[Paper Review] Jumping Numbers on Algebraic Surfaces with Rational Singularities

Kevin Tucker|ArXiv.org|Jan 4, 2008

Algebraic Geometry and Number Theory14 references4 citations

TL;DR

This paper presents a computational algorithm to determine jumping numbers of ideals on complex algebraic surfaces with rational singularities by analyzing contributions from reduced exceptional divisors in a fixed log resolution. The key contribution is a numerical criterion that identifies which divisor chains critically contribute jumping numbers, enabling explicit computation for plane curves and Du Val/toric singularities, with applications to smooth surfaces and ideal simplicity.

ABSTRACT

In this article, we study the jumping numbers of an ideal in the local ring at rational singularity on a complex algebraic surface. By understanding the contributions of reduced divisors on a fixed resolution, we are able to present an algorithm for finding of the jumping numbers of the ideal. This shows, in particular, how to compute the jumping numbers of a plane curve from the numerical data of its minimal resolution. In addition, the jumping numbers of the maximal ideal at the singular point in a Du Val or toric surface singularity are computed, and applications to the smooth case are explored.

Motivation & Objective

• To develop a systematic method for computing jumping numbers of ideals on algebraic surfaces with rational singularities.
• To extend the notion of jumping number contribution beyond prime divisors to include reduced divisor chains, ensuring all jumping numbers are captured.
• To provide a numerical criterion for identifying which exceptional divisor chains critically contribute a jumping number.
• To apply the method to compute jumping numbers for plane curves and ideals in smooth surfaces, including characterizing ideal simplicity.
• To demonstrate that every connected chain of exceptional divisors can critically contribute a jumping number for some ideal, generalizing prior results.

Proposed method

• Define jumping numbers as values of λ where multiplier ideals J(X, a^λ) change, using a log resolution π: Y → X with F = -div(a) and relative canonical divisor K_π.
• Introduce the concept of 'critical contribution' by a reduced divisor G = E₁ + ⋯ + Eₖ, where G is a connected chain of rational curves and contributes λ only if no proper subchain does.
• Use the condition that λ is a candidate jumping number for G if ord_Ei(K_π - λF) is integer for all i, and λ is realized as a jumping number iff J(X, a^λ) ≠ π_*O_Y(⌈K_π - λF⌉ + G).
• Establish a numerical criterion: λ is critically contributed by G iff the ends of G satisfy −⌊λF⌋·E = 1 and non-ends satisfy −⌊λF⌋·E′ = 0.
• Apply the adjunction formula to simplify the criterion, reducing it to checking intersection numbers between ⌊λF⌋ and exceptional divisors.
• Leverage Zariski-Lipman theory of complete ideals in 2D regular local rings, using the dual basis {Ê_i} to express ideals and their Rees valuations.

Experimental results

Research questions

• RQ1 Which reduced exceptional divisors critically contribute a jumping number for an ideal on a rational surface singularity?
• RQ2 How can one algorithmically compute the jumping numbers of a plane curve from the numerical data of its minimal resolution?
• RQ3 What is the precise condition under which a connected chain of exceptional divisors contributes a jumping number critically?
• RQ4 Can every connected chain of exceptional divisors be shown to critically contribute a jumping number for some ideal?
• RQ5 What is the relationship between the simplicity of a complete finite colength ideal and the presence of 1 as a jumping number?

Key findings

• Every jumping number is critically contributed by some reduced exceptional divisor G, and such G must be a connected chain of smooth rational curves.
• A jumping number λ is critically contributed by a reduced divisor G if and only if the ends of G satisfy −⌊λF⌋·E = 1 and the internal nodes satisfy −⌊λF⌋·E′ = 0.
• For a plane curve, the jumping numbers can be computed by checking finitely many inequalities between intersection numbers on the minimal resolution.
• The maximal ideal at a Du Val or toric surface singularity has explicitly computable jumping numbers using the algorithm.
• A complete finite colength ideal in the local ring of a smooth surface is simple if and only if 1 is not a jumping number of the ideal.
• Every connected chain of exceptional divisors can be shown to critically contribute a jumping number for some ideal, with 1 being such a jumping number for the product of two simple ideals.