Algebraically Skew Embeddings of Curves

Given a smooth complex variety $X$, an algebraically skew embedding of $X$ is an embedding of $X$ into a complex projective space ${\mathbb{P}}^N$ such that for any two points $x,y\in X$, their embedded tangent spaces in ${\mathbb{P}}^N$ do not intersect. In this work, we establish an upper bound and a lower bound of the minimal dimension $N$ such that there exists an algebraically skew embedding into ${\mathbb{P}}^N$ in terms of the dimension of the given smooth variety $X$. Then we further classify the algebraic curves in terms of their minimal skew embedding dimensions, and apply the same technique to other one-parameter family of lines.