Local Invariants Distinguishing Algebraic Spaces from Schemes and Their Applications to Moduli Spaces

Core Concepts

This paper introduces new local invariants, the schematic dimension and the schematic fiber, to characterize the difference between algebraic spaces and schemes, and applies these invariants to determine the schematicness of moduli spaces.

Abstract

Bibliographic Information:

Fernandez Herrero, A., Weißmann, D., & Zhang, X. (2024). Distinguishing Algebraic Spaces from Schemes. arXiv:2411.07169v1 [math.AG].

Research Objective:

This paper aims to develop criteria for distinguishing algebraic spaces from schemes and apply these criteria to determine when the moduli space of a stack is a scheme.

Methodology:

The authors introduce two new local invariants: the schematic dimension and the schematic fiber. These invariants measure how far an algebraic space or stack is from being a scheme. The schematic dimension quantifies the largest dimension of a scheme that can be locally obtained from the space, while the schematic fiber captures the "excess intersection" of Weil divisors.

Key Findings:

• The paper establishes a connection between local uniform base points (topological property) and schematically trivial points (geometric property) in specific settings like locally factorial spaces.
• The authors prove that the schematic fiber at a point is intrinsically linked to the schematicness of that point. A point is schematic if and only if its schematic fiber has dimension zero.
• The paper provides a criterion for the schematicness of a separated moduli space: a point in the moduli space is schematic if and only if its pre-image in the stack has a schematic fiber consisting of a single closed point.

Main Conclusions:

The introduced local invariants provide effective tools for studying the geometry of algebraic spaces and stacks. The schematic fiber criterion offers a practical way to determine the schematicness of moduli spaces, particularly useful in studying moduli problems in algebraic geometry.

Significance:

This research contributes significantly to the understanding of the subtle differences between algebraic spaces and schemes. The developed criteria have broad applications in moduli theory, potentially impacting the construction and analysis of moduli spaces in various geometric contexts.

Limitations and Future Research:

The paper primarily focuses on spaces and stacks over an algebraically closed field. Further research could explore generalizations of these concepts to more general base schemes. Additionally, investigating the behavior of these invariants under various geometric operations like base change and blowups could provide further insights.

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Key Insights Distilled From

Distinguishing Algebraic Spaces from Schemes

by Andr... at arxiv.org 11-12-2024

https://arxiv.org/pdf/2411.07169.pdf

Deeper Inquiries

How do the concepts of schematic dimension and schematic fiber extend to algebraic spaces and stacks over more general base schemes?

Extending the concepts of schematic dimension and schematic fiber to algebraic spaces and stacks over more general base schemes, beyond the case of an algebraically closed field k, requires careful consideration of the following aspects:
1. Finite Type Algebras: The definition of finite type algebras at a point can be naturally generalized to a relative setting. Given a morphism X → S of algebraic stacks, a finite type algebra at a geometric point x ∈ |X| lying over a point s ∈ |S| would be a pair (U, A), where U ⊆ X is an open substack containing x and A is a quasi-coherent OU-algebra such that A is finitely generated as an OU-algebra and the composition A → OU → κ(x) factors through OS,s → κ(x).
2. Schematic Dimension: The schematic dimension Sdimx(X/S) of X at x relative to S can be defined analogously as the maximum of the dimensions of Spec(A) as we vary over all finite type algebras (U, A) at x.
3. Schematic Fiber: The definition of the schematic fiber requires a bit more care. One possible approach is to consider the fiber product. Given a finite type algebra (U, A) at x, we can base change the morphism U → Spec(A) along the morphism Spec(κ(s)) → S to obtain a morphism Us → Spec(A ⊗OS,s κ(s)). The schematic fiber Fibx(X/S) can then be defined as the minimal closed substack of Xs containing x that arises as the image of the fiber over the point Spec(κ(x)) → Spec(A ⊗OS,s κ(s)) for some finite type algebra (U, A) at x.
Challenges and Considerations:

Non-Closed Points: Over a general base scheme, we need to consider non-closed points as well. The definitions would need to be adapted to handle this more general situation.
Properties: Key properties like semi-continuity of schematic dimension and upper semi-continuity of schematic fiber dimension might require additional assumptions on the morphism X → S, such as flatness or properness.
Geometric Interpretation: The geometric interpretation of these concepts in the relative setting would be more subtle and might require further investigation.

Could there be alternative geometric interpretations of the schematic fiber, potentially leading to new insights or connections with other geometric notions?

Yes, exploring alternative geometric interpretations of the schematic fiber could be very fruitful. Here are a few potential avenues:
1. Formal Geometry: Instead of just looking at Zariski open neighborhoods, one could consider the formal completion X̂x of X at x. The schematic fiber might be related to the structure of this formal scheme and how it compares to the formal completion of a scheme at a point.
2. Deformations: The schematic fiber could potentially be understood in terms of deformations. It might capture obstructions to extending infinitesimal deformations of x to deformations of a neighborhood of x that can be embedded in a scheme.
3. Descent Data: For an algebraic stack X, the schematic fiber might be related to the descent data defining X. It could capture the failure of this descent data to be effective in a schematic neighborhood of x.
4. Derived Algebraic Geometry: Viewing algebraic stacks and spaces in the context of derived algebraic geometry might offer new perspectives on the schematic fiber. It could be related to the cotangent complex or other derived invariants.
Connections to Other Notions:

Non-schematic Loci: The schematic fiber could provide a finer invariant for studying the non-schematic locus of an algebraic stack, complementing existing notions like the locus of schematic points.
Moduli Spaces: A deeper understanding of the schematic fiber could lead to more refined results about the structure of moduli spaces, particularly their non-schematic behavior.

How can the understanding of schematicness in moduli spaces be leveraged to study specific moduli problems in other areas of mathematics, such as representation theory or mathematical physics?

The interplay between schematicness and moduli spaces has significant implications for various areas of mathematics, including representation theory and mathematical physics. Here's how a deeper understanding can be leveraged:
Representation Theory:

Moduli of Representations: Moduli spaces of representations of quivers or algebras often arise in representation theory. Understanding when these moduli spaces are schematic can provide insights into the structure of the underlying representations and their deformations. For instance, schematicness might be related to the existence of certain stability conditions or the absence of "pathological" representations.
Geometric Langlands Program: The geometric Langlands program connects representation theory with the geometry of moduli spaces of bundles on curves. Schematicness of these moduli spaces plays a crucial role in establishing the correspondence between certain categories of sheaves and representations.
Mathematical Physics:

Gauge Theory: Moduli spaces of connections or instantons are fundamental objects in gauge theory. Their schematic properties are closely tied to the physical behavior of the theory. For example, non-schematic points might correspond to singularities in the moduli space, which can have physical interpretations as special solutions or phase transitions.
String Theory: Moduli spaces of curves, Calabi-Yau manifolds, or other geometric structures appear in string theory as parameter spaces for different string vacua. Understanding their schematic properties is essential for studying the landscape of string theory and its potential implications for particle physics and cosmology.
General Strategy:

Identify Moduli Problem: Start with a specific moduli problem arising in representation theory or mathematical physics.
Construct Moduli Space: Use appropriate techniques to construct a moduli space (algebraic stack or space) that parametrizes the objects of interest.
Analyze Schematicness: Apply the tools and criteria discussed in the context of schematic dimension and schematic fiber to investigate the schematic properties of the moduli space.
Interpret Results: Relate the findings about schematicness back to the original problem, drawing connections to representation-theoretic or physical phenomena.

By systematically following this approach, we can gain valuable insights into the structure of moduli spaces and their connections to other areas of mathematics.