[Paper Review] Constructible functions on schemes and stacks

Dominic Joyce|arXiv (Cornell University)|Mar 18, 2004

Polynomial and algebraic computation8 references10 citations

TL;DR

This paper introduces a framework for defining and analyzing constructible functions on algebraic schemes and stacks over an algebraically closed field of characteristic zero. By leveraging constructible sets and their associated functions, the work establishes foundational properties and closure under operations like pushforwards and pullbacks, contributing a systematic theory for constructible functions in algebraic geometry.

ABSTRACT

Let K be an algebraically closed field of characteristic zero, X a K-variety, and X(K) the set of closed points in X. A constructible set S ⊆ X(K) is a finite

Motivation & Objective

• To formalize the theory of constructible functions on algebraic schemes and Deligne–Mumford stacks over an algebraically closed field of characteristic zero.
• To define constructible functions via finite unions of locally closed subsets and establish their algebraic and topological properties.
• To investigate how constructible functions behave under geometric operations such as pushforwards and pullbacks.
• To provide a framework for extending classical constructible function theory to the setting of algebraic stacks.
• To lay the groundwork for applications in motivic integration, representation theory, and enumerative geometry.

Proposed method

• Define constructible functions as integer-linear combinations of characteristic functions of constructible subsets in the set of closed points X(K).
• Utilize the structure of schemes and stacks to define constructible sets as finite unions of locally closed subsets.
• Apply the theory of constructible sets to define pushforwards and pullbacks of functions under morphisms of schemes and stacks.
• Employ the characteristic function approach to ensure compatibility with algebraic geometry operations and topological invariants.
• Establish closure under addition and scalar multiplication, ensuring the set of constructible functions forms a group.
• Use the fact that K is algebraically closed and of characteristic zero to ensure good behavior of closed points and constructible topology.

Experimental results

Research questions

• RQ1 How can constructible functions be systematically defined on algebraic stacks, extending classical definitions on schemes?
• RQ2 What are the closure properties of constructible functions under geometric operations such as pushforwards and pullbacks?
• RQ3 How do constructible functions behave under morphisms of schemes and stacks in characteristic zero?
• RQ4 What structural properties do constructible functions inherit from the underlying scheme or stack?
• RQ5 In what ways can this theory support applications in motivic integration or representation theory?

Key findings

• Constructible functions on schemes and stacks form a well-defined abelian group under addition and scalar multiplication.
• The theory is stable under pushforwards and pullbacks along morphisms, preserving constructibility.
• Characteristic functions of constructible sets generate the group of constructible functions, ensuring a concrete representation.
• The framework extends naturally from schemes to Deligne–Mumford stacks, maintaining consistency with geometric intuition.
• The use of an algebraically closed field of characteristic zero ensures that closed points are dense and well-behaved, supporting the theory’s foundation.
• The construction provides a robust algebraic foundation for future applications in motivic integration and enumerative geometry.