Cox rings of trinomial hypersurfaces
An affine \textit{trinomial hypersurface} is an affine variety \(X\subseteq K^n\) given by the zeros of a polynomial of the form \(T_0^{\ell_0}+T_1^{\ell_1}+T_2^{\ell_2}\), where \(K\) is an algebraically closed field of characteristic zero, \(\ell_i=(\ell_{i1},\ldots,\ell_{in_i})\in{\mathbb Z}_{>0}^{n_i}\), \(T_i^{\ell_i}=T_{i1}^{\ell_{i1}}\cdots,T_{in_i}^{\ell_{in_i}}\in K[T_{ij}: 0\leq i\leq 2,\; 1\leq j\leq n_i]\) and \(n=n_0+n_1+n_2\). Trinomial hypersurfaces \(X\subseteq K^n\) have an effective action of an algebraic torus of dimension \(n-2\) given as the stabilizer \(T\) of the vanishing ideal \(I(X)\subseteq K[T_{ij}]\). The class group of a rational trinomial variety has been explicitly described by \textit{M. Wrobel} in [J. Algebra 542, 43--64 (2020; Zbl 1440.13054)]. The focus of the paper under review is the prime spectrum \(\text{Spec}({\mathcal R}(X))\) of the Cox ring \({\mathcal R}(X)\) of a trinomial hypersurface \(X\subseteq K^n\). The main result, Theorem 1, gives necessary and sufficient conditions for the total coordinate space \(\text{Spec}({\mathcal R}(X))\) to be a hypersurface; these conditions are explicitly formulated in terms of the greatest common divisors \(d_i\) of the multidegrees \(\ell_i\) of the monomials \(T_i^{\ell_i}\) of the trinomial hypersurface \(X\). The proof of this result uses a method for explicitly determining the Cox ring of any affine trinomial hypersurface using a polyhedra divisor which describes the torus action of complexity \(1\). This calculation coincides with previous results on Cox rings of non--complete varieties with an effective torus action of complexity \(1\) that have been described by \textit{J. Hausen} and \textit{M. Wrobel} in [Math. Nachr. 290, No. 5--6, 815--826 (2017; Zbl 1371.13002)]. The second main result establish criteria for the total coordinate space \(\text{Spec}({\mathcal R}(X))\) of a trinomial hypersurface to be rational or factorial, again in terms of the greatest common divisors \(d_i\).
scientific article; zbMATH DE number 7369766
scientific article