Domain

Definition

In algebraic geometry, the domain refers to the set of points in a variety where a rational map is defined and behaves well. It captures the concept of where you can evaluate the rational function without encountering undefined behavior, such as division by zero or discontinuities. Understanding the domain is crucial when analyzing rational maps between varieties, as it helps identify valid inputs for the functions involved.

5 Must Know Facts For Your Next Test

1. The domain of a rational map includes all points in the source variety where the map can be evaluated without any issues.
2. A rational map may have an indeterminacy locus where it fails to be defined; these points need special consideration when working with the map.
3. Understanding the domain helps identify how rational maps can be extended to larger spaces, possibly involving blow-ups or other techniques.
4. The images of points in the domain under a rational map may not cover all points in the target variety, leading to discussions about image and range.
5. Analyzing the domain is essential when studying morphisms between varieties and understanding their properties, such as being birational or dominant.

Review Questions

• How does the concept of the domain influence the evaluation of rational maps between varieties?
  • The domain plays a crucial role in evaluating rational maps because it defines the set of points where these maps are well-defined. Without a proper understanding of the domain, one might encounter undefined behavior, such as division by zero. Knowing the domain allows for effective application of these maps in algebraic geometry, ensuring that analysis is grounded on valid input values.
• What implications does an indeterminacy locus have for understanding the domain of a rational map?
  • The presence of an indeterminacy locus directly affects how we perceive the domain of a rational map, as it highlights points where the map is not defined. This requires careful handling when analyzing properties of the map and might necessitate methods such as resolving singularities or employing blow-ups. Understanding this relationship is vital for accurately studying how maps behave in algebraic geometry.
• Evaluate how understanding domains can impact higher-level concepts like birational maps and morphisms between varieties.
  • Grasping domains allows for deeper insights into higher-level concepts such as birational maps and morphisms between varieties. It aids in recognizing when two varieties are birationally equivalent by ensuring that the domains of their respective rational maps overlap meaningfully. Additionally, analyzing domains provides clarity on which varieties can be related through morphisms, shaping our understanding of their geometric properties and interactions in algebraic geometry.

Related terms

Rational Map: A rational map is a function between varieties that is given by ratios of polynomials. It is not necessarily defined everywhere due to potential singularities or points where the polynomials vanish.

Variety:

A variety is a fundamental object in algebraic geometry, representing solutions to polynomial equations. Varieties can be affine or projective and can have various dimensions.

Indeterminacy Locus: The indeterminacy locus consists of points in the domain where a rational map is not defined. This is important for understanding the behavior of the map and its properties.