On higher Gauss maps

TL;DR

This paper characterizes the fibers of higher Gauss maps in algebraic geometry, linking their dimensions to the structure of higher fundamental forms, and extends a classical theorem with new recursive formulas.

Contribution

It extends a classical theorem by establishing a recursive formula for higher fundamental forms and characterizes fibers of higher Gauss maps in algebraic geometry.

Findings

The fiber dimension of the i-th Gauss map equals m iff the (i+1)-th fundamental form consists of cones with a fixed vertex.

Introduces a recursive formula for higher fundamental forms.

Provides new insights into the structure of higher Gauss maps and their fundamental forms.

Abstract

We prove that the general fibre of the $i$-th Gauss map has dimension $m$ if and only if at the general point the $(i+1)$-th fundamental form consists of cones with vertex a fixed $\mathbb P^{m-1}$, extending a known theorem for the usual Gauss map. We prove this via a recursive formula for expressing higher fundamental forms. We also show some consequences of these results.