Embedding of Varieties

Definition

An embedding of varieties is a way of placing one algebraic variety into another in such a manner that the first variety retains its structure within the second. This is done by associating points in the first variety to points in the second variety via a morphism that is both injective and respects the algebraic structure, often allowing for a clearer understanding of geometric properties and relationships. This concept is crucial when examining how different varieties relate to one another and interact through their coordinate rings.

5 Must Know Facts For Your Next Test

1. An embedding can be seen as a way to study a variety within a larger context, often revealing deeper geometric insights.
2. The coordinate ring of an embedded variety can be derived from the coordinate ring of the larger variety by restricting the functions to those that are defined on the smaller variety.
3. An embedding ensures that properties like dimension and singularities are preserved when moving from one variety to another.
4. Common examples of embeddings include placing curves into projective spaces, where the curve can be analyzed with additional geometric tools.
5. The image of an embedding is always closed in the Zariski topology, which helps in understanding how varieties interact within larger spaces.

Review Questions

• How does an embedding help in understanding the geometric properties of varieties?
  • An embedding allows for one variety to be studied within another, making it easier to analyze its geometric properties using the richer structure of the ambient space. By viewing the smaller variety through this lens, we can leverage tools and techniques applicable to larger varieties, such as studying their coordinate rings or applying concepts from algebraic geometry. This perspective often leads to insights regarding dimensions, singularities, and other critical characteristics.
• Discuss how the coordinate rings of an embedded variety relate to those of the ambient variety.
  • When a variety is embedded into another, its coordinate ring can be viewed as a subring of the coordinate ring of the ambient variety. This means we can take functions from the larger ring and restrict them to functions that only make sense on the smaller variety. This relationship not only helps in understanding how the embedded variety behaves within its context but also allows us to explore properties like algebraic relations and ideal structures that may differ between the two varieties.
• Evaluate the implications of embedding on studying singularities within varieties.
  • Embedding provides a powerful framework for investigating singularities by placing a variety within an environment where singular points can be analyzed more effectively. When an embedded variety has singularities, these points can often reveal insights about its behavior in relation to surrounding varieties. Furthermore, examining how these singularities interact with the broader geometric space can lead to discoveries about deformation, resolution techniques, and how local properties reflect global structures across both varieties.

Related terms

Affine Variety: An affine variety is a subset of an affine space defined as the solution set to a system of polynomial equations, where each point corresponds to a maximal ideal in the coordinate ring.

Morphisms: Morphisms are structure-preserving maps between varieties that allow for the comparison and transformation of algebraic objects while maintaining their inherent properties.

Isomorphism: An isomorphism is a special type of morphism between varieties that establishes a one-to-one correspondence, indicating that the two varieties are structurally identical.