Non-Isolated Singularity

Definition

A non-isolated singularity is a point on an algebraic variety where the singularity is not a standalone feature; instead, it exists in a broader context of other singular points. This means that there are infinitely many nearby points that also exhibit singular behavior. Non-isolated singularities often arise in situations where multiple curves or surfaces intersect or overlap, leading to more complex geometric and algebraic structures.

5 Must Know Facts For Your Next Test

1. Non-isolated singularities can occur in varieties defined by equations that have multiple solutions concentrated around the same region.
2. In algebraic geometry, non-isolated singularities can complicate the process of resolution, as they require dealing with an entire family of singular points rather than just one.
3. The local structure around a non-isolated singularity can often be described using algebraic techniques like the study of local rings and their properties.
4. One common example of non-isolated singularities occurs in the intersection of two curves at a common point, which creates additional tangential intersections.
5. Non-isolated singularities can lead to interesting topological phenomena, affecting the overall properties of the variety they belong to.

Review Questions

• How do non-isolated singularities differ from isolated singularities in terms of their surrounding structure?
  • Non-isolated singularities differ from isolated singularities primarily in that they exist within a larger family of singular points nearby. While isolated singularities are unique and have no other singular behavior around them, non-isolated singularities are characterized by having many nearby points that also exhibit similar singular features. This creates a more complex landscape for understanding the geometry and algebraic properties of the variety.
• What challenges do non-isolated singularities present when attempting to perform resolution of singularities?
  • Non-isolated singularities pose significant challenges during resolution processes because they require handling an entire set of singular points instead of just one. This means that traditional methods for resolving isolated singularities may not be directly applicable. Instead, specialized techniques must be employed to address the collective nature of these singularities and ensure that the resolution process accurately reflects the intricate structure of the variety.
• Evaluate the role of blow-ups in managing non-isolated singularities and their impact on the overall geometry of an algebraic variety.
  • Blow-ups play a crucial role in managing non-isolated singularities by replacing problematic points with projective spaces, thereby allowing for a clearer understanding of local geometry. This technique effectively transforms non-isolated behavior into a more manageable form, often leading to the creation of new varieties that are easier to analyze. By resolving these complex structures, blow-ups help clarify the relationships between different components of the variety and their implications for its geometric properties.

Related terms

Isolated Singularity: A point on an algebraic variety where the singularity is unique in its local neighborhood, meaning there are no other singular points in the vicinity.

Resolution of Singularities: A process by which singular points of a variety are systematically 'resolved' or 'smoothed out' to produce a non-singular variety.

Blow-Up: An operation in algebraic geometry that replaces a point (or a subvariety) with a projective space to resolve singularities and study their local structure.