TY - JOUR
T1 - On separable higher Gauss maps
AU - Furukawa, Katsuhisa
AU - Ito, Atsushi
N1 - Funding Information:
Received June 24, 2017. Revision received October 30, 2017. The first author was supported by the Grant-in-Aid for JSPS fellows, No. 16J00404. The second author was supported by the Grant-in-Aid for JSPS fellows, No. 14J01881.
Funding Information:
The first author was supported by the Grant-in-Aid for JSPS fellows, No. 16J00404. The second author was supported by the Grant-in-Aid for JSPS fellows, No. 14J01881. The authors would like to thank Professors Satoru Fukasawa and Hajime Kaji for their valuable comments and advice.
Publisher Copyright:
© 2019 University of Michigan. All rights reserved.
PY - 2019
Y1 - 2019
N2 - We study the mth Gauss map in the sense of F. L. Zak of a projective variety X ⊂ PN over an algebraically closed field in any characteristic. For all integers m with n := dim(X) ≤ m < N, we show that the contact locus on X of a general tangent m-plane is a linear variety if the mth Gauss map is separable. We also show that for smooth X with n < N − 2, the (n + 1)th Gauss map is birational if it is separable, unless X is the Segre embedding P1 × Pn ⊂ P2n−1. This is related to Ein's classification of varieties with small dual varieties in characteristic zero.
AB - We study the mth Gauss map in the sense of F. L. Zak of a projective variety X ⊂ PN over an algebraically closed field in any characteristic. For all integers m with n := dim(X) ≤ m < N, we show that the contact locus on X of a general tangent m-plane is a linear variety if the mth Gauss map is separable. We also show that for smooth X with n < N − 2, the (n + 1)th Gauss map is birational if it is separable, unless X is the Segre embedding P1 × Pn ⊂ P2n−1. This is related to Ein's classification of varieties with small dual varieties in characteristic zero.
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U2 - 10.1307/mmj/1555574416
DO - 10.1307/mmj/1555574416
M3 - Article
AN - SCOPUS:85071733551
SN - 0026-2285
VL - 68
SP - 483
EP - 503
JO - Michigan Mathematical Journal
JF - Michigan Mathematical Journal
IS - 3
ER -