $X\dashrightarrow G(k,n)$.