Algebraically Skew Embeddings of Curves

TL;DR

The paper defines algebraically skew embeddings of smooth complex varieties into projective spaces and links skewness to Gauss maps and Terracini loci. It establishes a universal upper bound $ ext{msdim} olimits_X\, efell o 4n+1$ via the tangent variety of the secant variety and a lower bound $ ext{msdim} olimits_X\ge 3n$ through a degeneracy-loci framework, then analyzes curves by blowing up Grassmannian products to resolve diagonal intersections. For curves in $\P^3$, the twisted cubic is the only skew curve, while in $\P^4$ the only algebraically skew curves are the rational normal and elliptic normal curves, with precise excess-intersection calculations confirming these classifications. The work synthesizes Veronese skewness, Terracini loci, and excess-intersection techniques to derive exact classifications and bounds, highlighting deep connections between higher-ampleness and skew embeddings. Overall, the results sharpen bounds on skew embeddings and furnish explicit curve classifications in low dimensions, providing a blueprint for studying higher-dimensional cases via degeneracy loci and blowups.

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.

Theorems & Definitions (60)

• Definition 1.1
• Definition 1.2
• Lemma 2.1
• Lemma 2.2
• proof
• Definition 2.3
• Proposition 2.4
• proof
• Corollary 2.5
• proof