A characterization of A-discriminantal hypersurfaces in terms of logarithmic Gauss map
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)].
scientific article; zbMATH DE number 4175148