Algebraically Skew Embeddings of Curves

TL;DR

This paper investigates the minimal embedding dimensions for algebraically skew embeddings of smooth complex varieties, establishing bounds and classifying curves based on their skew embedding properties.

Contribution

It provides bounds on the minimal dimension for algebraically skew embeddings and classifies algebraic curves according to their skew embedding dimensions.

Findings

Established upper and lower bounds for embedding dimensions.

Classified algebraic curves by their minimal skew embedding dimensions.

Applied techniques to one-parameter families of lines.

Abstract

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.