Quasilinear Tropical Compactifications: Extending the Geometry of Hyperplane Arrangement Compactifications

Core Concepts

This paper introduces the concept of "quasilinear tropical compactifications," a class of compactifications of subvarieties of algebraic tori that share desirable properties with compactifications of hyperplane arrangement complements, such as being "schön" and having a Chow ring determined by the tropical fan.

Abstract

• Bibliographic Information: SCHOCK, N. (2024). QUASILINEAR TROPICAL COMPACTIFICATIONS. arXiv preprint arXiv:2112.02062v3.

• Research Objective: To investigate the geometry of tropical compactifications and introduce a broader class, termed "quasilinear tropical compactifications," that exhibit desirable properties analogous to those found in compactifications of complements of hyperplane arrangements.

• Methodology: The paper employs tools from algebraic geometry, particularly toric geometry and tropical geometry. It leverages the concept of tropical modifications and the theory of Chow rings to analyze the properties of quasilinear tropical fans and their corresponding compactifications.

• Key Findings:

  • Quasilinear tropical compactifications are "schön," meaning they admit a smooth and surjective multiplication map from the torus to the compactification.
  • The Chow ring of a quasilinear tropical compactification is isomorphic to the Chow ring of the ambient toric variety.
  • The moduli spaces of 6 lines in P2 and marked cubic surfaces are proven to be quasilinear, yielding insights into the geometry of their stable pair compactifications.

• Main Conclusions: The introduction of quasilinear tropical compactifications provides a framework for studying a wider class of compactifications that retain key geometric properties of linear compactifications. The results have implications for understanding the geometry of moduli spaces, particularly those related to line arrangements and cubic surfaces.

• Significance: This research contributes significantly to the field of tropical geometry by expanding the understanding of tropical compactifications beyond the well-studied case of hyperplane arrangements. The quasilinearity property offers a new perspective on the geometry of certain moduli spaces and their compactifications.

• Limitations and Future Research: The paper primarily focuses on specific examples of quasilinear tropical compactifications. Further research could explore the properties and applications of this concept in a broader context, investigating other moduli spaces or geometric objects that might exhibit quasilinearity.

Stats

Quotes

• "The prototypical examples of tropical compactifications are compactifications of complements of hyperplane arrangements, which posses a number of remarkable properties not satisfied by more general tropical compactifications of closed subvarieties of tori."
• "We call complements of hyperplane arrangements linear varieties, and their tropical compactifications linear (tropical) compactifications."
• "We define a broader class of closed subvarieties of tori, called quasilinear varieties, and their tropical compactifications, called quasilinear (tropical) compactifications, which includes the class of linear varieties (and linear tropical compactifications), for which the analogues of the above properties hold."

Key Insights Distilled From

Quasilinear tropical compactifications

by Nolan Schock at arxiv.org 11-25-2024

https://arxiv.org/pdf/2112.02062.pdf

Deeper Inquiries

How does the concept of quasilinear tropical compactifications contribute to the understanding of other moduli spaces beyond those explored in the paper?

The concept of quasilinear tropical compactifications provides a powerful new lens through which to study moduli spaces in algebraic geometry. The paper focuses on the moduli spaces $M(3,6)$ and $Y(E_6)$, but the techniques and insights have broader implications:
1. Identifying Potential Quasilinear Moduli Spaces: The paper provides criteria for checking the quasilinearity of algebraic varieties (Theorem 1.5). These criteria can be applied to other moduli spaces, potentially revealing new examples where quasilinear tropical compactifications exist. Prime candidates are moduli spaces with known or conjectured "nice" combinatorial structures, such as those arising from matroids, hyperplane arrangements, or cluster algebras.
2. Understanding Degenerations and Compactifications: Quasilinear tropical compactifications offer a controlled way to understand how objects in a moduli space degenerate. The combinatorial nature of tropical geometry allows for explicit descriptions of the boundary strata, shedding light on the possible limiting behaviors of families of varieties. This can be particularly useful for moduli spaces where other compactification methods, such as Geometric Invariant Theory (GIT), lead to complicated or poorly understood boundaries.
3. Computing Invariants: The isomorphism between the Chow rings of a quasilinear tropical compactification and its ambient toric variety (Theorem 1.2) provides a powerful tool for computing intersection-theoretic invariants. This can be used to study enumerative problems, such as counting curves on surfaces or understanding the cohomology of moduli spaces.
4. Exploring Connections to Mirror Symmetry: The "K"ahler package" satisfied by quasilinear tropical fans suggests potential connections to mirror symmetry. Mirror symmetry predicts deep relationships between the symplectic geometry of one algebraic variety and the algebraic geometry of its mirror. The combinatorial nature of tropical geometry, combined with the rich structure of quasilinear fans, could provide a fruitful setting for exploring these connections.

Could there be alternative geometric conditions that imply some of the desirable properties of quasilinear tropical compactifications, potentially leading to an even broader class of well-behaved compactifications?

Yes, it's plausible that alternative geometric conditions could lead to a broader class of well-behaved compactifications sharing desirable properties with quasilinear tropical compactifications. Here are some potential avenues for exploration:
1. Relaxing Combinatorial Restrictions: Quasilinear tropical fans are built iteratively from complete fans using tropical modifications along quasilinear divisors. One could explore relaxing these conditions, for example, by allowing modifications along divisors satisfying weaker combinatorial properties. This might lead to a broader class of "almost quasilinear" compactifications that retain some, but not all, of the desirable properties.
2. Exploring Other Tropical Constructions: Tropical modifications are just one way to construct new tropical fans from old ones. Other operations, such as tropical quotients or base changes, could be investigated to see if they preserve relevant geometric properties and lead to new classes of well-behaved compactifications.
3. Weakening the Schön Property: The "schön" property, while powerful, is quite strong. One could investigate weaker notions of "tameness" for the multiplication map that still imply desirable properties like smoothness of strata or well-behaved intersection theory. This might involve studying the singularities of the multiplication map and their resolutions.
4. Connections to Minimal Model Program: The notion of a "minimal model" in birational geometry seeks to find the simplest model in a birational equivalence class. It's possible that certain classes of well-behaved compactifications could be characterized as minimal models within a suitable birational class, potentially leading to new connections between tropical geometry and the Minimal Model Program.

What are the implications of the "schön" property for the topology and arithmetic of quasilinear tropical compactifications, and how do these relate to the combinatorial properties of the associated tropical fans?

The "schön" property of quasilinear tropical compactifications has profound implications for their topology and arithmetic, largely stemming from the smooth and surjective nature of the multiplication map:
Topological Implications:

Smoothness of Strata: The "schön" property implies that all strata of a quasilinear tropical compactification are smooth. This leads to a well-behaved stratification, simplifying topological analysis.
Cohomology and Betti Numbers: The smooth and surjective multiplication map allows for the application of techniques from toric geometry to study the cohomology of quasilinear tropical compactifications. The combinatorial structure of the associated tropical fan governs the Betti numbers and other topological invariants.
Poincaré Duality: Quasilinear tropical fans satisfy Poincaré duality, reflecting a deep connection between their combinatorial structure and the topology of the corresponding compactifications.
Arithmetic Implications:

Rationality: The "schön" property implies that quasilinear varieties are rational, meaning they are birationally equivalent to projective space. This has significant implications for their arithmetic properties, such as the behavior of rational points.
Chow-Freeness: Quasilinear varieties are Chow-free, meaning their Chow groups are generated by classes of linear cycles. This simplifies intersection theory and facilitates computations of arithmetic invariants.
Connections to Combinatorics:

Tropical Fan Structure: The combinatorial properties of the associated tropical fan dictate the topology and arithmetic of quasilinear tropical compactifications. For example, the number and arrangement of cones in the fan determine the Betti numbers of the compactification.
Shellability and Hodge Theory: Quasilinear tropical fans are shellable, a combinatorial property that leads to strong restrictions on their homology. This shellability property is also intimately connected to the "K"ahler package" satisfied by these fans, highlighting the interplay between combinatorics, topology, and Hodge theory.
In summary, the "schön" property of quasilinear tropical compactifications provides a bridge between the combinatorial world of tropical fans and the geometric and arithmetic properties of algebraic varieties. This connection allows for the application of powerful tools from each realm to study the other, leading to a deeper understanding of both.