[Paper Review] Toroidalization of birational morphisms of 3-folds

Steven Dale Cutkosky|ArXiv.org|Jul 14, 2004

Algebraic Geometry and Number Theory32 references3 citations

TL;DR

This paper proves that any birational morphism of 3-folds over a field of characteristic zero can be toroidalized via a sequence of blow-ups of nonsingular curves and points, resulting in a commutative diagram where the induced morphism becomes toroidal with respect to simple normal crossing divisors. The key contribution is a constructive proof using local power series analysis and resolution techniques, establishing toroidalization for 3-folds and yielding new proofs of weak and strong factorization in this dimension.

ABSTRACT

We prove that a birational morphism of projective 3-folds, over a field of characteristic zero, can be made toroidal by performing a sequence of blow ups of points and nonsingular curves above the domain and target.

Motivation & Objective

• To resolve the toroidalization problem for birational morphisms of 3-folds in characteristic zero.
• To extend the known results of toroidalization from surfaces and curves to 3-folds, where previous methods fail due to increased geometric complexity.
• To provide a constructive method using blow-ups of nonsingular curves and points to transform a birational morphism into a toroidal one.
• To establish a new proof of weak factorization for 3-folds by combining toroidalization with existing weak factorization theorems.
• To reduce the strong factorization conjecture for 3-folds to the case of toroidal morphisms, assuming the Oda conjecture holds.

Proposed method

• Use resolution of singularities and resolution of indeterminacy to construct an initial commutative diagram with nonsingular, projective 3-folds.
• Apply a preparation process to ensure the morphism is prepared and the total transform of the divisor is cuspidal.
• Employ descending induction on a numerical invariant τ(X) to reduce the complexity of the morphism step by step.
• Analyze local power series expansions of the morphism at singular points (1-points and 2-points) to verify toroidal structure.
• Use the λ-invariant (defined via Jacobian determinants and exponents) to detect toroidal behavior in local charts.
• Verify toroidal form by showing that the Jacobian determinant has exponent 1 and that the morphism is locally monomial after coordinate changes.

Experimental results

Research questions

• RQ1 Can every birational morphism of 3-folds over a field of characteristic zero be toroidalized via blow-ups of nonsingular curves and points?
• RQ2 Does toroidalization of birational morphisms of 3-folds imply weak factorization in this dimension?
• RQ3 Can the strong factorization conjecture for 3-folds be reduced to the case of toroidal morphisms?
• RQ4 What local invariants or conditions ensure that a morphism becomes toroidal after blow-ups?
• RQ5 How does the structure of the Jacobian determinant and the λ-invariant detect toroidal behavior in local power series expansions?

Key findings

• The paper establishes that every birational morphism of 3-folds over a field of characteristic zero admits a toroidalization via blow-ups of nonsingular curves and points.
• The morphism becomes toroidal after a finite sequence of blow-ups, with the target and source 3-folds becoming nonsingular and equipped with simple normal crossing divisors.
• The λ-invariant condition λ(E) = 1 at each exceptional divisor component E is shown to imply toroidal form after local coordinate changes.
• The proof uses descending induction on τ(X), a numerical invariant measuring the complexity of the morphism, to reduce to a case where f is toroidal.
• The result implies a new proof of weak factorization for 3-folds by combining toroidalization with the weak factorization theorem of AKMW and Włodarczyk.
• Corollary 0.3 shows that if the Oda conjecture on strong factorization of toroidal morphisms holds, then the Abhyankar–Hironaka strong factorization conjecture for 3-folds is true.