A characterization of A-discriminantal hypersurfaces in terms of logarithmic Gauss map
type: Item
instance of: scholarly article
title: A characterization of A-discriminantal hypersurfaces in terms of logarithmic Gauss map
zbMATH Open document ID: 0714.14031
DOI: 10.1007/BF01459245
author: Mikhail Kapranov
published in: Mathematische Annalen
publication date: 1991-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164818
review text: Let G be a complex algebraic group with Lie algebra \({\mathcal G}\) and \(Z\subset G\) a hypersurface. The (left) Gauss map \(\gamma_ Z: Z\to {\mathbb{P}}({\mathcal G}^*)\) takes a smooth point \(z\in Z\) to the left translation to unity of the tangent hyperplane \(T_ zZ\subset T_ zG\). It is a natural question to classify algebraic hypersurfaces \(Z\in G\) for which \(\gamma_ Z\) is birational. In the paper this is done for the case when \(G=({\mathbb{C}}^*)^ N\) is an algebraic torus. It turns out that hypersurfaces with the above properties are (up to killing inessential quasi-homogeneities) so-called A-discriminantal hypersurfaces \(\nabla_ A\), where A is a certain finite set of (Laurent) monomials. More precisely, \(\nabla_ A\) is the closure of the set of Laurent polynomials on the monomials from A which have a critical point with critical \(value\quad 0.\) The calculations involved in the proof of this theorem go back to the classical paper by \textit{J. Horn} [Math. Ann. 34, 577-600 (1889)].
Mathematics Subject Classification ID: 14M12
Mathematics Subject Classification ID: 14J70
zbMATH DE Number: 4175148
zbMATH Keywords: Gauss map
zbMATH Keywords: algebraic torus
zbMATH Keywords: discriminantal hypersurfaces
MaRDI profile type: Publication
cites work: Generating functional of correlation functions for the nonlinear Schrödinger equation
cites work: Discriminants of polynomials in several variables
description: scientific article; zbMATH DE number 4175148