Versality in toric geometry (Q2168806)

This paper studies the deformation theory of affine toric singularities. The versal base space inherits a torus action, and thus a lattice grading. The aim is to construct a maximal deformation in a given primitive degree \(-R\). For isolated Gorenstein toric singularities the whole versal deformation is concentrated in a single degree. The versal deformation for such toric singularities was obtained by the first author [Invent. Math. 128, No. 3, 443--479 (1997; Zbl 0894.14025)]. The present paper drops both the Gorenstein and the smoothness in codimension two assumptions; now the theory applies to 2-dimensional cyclic quotients. Let \(N\) and \(M\) be dual lattices and \(\sigma\subset (N \oplus \mathbb{Z}) \otimes_{\mathbb{Z}} \mathbb{R}\) a polyhedral cone. The toric singularity is \(\operatorname{Spec} k[S]\), where \(S=\sigma^\vee \cap (M\oplus \mathbb Z)\). Given an \( R \in M \oplus \mathbb{Z}\) the cross-cut of \(\sigma\) with the affine hyperplane \([R = 1]\) gives a rational polyhedron \(P\). Starting with such a \(P\) the Authors construct a pair of monoids \(\widetilde T \subset \widetilde S\) extending the map \(R\colon \operatorname{Spec} k[S] \to \operatorname{Spec} k[\mathbb N]\). The monoid \(\widetilde T\) is a generalization of the Minkowski scheme of a lattice polytope and the monoid \(\widetilde S\) is a generalization of the monoid corresponding to a tautological cone. According to the main result of the preprint [\textit{K. Altmann} et al., ``Polyhedra, lattice structures, and extensions of semigroups'', Preprint, \url{arXiv:2004.07377}] of the authors the pair \((\widetilde T ,\widetilde S)\) is a universal extension of the pair \((\mathbb{N}\cdot R, S)\). The flat map \( \operatorname{Spec} k[\widetilde S] \to \operatorname{Spec} k[\widetilde T]\) is a deformation of the hyperplane section \(R\) of \(X\). Under the assumption that \(\widetilde T\) is generated by elements of degree \(1\) (in fact under a slightly stronger assumption), which translates in length assumptions on the edges of the polyhedron, it is shown that there is a maximal subspace of \( \operatorname{Spec} k[\widetilde T]\) invariant under the diagonal action of \(k\), and the quotient is the base space of the versal deformation of \(X\) in the fixed primitive degree \(-R\). The construction is explained with many examples. The proof of versality depends on a detailed description of all the maps involved, covering the last three sections.