Smooth points

Definition

Smooth points on a variety are points where the local geometry behaves nicely, meaning that the tangent space is well-defined and has the expected dimension. These points do not exhibit any singular behavior, which contrasts with singular points where the geometry can be more complicated and irregular. In the context of canonical and terminal singularities, identifying smooth points is crucial as they often play a key role in understanding the structure and classification of singularities.

5 Must Know Facts For Your Next Test

1. Smooth points are critical for various geometric constructions since they allow for well-defined notions of tangents and curves.
2. The presence of smooth points ensures that local properties can be studied using calculus, facilitating techniques such as differentiation and integration.
3. In algebraic geometry, determining whether a point is smooth can significantly impact the classification of singularities, influencing resolutions and birational geometry.
4. A variety is considered smooth if every point on it is smooth, while a variety containing even a single singular point is referred to as singular.
5. Smooth points can often be analyzed using tools from differential geometry, such as the study of manifolds and their properties.

Review Questions

• How do smooth points contribute to our understanding of the local structure of varieties?
  • Smooth points provide a clear and predictable local structure that allows mathematicians to apply calculus and differential geometry. At these points, one can define tangent spaces that accurately reflect the local behavior of the variety. This predictability is essential when analyzing how varieties behave near singularities, as it helps clarify how different regions relate to one another.
• Compare and contrast smooth points with singular points in terms of their implications for canonical and terminal singularities.
  • Smooth points indicate a nice structure where local geometry is manageable, while singular points represent complications in this structure. In studying canonical and terminal singularities, identifying smooth points helps classify how 'bad' a singularity is. Canonical singularities often arise near smooth points, suggesting that these areas can be resolved or simplified, while terminal singularities may indicate more severe issues that require more complex resolutions.
• Evaluate the role of smooth points in the resolution of singularities and their significance in algebraic geometry.
  • Smooth points play an essential role in resolving singularities by providing reference structures where normal geometric methods can be applied. The presence of these points allows mathematicians to perform blow-ups or other transformations that can simplify or eliminate singular behavior. Understanding where these smooth points lie is crucial for effective resolution strategies, as they form the backbone for constructing modifications that lead to smoother varieties. This highlights their importance in both theoretical frameworks and practical applications within algebraic geometry.

Related terms

Singular Points: Points on a variety where the geometry fails to be smooth, often characterized by the failure of certain derivatives to exist or be well-defined.

Tangent Space: A vector space that approximates the local behavior of a variety at a given point, providing insight into the geometry and properties of that point.

Canonical Singularity: A type of singularity that behaves well in terms of deformation and can often be resolved through blow-ups or other geometric methods.