Smooth toric variety

Definition

A smooth toric variety is a type of algebraic variety that is defined by a fan, which consists of strongly convex polyhedral cones. These varieties are characterized by their nice geometric and algebraic properties, particularly the absence of singularities. Smooth toric varieties serve as an important tool in studying resolutions of singularities and understanding how complex algebraic structures can be simplified.

5 Must Know Facts For Your Next Test

1. Smooth toric varieties are defined by fans consisting entirely of smooth cones, meaning each cone corresponds to a well-defined local structure.
2. These varieties can often be constructed from the affine spaces associated with the cones in the fan, allowing for explicit examples and easier computations.
3. Smooth toric varieties have well-behaved cohomology properties, which can simplify calculations in algebraic geometry.
4. The concept of smooth toric varieties is crucial for understanding toric resolutions, which help to resolve singularities in more complex varieties.
5. Examples of smooth toric varieties include projective spaces and many torus-invariant subvarieties, highlighting their role in both algebraic geometry and combinatorial geometry.

Review Questions

• How do smooth toric varieties relate to the study of singularities in algebraic geometry?
  • Smooth toric varieties play a critical role in the study of singularities because they provide models that are free from such issues. By constructing resolutions of singularities using smooth toric varieties, mathematicians can better understand how to handle and resolve points where varieties may fail to behave nicely. This connection allows for a more systematic approach to studying complex varieties by analyzing simpler, smooth structures.
• In what ways do smooth toric varieties exhibit nice geometric properties that facilitate computations in algebraic geometry?
  • Smooth toric varieties exhibit nice geometric properties due to their well-defined local structures, which stem from the fan's smooth cones. This leads to regular cohomological behavior, making it easier to compute invariants and study their geometric features. The clear combinatorial structure provided by the fan also simplifies many calculations and allows for explicit construction methods when working with these varieties.
• Evaluate the significance of smooth toric varieties in constructing toric resolutions and how this impacts the broader field of algebraic geometry.
  • Smooth toric varieties are significant in constructing toric resolutions because they serve as the 'nice' spaces needed to replace singular points in more complex varieties. By using smooth models, researchers can systematically resolve singularities while preserving important geometric properties. This has broad implications for algebraic geometry as it opens pathways to analyze and classify varieties based on their underlying structure, ultimately enriching our understanding of algebraic phenomena and paving the way for further developments in the field.

Related terms

Toric Variety: An algebraic variety that is combinatorially and geometrically defined by fans and their associated cones, providing a connection between geometry and combinatorics.

Singularity: A point on an algebraic variety where the variety fails to be well-behaved, such as having a non-smooth structure or not being locally isomorphic to affine space.

Fan: A collection of cones in a vector space that meet certain conditions, used to define toric varieties and provide a combinatorial framework for their study.