Gauss maps of toric varieties (Q1683771)

For a projective variety \(X\subset\mathbb P^N\), of dimension \(n\), defined over an algebraically closed field \(K\) (of arbitrary characteristic), the Gauss map is the rational map \(\gamma: X\dashrightarrow G(n,N)\) which sends a smooth point of \(X\) to the point of the Grassmannian which parameterizes the tangent \(n\)-space to \(X\). Several properties of the Gauss map are known when \(X\) is a general variety. The authors investigate the special case in which \(X\) is a toric variety, whose torus action extends to \(\mathbb P^N\). Such varieties \(X\) are associated to the choice of a free abelian group \(M\) of rank \(n\) and a finite subset \(A =\{u_0,\dots,u_N\}\) of \(M\). The authors prove that, by setting \(B=\{u_{i_0}+\dots+u_{i_n}: u_{i_0},\dots,u_{i_n}\) span \(M\otimes K\}\), then the image of \(\gamma\) is the toric variety associated to \(B\). The authors also describe, in terms of the toric structure, the restriction of \(\gamma\) to the torus \(T_M\subset X\). As a consequence, the authors find that the Gauss map of a toric variety \(X\) as above is degenerate if and only if \(X\) is the join of torus invariant subvarieties. The authors also prove that given two toric varieties, \(Y\subset\mathbb P^M\) and \(Z\), such that \(\dim(Y)+\dim(Z)\geq M\), then there exists a toric variety \(X\) whose Gauss image \(\gamma(X)\) is projectively equivalent to \(Z\) and such that the general fibers of \(\gamma\) are projectively equivalent to \(Y\).