Indeterminate Points

Definition

Indeterminate points are specific points in the context of rational maps between varieties where the map is not well-defined, often resulting from a lack of correspondence due to the map's algebraic structure. These points arise when the denominator of a rational function equals zero, leading to undefined behavior in the mapping from one variety to another. Understanding indeterminate points is crucial for analyzing the properties and behaviors of rational maps in algebraic geometry.

5 Must Know Facts For Your Next Test

1. Indeterminate points occur when the rational map has poles, specifically when the denominators of the defining polynomials vanish at certain coordinates.
2. To manage indeterminate points, mathematicians often employ techniques like blow-ups or modifications of varieties to create a well-defined map.
3. The behavior at indeterminate points can reveal important information about the geometry and topology of varieties involved in the mapping process.
4. In projective space, indeterminate points can be interpreted as points at infinity, which play a significant role in understanding rational maps between projective varieties.
5. Indeterminate points can lead to complications in computing intersection theory, making it vital to address them when working with rational maps.

Review Questions

• How do indeterminate points affect the behavior of rational maps between varieties?
  • Indeterminate points affect rational maps by introducing locations where the map fails to be defined due to the vanishing of denominators in polynomial ratios. This can result in undefined mappings, which complicate our understanding of how one variety relates to another. By identifying these points, mathematicians can take necessary steps to modify the mapping, ensuring that it remains well-defined and continuous across relevant domains.
• Discuss strategies for resolving indeterminate points in the context of rational maps.
  • Strategies for resolving indeterminate points include techniques such as blow-ups, where an indeterminate point is replaced with a new geometric object that allows for better handling of singularities. This method enables mathematicians to create a modified variety that behaves well under rational maps. Additionally, examining local coordinates and adjusting the definitions of maps can help mitigate issues caused by these problematic points.
• Evaluate the implications of ignoring indeterminate points when studying rational maps and their effects on algebraic varieties.
  • Ignoring indeterminate points when studying rational maps can lead to significant misinterpretations of the relationships between algebraic varieties. Without properly addressing these undefined locations, researchers may overlook key geometric properties or encounter erroneous conclusions regarding morphisms and mappings. Furthermore, failing to resolve indeterminacies can obscure important features such as intersection behaviors and singularities, ultimately hindering a comprehensive understanding of the underlying algebraic structures.

Related terms

Rational Map: A rational map is a function between varieties that can be expressed as a ratio of polynomials, where both the numerator and denominator are polynomials in several variables.

Variety:

A variety is a fundamental concept in algebraic geometry, representing the solution set of a system of polynomial equations, and can be either affine or projective.

Blow-up:

A blow-up is a geometric operation that replaces an indeterminate point with a new space, allowing for better control over singularities and providing a way to resolve indeterminacies.