Normal Varieties

Definition

Normal varieties are algebraic varieties that satisfy a particular property of local rings: they are integrally closed in their function fields. This means that every element that is integral over the coordinate ring of the variety is already contained in that ring. A key aspect of normal varieties is their nice geometric properties, particularly that they do not have certain types of singularities, which relates closely to Cohen-Macaulay varieties and their role in algebraic geometry.

5 Must Know Facts For Your Next Test

1. A variety is normal if every local ring at a point is integrally closed in its fraction field, which reflects good geometric properties.
2. All smooth varieties are normal, but not all normal varieties are smooth; some may still possess certain singular points.
3. Normal varieties can be constructed as a minimal model of a variety after resolving singularities, linking them with the concept of birational geometry.
4. If a variety has only rational singularities, it will also be normal, indicating a strong relationship between the two concepts.
5. Normal varieties behave well under various operations such as taking products and forming blow-ups, making them fundamental in algebraic geometry.

Review Questions

• What criteria must a variety meet to be classified as normal, and how does this classification affect its local rings?
  • For a variety to be classified as normal, each of its local rings must be integrally closed in its fraction field. This implies that any element that is integral over the local ring can be expressed within it. This classification affects the local rings by ensuring they exhibit good behavior and do not have certain singularities, which helps maintain desirable geometric properties across the variety.
• Discuss the relationship between normal varieties and Cohen-Macaulay varieties, especially in terms of their structural properties.
  • Normal varieties and Cohen-Macaulay varieties share an important relationship in algebraic geometry as both concepts help describe specific structural properties of varieties. While normality focuses on the integrality condition of local rings, Cohen-Macaulay properties require a depth condition related to dimension. Many normal varieties exhibit Cohen-Macaulay properties, but being Cohen-Macaulay does not guarantee normality. Understanding this interplay can provide deeper insights into their respective geometric characteristics.
• Evaluate how the property of being normal influences the study and resolution of singularities in algebraic geometry.
  • The property of being normal significantly influences the study and resolution of singularities in algebraic geometry because it sets a framework for understanding how these singularities can be managed. Normal varieties are often constructed as resolutions of singularities, allowing mathematicians to apply techniques like minimal models and birational transformations effectively. By ensuring that the resulting variety avoids certain problematic singular structures, researchers can focus on more regular cases, facilitating further exploration into their geometrical and topological features.

Related terms

Cohen-Macaulay Varieties: These are varieties whose local rings are Cohen-Macaulay, meaning they satisfy a specific depth condition related to their dimension and ideal structure.

Integral Closure: The integral closure of a ring is the set of elements in a larger ring that are roots of monic polynomials with coefficients from the original ring, ensuring certain regularity in varieties.

Singularities: Points on a variety where it fails to be well-behaved or smooth; normal varieties are defined to avoid certain types of singularities.