Cubic hypersurfaces admitting an embedding with Gauss map of rank 0
type: Item
instance of: scholarly article
title: Cubic hypersurfaces admitting an embedding with Gauss map of rank 0
zbMATH Open document ID: 1281.14043
author: Katsuhisa Furukawa
published in: Advances in Mathematics
publication date: 2012-07-20T00:00:00Z
review text: Let \(X\) be an \(n\)-dimensional projective variety defined over an algebraically closed field \(K\). Each embedding of \(i:\;X\hookrightarrow{\mathbb P}^M\) gives rise to a Gauss map, the rational map that associates to each non-singular point \(x\) of \(X\) the element of the Grassmannian corresponding to the embedded tangent space to \(i(X)\) at \(i(x)\). A projective variety \(X\) is said to satisfy property (GMRZ) if it admits an embedding into projective space such that the rank of the differential of the Gauss map at a general point is \(0\). In the paper [\textit{S. Fukasawa, K. Furukawa} and \textit{H. Kaji}, Adv. Math. 224, No. 6, 2645--2661 (2010; Zbl 1202.14049)] it was shown that a smooth cubic hypersurface of dimension \(n\geq 4\) satisfies (GMRZ) if and only if it is isomorphic to the Fermat cubic hypersurface and the characteristic of the base field \(K\) is \(2\). In the paper under review, this result is extended to cubic threefolds. Moreover, it is shown that every smooth cubic surface over a field of characteristic \(2\) satisfies (GMRZ). To show this, the author proves that, in characteristic \(2\), property (GMRZ) is preserved under blowing up at points. This property is also used to construct further examples of varieties satisfying (GMRZ) in characteristic \(2\).
reviewed by: Orsola Tommasi
Mathematics Subject Classification ID: 14N05
Mathematics Subject Classification ID: 14J70
Mathematics Subject Classification ID: 14G17
zbMATH DE Number: 6059024
zbMATH Keywords: Gauss map
zbMATH Keywords: cubic hypersurfaces
zbMATH Keywords: blow up
zbMATH Keywords: positive characteristic
MaRDI profile type: Publication
full work available at URL: j.aim.2012.03.006
OpenAlex ID: W2050408693
cites work: Q3048935
cites work: Higher-dimensional algebraic geometry
cites work: Projective varieties admitting an embedding with Gauss map of rank zero
cites work: Existence of a non-reflexive embedding with birational Gauss map for a projective variety
cites work: Any algebraic variety in positive characteristic admits a projective model with an inseparable Gauss map
cites work: A connectedness theorem for projective varieties, with applications to intersections and singularities of mappings
cites work: On the Tangentially Degenerate Curves
cites work: Q3033882
cites work: Q3971595
cites work: The uniform position principle for curves in characteristic p
cites work: Q3798785
cites work: Q3140224
DOI: 10.1016/J.AIM.2012.03.006
Recommended article: Projective varieties admitting an embedding with Gauss map of rank zero
Recommended article: Existence of a non-reflexive embedding with birational Gauss map for a projective variety
Recommended article: On Kleiman–Piene's question for Gauss maps
Recommended article: Any algebraic variety in positive characteristic admits a projective model with an inseparable Gauss map
Recommended article: The injectivity of the extended Gauss map of general projections of smooth projective varieties
Recommended article: Varieties with nonconstant Gauss fibers
Recommended article: The existence of embeddings for which the Gauss map is an embedding
Recommended article: On Gauss Maps in Positive Characteristic in View of Images, Fibers, and Field Extensions
Recommended article: The separability of the Gauss map and the reflexivity for a projective surface
Recommended article: Gauss maps with nontrivial separable degree in positive characteristic
description: scientific article; zbMATH DE number 6059024