9.4 Toric resolutions and singularities

Toric resolutions are powerful tools for fixing singularities in toric varieties. By tweaking the fan structure, we can smooth out rough spots in these geometric objects. It's like using a magic eraser on a bumpy surface!

This process bridges the gap between combinatorics and algebraic geometry. By studying how fans change, we gain insights into the nature of singularities and how to resolve them, even in more complex varieties.

Toric Resolutions and Singularities

Definition and Role of Toric Resolutions

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• Toric resolutions are birational morphisms from smooth toric varieties to singular toric varieties that resolve singularities
• Constructed by subdividing the fan of the singular toric variety to obtain a smooth fan, corresponding to a smooth toric variety
• Resolving singularities using toric methods involves modifying the combinatorial data of the fan while preserving the toric structure
  • Allows for systematic study and resolution of singularities in toric varieties using combinatorial properties of fans

Properties and Significance of Toric Resolutions

• Provide a powerful tool for understanding and resolving singularities in algebraic geometry
  • Toric varieties serve as important examples and building blocks in algebraic geometry (affine spaces, projective spaces)
• Toric resolutions offer a combinatorial approach to studying singularities
  • Singularities are encoded in the fan structure of the toric variety
  • Resolving singularities corresponds to subdividing the fan to achieve smoothness
• Toric methods have applications beyond toric varieties
  • Techniques and insights from toric resolutions can be applied to more general algebraic varieties
  • Toric resolutions provide a bridge between combinatorics and algebraic geometry

Constructing Toric Resolutions

Subdivision Process

• Begin with the fan of the singular toric variety
• Subdivide the fan by adding new rays and cones until the resulting fan is smooth
  • Introduce new rays generated by primitive vectors in the lattice
  • Refine the cones to ensure they are all smooth
• Different choices of subdivision may lead to different toric resolutions with distinct properties
  • Subdivision process is not unique

Toric Resolution Map

• Determined by the inclusion of the original fan into the subdivided fan
• Induces a birational morphism between the corresponding toric varieties
  • Maps the smooth toric variety onto the singular toric variety
  • Resolves the singularities by "blowing up" the singular points
• Toric resolution map preserves the toric structure
  • Compatibility with the torus action
  • Allows for the study of the resolution using toric methods

Analyzing Toric Resolutions

Exceptional Locus

• The preimage of the singular locus under the toric resolution map
• Determined by studying the newly added rays and cones in the subdivided fan
  • Exceptional locus corresponds to the new rays and cones introduced during subdivision
• Geometry of the exceptional locus is encoded in the combinatorial structure of the subdivided fan
  • Irreducible components and their intersections can be understood through the fan structure
  • Provides insights into the structure of the resolved singularities

Intersection Theory and Discrepancies

• Self-intersection numbers of exceptional divisors can be computed using linear relations among rays in the subdivided fan
  • Intersection theory on the smooth toric variety can be studied combinatorially
• Discrepancies of exceptional divisors measure the difference between canonical divisors of smooth and singular toric varieties
  • Determined by the combinatorial data of the fans
  • Important invariants in the study of singularities and birational geometry
• Toric methods provide a computational framework for intersection theory and discrepancy calculations
  • Allows for explicit computations and understanding of these invariants in the toric setting

Classifying Singularities for Toric Resolution

Types of Singularities

• Toric methods can resolve a wide range of singularities
  • Quotient singularities arise from the action of a finite abelian group on a smooth toric variety
    • Resolved by subdividing the fan according to the group action
  • Gorenstein singularities are characterized by the existence of a Cartier divisor that is a multiple of the canonical divisor
    • Toric resolutions are related to the combinatorics of the Gorenstein cone
  • Terminal singularities are the mildest type of singularities in the minimal model program
    • Toric resolutions can be studied using properties of the canonical divisor and singularity index

Active Research Areas

• Classification of singularities that admit toric resolutions is an active area of research
  • New classes of singularities, such as complexity-one T-varieties, are being investigated using toric methods
• Toric resolutions provide a framework for studying and classifying singularities
  • Combinatorial approach allows for explicit constructions and computations
  • Connections to other areas of algebraic geometry and commutative algebra
• Further developments in toric methods may lead to new insights and resolutions of singularities
  • Potential for applications in birational geometry, mirror symmetry, and other areas of mathematics