Projective varieties admitting an embedding with Gauss map of rank zero
type: Item
instance of: scholarly article
title: Projective varieties admitting an embedding with Gauss map of rank zero
zbMATH Open document ID: 1202.14049
author: Satoru Fukasawa
author: Katsuhisa Furukawa
author: Hajime Kaji
published in: Advances in Mathematics
publication date: 2010-07-20T00:00:00Z
review text: Let \(X\) be a projective variety of dimension \(n\) in \({\mathbb P}^N\) defined over an algebraically closed field \(k\). Then the Gauss map associated with this embedding of \(X\) is the rational map that associates with every non-singular point \(x\in X\) the embedded tangent space to \(X\) at \(x\), viewed as a point of the Grassmannian of \(n\)-dimensional subspaces of \({\mathbb P}^N\). When the field \(k\) has characteristic \(0\), a variety whose Gauss map has rank \(0\) is automatically a linear space. The paper under review deals with the case where \(\text{char} k=p>0\). Let us call a variety \(X\) a GMRZ variety if it admits an embedding in projective space such that the rank of the differential of the Gauss map is \(0\) at a general point of \(X\). Then the paper under review shows that this condition imposes strong restrictions on the rational curves on \(X\). In particular, one of the main results is a precise characterization of minimal free rational curves on GMRZ varieties, i.e. rational curves \({\mathbb P}^1\rightarrow X\) such that the pull-back of the tangent bundle \(T_X\) is of the form \({\mathcal O}_{{\mathbb P}^1}(2)\oplus{\mathcal O}_{{\mathbb P}^1}(1)^{d-2}\oplus{\mathcal O}_{{\mathbb P}^1}^{n-d+1}\), in terms of the degree of the pull-back to \({\mathbb P}^1\) of \(K_X\) and of the line bundle defining the GMRZ embedding. Furthermore, a characterization of GMRZ varieties is given inside the classes of Segre and Grassmann varieties, for cubic hypersurfaces and for general hypersurfaces of low degree. As an application, the authors prove that Fermat hypersurfaces of degree \(mp+1\) and dimension \(\geq 2mp\) contain free rational curves but no minimal free rational curves. This is in contradiction with the situation in characteristic zero, where the existence of free rational curves on a variety \(X\), i.e. rational curves such that the pull-back of \(T_X\) is generated by global sections, implies the existence of minimal rational curves…
reviewed by: Orsola Tommasi
Mathematics Subject Classification ID: 14N05
Mathematics Subject Classification ID: 14C05
Mathematics Subject Classification ID: 14J45
Mathematics Subject Classification ID: 14J70
zbMATH DE Number: 5758038
zbMATH Keywords: Gauss map
zbMATH Keywords: inseparable
zbMATH Keywords: minimal free rational curve
zbMATH Keywords: normal bundle
zbMATH Keywords: hypersurface
MaRDI profile type: Publication
full work available at URL: j.aim.2010.02.017
OpenAlex ID: W2079176455
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DOI: 10.1016/J.AIM.2010.02.017
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description: scientific article; zbMATH DE number 5758038