equi dimensional model BKE5X4UL

equi-dimensional model

---

Definition-Theorem 5.1.2 ([LLM23, Definition-Theorem 6.5], [ACSS21, Theorem 2.2]). Let X be a normal quasi-projective variety, X → U a projective morphism, X → Z a contraction, B an R-divisor on X, M a nef/U b-divisor on X, D1, . . . , Dm prime divisors over X, and DZ,1, . . . , DZ,n prime divisors over Z. Then there exist a toroidal g-pair (X′, ΣX′ , M)/U , a log smooth pair (Z′, ΣZ′), and a commutative diagram X′ h / f′ X f Z′ hZ / Z satisfying the following. (1) h and hZ are projective birational morphisms. (2) f ′ : (X′, ΣX′ , M) → (Z′, ΣZ′) is a toroidal contraction. (3) Supp(h∗−1B) ∪ Supp Exc(h) is contained in Supp ΣX′ . (4) X′ has at most toric quotient singularities. (5) f ′ is equi-dimensional. (6) M descends to X′. (7) X′ is Q-factorial klt. (8) The center of each Di on X′ and the center of each DZ,i on Z′ are divisors. We call any such f ′ : (X′, ΣX′, M) → (Z′, ΣZ′) (associated with h and hZ ) which satisfies (1-7) an equi-dimensional model of f : (X, B, M) → Z. (pdf) (Chen 等, 2023, p. 37)