toroidal morphism AXAL6IDE

toroidal morphism

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Definition 5.1.1 (cf. [ACSS21, Definition 2.1]). Let (X, ΣX , M)/U be a g-pair. We say that (X, ΣX , M) is toroidal if ΣX is a reduced divisor, M descends to X, and for any closed point x ∈ X, there exists a toric variety Xσ, a closed point t ∈ Xσ, and an isomorphism of complete local algebras φx : ̂OX,x ∼= ̂OXσ,t such that the ideal of ΣX maps to the invariant ideal of Xσ\Tσ, where Tσ ⊂ Xσ is the maximal torus of Xσ. Any such (Xσ, t) will be called as a local model of (X, ΣX , M) at x ∈ X. Let (X, ΣX , M)/U and (Z, ΣZ , MZ )/U be toroidal g-pairs and f : X → Z a surjective morphism/U . We say that f : (X, Σ, M) → (Z, Σ, MZ ) is toroidal, if for every closed point x ∈ X, there exist a local model (Xσ, t) of (X, ΣX , M) at x, a local model (Zτ , s) of (Z, ΣZ , MZ ) at z := f (x), and a toric morphism g : Xσ → Zτ , so that the diagram of algebras commutes. ̂OX,x ∼= / ̂OXσ,t ̂OZ,z ∼= / O ̂OZτ ,s O Here the vertical maps are the algebra homomorphisms induced by f and g respectively. (pdf) (Chen 等, 2023, p. 37)