Normality

Definition

Normality is a property of a variety that indicates whether it behaves like a 'nice' object in algebraic geometry, specifically meaning that it is not too singular and has certain desirable dimensional characteristics. In the context of projective varieties, normality ensures that the variety does not have 'wild' behaviors, such as having points with high-dimensional local rings or irregular singularities, thus maintaining coherence in the structure and dimensionality of the space.

5 Must Know Facts For Your Next Test

1. A variety is called normal if every local ring at a point of the variety is integrally closed in its field of fractions.
2. Normality is particularly important in dimension theory because normal varieties have well-defined dimensions and are easier to study using techniques from algebraic geometry.
3. The normalization process can be used to transform a singular variety into a normal one, effectively resolving singularities.
4. In projective geometry, normal varieties ensure that their structure sheaves behave well and do not exhibit pathological properties.
5. Not all varieties are normal; for example, certain curves with self-intersections or specific types of singular points can violate this property.

Review Questions

• How does normality relate to the concepts of singularities and local rings in algebraic geometry?
  • Normality is closely tied to singularities and local rings since a variety is considered normal when its local rings at any point are integrally closed. This means that around singular points, normal varieties maintain a level of coherence that prevents irregular behavior. By ensuring that local rings behave properly, we can study the local properties of the variety without encountering the complications that arise from singularities.
• Discuss why normality is essential for dimension theory in projective varieties and how it impacts their geometric properties.
  • Normality is essential for dimension theory because it guarantees that projective varieties have consistent dimensions and allows for effective application of tools from algebraic geometry. When dealing with normal varieties, we can use techniques like intersection theory without worrying about pathological behaviors. This consistency also aids in understanding the overall geometric structure and helps classify these varieties accurately within projective space.
• Evaluate how the process of normalization can change the properties of a given variety and its implications for studying algebraic structures.
  • Normalization transforms a potentially singular variety into a normal one by resolving its singularities, which significantly alters its properties. This process not only makes the variety easier to study but also preserves key algebraic structures that may otherwise be obscured by irregularities. Understanding how normalization impacts these properties helps mathematicians identify underlying relationships between different varieties and classify them more effectively within the broader context of algebraic geometry.

Related terms

Singularity: A point on a variety where it fails to be well-behaved, such as where it may not be smooth or where it exhibits unusual local behavior.

Projective Variety: A subset of projective space that is defined by homogeneous polynomials and has a structure that allows for geometric interpretations in higher dimensions.

Local Ring: A type of ring that focuses on the behavior of functions near a specific point in a variety, crucial for studying the properties of varieties at singularities.