Higher Gauss Maps of Veronese Varieties---a generalization of Boole's formula and degree bounds for higher Gauss map images---

TL;DR

This paper generalizes Boole's classical formula by deriving a closed degree formula for higher Gauss maps of Veronese varieties, providing new bounds and insights into their geometric properties.

Contribution

It introduces a closed formula for the degree of higher Gauss map images of Veronese varieties, extending classical results and offering degree bounds.

Findings

Derived a closed degree formula for higher Gauss maps of Veronese varieties.

Generalized Boole's 1844 formula for dual varieties.

Provided degree bounds for higher Gauss map images.

Abstract

The image of the higher Gauss map for a projective variety is discussed. The notion of higher Gauss maps here was introduced by Fyodor L. Zak as a generalization of both ordinary Gauss maps and conormal maps. The main result is a closed formula for the degree of those images of Veronese varieties. This yields a generalization of a classical formula by George Boole on the degree of the dual varieties of Veronese varieties in 1844. As an application of our formula, degree bounds for higher Gauss map images of Veronese varieties are given.