Dominance

Definition

Dominance refers to the relationship between algebraic varieties where one variety can be considered as a 'larger' or 'more complex' entity compared to another, often through the existence of a rational map that facilitates a morphism in one direction. This concept is crucial in understanding how varieties relate to each other, particularly in contexts such as birational equivalence, where two varieties can be viewed as equivalent under certain rational maps, and Krull dimension, where dominance helps determine the structural complexity of varieties based on their dimensions.

5 Must Know Facts For Your Next Test

1. Dominance is a key concept when discussing rational maps, as it indicates whether one variety can effectively influence or relate to another through these maps.
2. In birational geometry, dominance often leads to the classification of varieties, allowing mathematicians to understand their similarities and differences.
3. If a variety X dominates another variety Y, it often implies that the dimension of X is greater than or equal to the dimension of Y.
4. Dominance can also be reflected in the behavior of morphisms between varieties, indicating which varieties serve as 'targets' for other varieties.
5. In the context of Krull dimension, if X dominates Y, then the Krull dimension of X will be greater than or equal to that of Y, helping us analyze their geometric properties.

Review Questions

• How does the concept of dominance facilitate the understanding of relationships between varieties in algebraic geometry?
  • The concept of dominance allows us to see how one variety can influence or relate to another through rational maps. If one variety dominates another, it suggests that there are meaningful geometric or structural connections between them. This can lead to insights about birational equivalence, where we classify varieties based on their rational maps and identify similarities despite differences in their forms.
• In what ways does dominance play a role in birational equivalence when comparing two algebraic varieties?
  • Dominance is fundamental in establishing birational equivalence because it allows us to understand when two varieties are essentially 'the same' outside of lower-dimensional subsets. When one variety dominates another, we can often find rational maps that connect them, highlighting their shared properties. This connection enables mathematicians to classify varieties and determine their relationships despite apparent differences.
• Analyze how the concept of dominance relates to Krull dimension and its implications for understanding the geometric complexity of varieties.
  • Dominance directly impacts the analysis of Krull dimension because it provides insight into the geometric complexity of varieties. When one variety dominates another, it indicates that the dominating variety has at least the same level of complexity as the dominated one, reflected in their respective dimensions. This relationship helps mathematicians gauge not just the sizes of these varieties but also their overall structure and how they might interact within larger frameworks in algebraic geometry.

Related terms

Rational Map:

A rational map is a function between two varieties that is defined by polynomials but may not be defined everywhere, often representing a 'partial' morphism.

Birational Equivalence:

Birational equivalence is a relationship between two varieties that indicates they are isomorphic outside of lower-dimensional subsets, often established through rational maps.

Krull Dimension:

Krull dimension is a measure of the 'size' or complexity of a variety, determined by the maximum length of chains of prime ideals in its coordinate ring.