Higher Gauss maps of Veronese varieties—A generalization of Boole’s formula and 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.

KEYWORDS:

• Boole’s formula
• degree bound
• dual variety
• higher Gauss map
• Veronese variety

2000 MATHEMATICS SUBJECT CLASSIFICATION:

• 14N05

Acknowledgments

The author would like to thank Professor Fyodor L. Zak, who gave me detailed comments and expert advice. The author would like to thank Professor Shigeharu Takayama, too: the present work was started by his question on higher Gauss maps. The author wishes to thank Professor Satoru Fukasawa for useful comments and invaluable advice, and Professor Tomohide Terasoma for useful discussion. Finally the author would like to thank Professor Wu-yen Chuang, Professor Jiun-Cheng Chen, Professor Jungkai Chen and Professor Katsuhisa Furukawa for inviting me to the mini-conference on algebraic geometry (March 6, 2015) at National Center for Theoretical Sciences (NCTS), and for their warm hospitality throughout his stay in Taipei: In fact, the present work was done partly at NCTS.