The separability of the Gauss map and the reflexivity for a projective surface
type: Item
instance of: scholarly article
title: The separability of the Gauss map and the reflexivity for a projective surface
zbMATH Open document ID: 1118.14058
author: Satoru Fukasawa
author: Hajime Kaji
published in: Mathematische Zeitschrift
publication date: 2007-06-07T00:00:00Z
review text: Let \(X\subset{\mathbb P}^N\) be a projective variety of dimension \(n\) over an algebraically closed field. The Gauss map of \(X\) is the map associating to every smooth point of \(X\) the embedded tangent space at that point, viewed as an element of the Grassmannian of \(n\)-dimensional linear subspaces of \({\mathbb P}^N\). The variety \(X\) is called reflexive if its conormal variety \(CX\) is isomorphic to the conormal variety of its dual variety \(X^*\). An equivalent condition is that the projection from \(CX\) to \(X^*\) is separable. It is well known that if a variety \(X\) is reflexive, then its Gauss map is separable. The main result of this note is that also the converse holds if \(X\) is a projective surface. This property was already known in the case of curves, but is false if the dimension of \(X\) is bigger than three [see \textit{H. Kaji}, Geom. Dedicata 99, 221--229 (2003; Zbl 1027.14025) and \textit{S. Fukasawa}, Compos. Math. 142, No. 5, 1305--1307 (2006; Zbl 1107.14044)]. In the case of curves, the property is related to the fact that the Gauss map of \(X\) and the natural map \(CX\rightarrow X^*\) always have the same inseparable degree [\textit{S. Kleiman} and \textit{R. Piene}, Contemp. Math. 123, 107--129 (1991; Zbl 0758.14032)]. This does not hold for projective surfaces with \(2\)-dimensional Gauss image, as the authors show by giving a counterexample.
reviewed by: Orsola Tommasi
Mathematics Subject Classification ID: 14N05
Mathematics Subject Classification ID: 14M15
Mathematics Subject Classification ID: 53A20
zbMATH DE Number: 5162364
MaRDI profile type: Publication
full work available at URL: s00209-006-0085-0
OpenAlex ID: W2017644634
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DOI: 10.1007/S00209-006-0085-0
description: scientific article; zbMATH DE number 5162364