Exceptional Hereditary Curves over Arbitrary Fields and Their Relation to Real Curve Orbifolds

Core Concepts

This paper explores the theory of exceptional hereditary curves over arbitrary fields, focusing on the construction of tilting objects in their derived categories and establishing a connection between wallpaper groups and real hereditary curves.

Abstract

• Bibliographic Information: Burban, I. (2024). Exceptional hereditary curves and real curve orbifolds. arXiv preprint arXiv:2411.06222.
• Research Objective: This paper aims to elaborate the theory of exceptional hereditary curves over arbitrary fields, going beyond the limitations of previous studies that primarily focused on algebraically closed fields.
• Methodology: The author utilizes techniques from algebraic geometry, particularly focusing on the study of coherent sheaves on non-commutative projective curves. Key tools include derived categories, tilting objects, skew group rings, and Brauer groups.
• Key Findings:
  • The paper provides a straightforward construction of a tilting complex in the derived category of coherent sheaves on a projective hereditary curve of a special type over an arbitrary field.
  • It demonstrates that hereditary curves arising from finite group actions on regular projective curves, where the quotient curve has genus zero, are exceptional.
  • The paper establishes a link between wallpaper groups and real hereditary curves, showing that in 13 out of 17 cases, the derived category of coherent sheaves on the hereditary curve associated with a wallpaper group admits a tilting object.
• Main Conclusions: The results presented in this paper significantly contribute to the understanding of exceptional hereditary curves over arbitrary fields. The construction of tilting objects and the connection to real curve orbifolds provide valuable insights into the structure and properties of these mathematical objects.
• Significance: This research has implications for the broader field of non-commutative algebraic geometry, particularly in the study of derived categories and their applications to the classification and representation theory of algebraic varieties.
• Limitations and Future Research: The paper primarily focuses on a specific class of exceptional hereditary curves. Further research could explore the properties of these curves in more general settings or investigate other classes of non-commutative curves and their connections to different geometric structures.

Stats

gcd(|G|, char(k)) = 1, where |G| is the order of the finite group G and char(k) is the characteristic of the field k.
The quotient curve X = Y/G has genus zero.
13 out of 17 wallpaper groups correspond to hereditary curves whose derived categories admit a tilting object.

Quotes

"Since there is at present no “geometric” definition available for coherent sheaves on a weighted projective line over an arbitrary field, the formulation of our main result will be somewhat different from the formulation for an algebraically closed field k."
"In this work, we give a further elaboration of this theory starting with a ringed space X = (X, H) as a primary object."
"Wallpaper groups lead to a very interesting class of finite group actions over R on complex elliptic curves, what makes a link to the so-called real tubular curves. This striking observation was made by Lenzing many years ago [31], although the underlying details were never published."

Key Insights Distilled From

Exceptional hereditary curves and real curve orbifolds

by Igor Burban at arxiv.org 11-12-2024

https://arxiv.org/pdf/2411.06222.pdf

Deeper Inquiries

How do the findings of this paper relate to the representation theory of finite-dimensional algebras?

This paper establishes a strong link between the theory of exceptional hereditary curves and the representation theory of finite-dimensional algebras, particularly those of tame representation type. Here's how:

Tilting Objects and Derived Equivalences: The central notion of exceptional hereditary curves is intrinsically tied to the existence of tilting objects in their derived categories of coherent sheaves. These tilting objects induce derived equivalences between the derived category of coherent sheaves on the curve and the derived category of modules over a finite-dimensional algebra, denoted as Σ or Π in the paper.

Tame Algebras and Classification: The derived equivalences mentioned above often relate exceptional hereditary curves to finite-dimensional algebras of tame representation type. This connection is significant because tame algebras admit a classification of their indecomposable modules, which are generally well-understood. Examples include canonical algebras and squid algebras, both featured prominently in the paper.

Explicit Constructions and Classifications: The paper leverages these connections to provide explicit constructions of tilting objects and the corresponding tilted algebras for specific classes of exceptional hereditary curves. This leads to a deeper understanding of their module categories and, consequently, the categories of coherent sheaves on the curves themselves.

Wallpaper Groups and Tubular Algebras: The paper highlights a fascinating connection between wallpaper groups, which are groups of isometries of the Euclidean plane, and real hereditary curves. This connection arises from considering the action of wallpaper groups on elliptic curves. The associated tilted algebras in these cases turn out to be tubular canonical algebras, a class of tame algebras with a rich representation theory.

In summary, the paper's findings demonstrate how the representation theory of finite-dimensional algebras, particularly those of tame type, provides powerful tools for studying and classifying exceptional hereditary curves. The derived equivalences induced by tilting objects act as a bridge, translating geometric questions about curves into algebraic questions about modules over finite-dimensional algebras.

Could there be alternative geometric interpretations of exceptional hereditary curves over arbitrary fields that do not rely on tilting objects?

While the definition of exceptional hereditary curves hinges on the existence of tilting objects, exploring alternative geometric interpretations that might not directly rely on them is an intriguing question. Here are some potential avenues for exploration:

Moduli Spaces of Vector Bundles: Exceptional hereditary curves might correspond to special points or subspaces within moduli spaces of vector bundles on the underlying commutative curve. These moduli spaces parameterize vector bundles of a fixed rank and degree, and the exceptional curves could be associated with bundles possessing unique properties reflected in the moduli space's geometry.

Non-Commutative Geometry: Interpreting exceptional hereditary curves through the lens of non-commutative geometry could offer valuable insights. The data defining these curves, including the sheaf of orders and the Brauer class, naturally lend themselves to a non-commutative perspective. Exploring notions like non-commutative tangent spaces or differential calculus might reveal geometric features not immediately apparent from the tilting object viewpoint.

Special Metrics or Connections: The existence of a tilting object might have implications for the existence of special metrics or connections on the underlying commutative curve. These metrics or connections could be related to the stability conditions on the derived category of coherent sheaves, which are known to be connected to tilting objects.

Mirror Symmetry: In the context of mirror symmetry, exceptional collections of objects, which are closely related to tilting objects, often have counterparts on the mirror side. Investigating potential mirror partners to exceptional hereditary curves could lead to alternative geometric interpretations.

It's important to note that these are speculative directions, and further research is needed to determine if they yield fruitful alternative geometric interpretations. Nonetheless, exploring these avenues could deepen our understanding of exceptional hereditary curves and their connections to various geometric structures.

How can the connection between wallpaper groups and real hereditary curves be utilized to study other geometric structures arising from group actions?

The connection between wallpaper groups and real hereditary curves provides a compelling example of how group actions can lead to interesting geometric structures. This connection can potentially be leveraged to study other geometric objects arising from group actions in the following ways:

Orbifold Tilting and Derived Equivalences: The paper demonstrates that the orbifold structure associated with a wallpaper group is reflected in the type of tilting object for the corresponding real hereditary curve. This suggests a broader principle: group actions with specific properties might lead to geometric structures whose derived categories admit tilting objects, potentially revealing hidden symmetries or equivalences.

Classifying Geometric Quotients: Wallpaper groups provide a classification of possible symmetries of two-dimensional periodic patterns. Analogously, the connection to real hereditary curves might hint at a classification of certain geometric quotients arising from group actions on curves. Studying the properties of the tilted algebras associated with these quotients could provide invariants for classifying the quotients themselves.

Generalizing to Higher Dimensions: While the paper focuses on curves, the underlying ideas could potentially generalize to higher-dimensional geometric structures. For instance, considering actions of crystallographic groups on complex tori might lead to interesting higher-dimensional analogues of real hereditary curves and their associated tilted algebras.

Connections to Other Areas: The interplay between group actions, derived categories, and tilting objects transcends specific examples. This interplay has connections to various areas, including:

Geometric Representation Theory: The study of representations of groups arising from geometric contexts.
Algebraic Geometry and Topology: The study of geometric structures using algebraic and topological methods.
Mirror Symmetry: The duality between certain geometric spaces, often mediated by group actions.

By exploring these connections, we can gain a deeper understanding of how group actions give rise to rich geometric structures and how tools from representation theory, such as tilting objects and derived equivalences, can be used to study them.