Flatness

Definition

Flatness is a property of morphisms between algebraic varieties, indicating that the fibers over points are 'uniform' in a certain sense. It describes how a family of varieties varies smoothly, ensuring that the structure of the fibers does not vary too wildly as one moves through the base space. Flatness is crucial for ensuring that certain geometric and algebraic properties are preserved under base change, playing an essential role in various constructions such as Chow rings and Hilbert schemes.

5 Must Know Facts For Your Next Test

1. A morphism is flat if the fibers are 'locally constant' in terms of dimension, meaning their dimensions do not vary too much as one moves through the base.
2. Flatness ensures that certain algebraic properties, like dimension and smoothness, can be preserved when performing base changes or working with families of varieties.
3. In intersection theory, flatness allows for well-defined intersection products in Chow rings, helping to calculate classes corresponding to intersections.
4. Flat families can be thought of as families of varieties that behave similarly, which is crucial for defining things like Hilbert schemes that parameterize families of subschemes.
5. In practical terms, checking flatness often involves examining the behavior of local rings and their modules over a base ring.

Review Questions

• How does flatness affect the properties of fibers in algebraic varieties?
  • Flatness ensures that as one varies through the base space in a morphism, the fibers remain consistent in dimension. This uniformity allows for various geometric properties to be maintained across the family of varieties. If a morphism is flat, then each fiber behaves similarly, which is essential for working with concepts like Chow rings and understanding intersections.
• Discuss the importance of flatness in relation to base change and its implications for intersection theory.
  • Flatness plays a vital role in base change because it guarantees that certain desirable properties are preserved when changing from one base scheme to another. In intersection theory, flat morphisms allow us to define intersection products consistently within Chow rings. Without flatness, these products could vary unpredictably with changes in the base scheme, complicating the study of intersections.
• Evaluate the relationship between flatness and Hilbert schemes regarding families of subschemes.
  • Flatness is crucial when considering families of subschemes parameterized by Hilbert schemes because it ensures that as we vary within the Hilbert scheme, the corresponding subschemes behave uniformly. This uniformity allows for smooth deformations and simplifies the analysis of how subschemes can be continuously transformed within families. The relationship between flatness and Hilbert schemes highlights how geometric properties persist across various forms of subschemes, facilitating deeper investigations into their structure and classification.

Related terms

Fiber: The fiber over a point in a morphism is the preimage of that point, representing the variety or set of points in the source variety corresponding to that point in the target variety.

Base Change: A process that allows one to change the base scheme over which a family of varieties is defined, enabling the study of how properties of varieties behave under different settings.

Chow Ring: An algebraic invariant associated with a variety that encodes information about its cycles and intersection theory, providing a way to study its geometric properties.