[Paper Review] Logarithmic Jet Spaces and Intersection Multiplicities

Seth Dutter|arXiv (Cornell University)|Feb 27, 2010

Algebraic Geometry and Number Theory2 references3 citations

TL;DR

This paper develops an algebraic theory of relative logarithmic jet spaces for log schemes, using Vojta's framework to establish uniform bounds on intersection multiplicities of curves and divisors. It generalizes Noguchi and Winkelmann's results by replacing the semi-abelian condition with a differential condition, proving that for morphisms from affine curves to log schemes, the order of vanishing of a divisor is bounded unless the image lies entirely in the divisor's support.

ABSTRACT

The theory of relative logarithmic jet spaces is developed for log schemes. With this theory the existence of bounds of intersection multiplicities of curves and divisors on certain log schemes is established. This result extends those of Noguchi and Winkelmann by replacing the semi-abelian condition by a differential one.

Motivation & Objective

• To establish a purely algebraic theory of relative logarithmic jet spaces for log schemes, avoiding reliance on complex topology.
• To generalize Noguchi and Winkelmann's results on intersection multiplicities by replacing the semi-abelian condition with a differential condition.
• To demonstrate the utility of log geometry, compactification, and jet spaces in bounding intersection multiplicities.
• To provide a geometric interpretation of Mason’s Theorem and its corollaries in higher-dimensional schemes.
• To prove that for morphisms from affine curves to log schemes, the order of vanishing of a divisor is uniformly bounded unless the image lies in the divisor's support.

Proposed method

• Develops relative logarithmic jet spaces using Vojta’s algebraic framework, avoiding complex analytic methods.
• Uses log differential forms and jet space constructions to analyze the vanishing order of sections.
• Applies the log derivative trick via rational functions and partial fractions to bound orders of vanishing.
• Constructs a decreasing sequence of closed subsets $ H_n $ in the jet space to control the multiplicity of pullbacks.
• Employs the projection $ \pi_n $ from the jet space to the base to define the sets $ H_n $, which stabilize due to the noetherian property.
• Uses the isomorphism between jet space ideals and vanishing orders to link geometric conditions to algebraic multiplicity bounds.

Experimental results

Research questions

• RQ1 Can intersection multiplicities of curves and divisors on log schemes be bounded using an algebraic jet space theory?
• RQ2 How does the differential condition in this paper generalize the semi-abelian condition used by Noguchi and Winkelmann?
• RQ3 What role does log geometry play in compactifying jet spaces and controlling multiplicities?
• RQ4 Can the bound on multiplicity be derived purely algebraically without complex analytic tools?
• RQ5 Under what conditions does the image of a curve lie entirely in the support of a divisor, and when is the multiplicity bounded?

Key findings

• The order of vanishing of a divisor $ D $ under pullback by a morphism $ j: C \to X $ is bounded by $ N $, where $ N $ is the number of points removed from the affine line, unless $ j(C) \subset \operatorname{Supp}(D) $.
• The bound $ \operatorname{ord}_p j^*(D) \leq N $ holds uniformly for all points $ p \in C $, generalizing Corollary 1.4 to higher dimensions.
• The construction of the decreasing sequence $ H_n $ stabilizes due to the noetherian property of the base scheme, ensuring a finite bound on multiplicity.
• The theory applies in arbitrary characteristic, though the hypotheses are only non-trivial in characteristic zero.
• The result holds for morphisms from any affine curve to the complement of a divisor defined by regular functions whose differentials generate the sheaf of differentials.
• A global log 1-form can be constructed on projective space using homogeneous polynomials and log structures, enabling the application of the main theorem in affine patches.