Blowing Up

Definition

Blowing up is a process in algebraic geometry where a point or a subvariety of a scheme is replaced with a projective space or a more complex structure to resolve singularities. This technique helps transform spaces with undesirable features into smoother varieties, often allowing for a better understanding of their geometric properties and facilitating further analysis of their structure.

5 Must Know Facts For Your Next Test

1. Blowing up can be performed at points, curves, or more complex subvarieties, leading to different types of resolutions depending on the context.
2. The blow-up introduces an exceptional divisor, which captures the information about the singularity being resolved and plays a crucial role in the overall structure.
3. The process is not just a local operation; it affects the global geometry of the variety and can lead to new insights about its properties.
4. In toric geometry, blowing up is particularly useful for resolving singularities that arise from combinatorial configurations of cones and fan structures.
5. After blowing up, the resulting space retains many properties of the original variety but becomes easier to study, especially in terms of its intersection theory and cohomology.

Review Questions

• How does blowing up transform a singular point into a non-singular structure, and what role does the exceptional divisor play in this process?
  • Blowing up transforms a singular point by replacing it with a projective space that smooths out the singularity. The exceptional divisor created during this process represents the new geometry introduced at that location. This divisor carries important information about the original singular point, allowing mathematicians to study its properties in greater depth and providing insights into the overall structure of the variety.
• Discuss the connection between blowing up and toric varieties in resolving singularities, highlighting how combinatorial data influences this relationship.
  • In toric varieties, blowing up serves as a powerful tool for resolving singularities that arise from specific combinatorial configurations. The data from fans and cones directly influences how these blow-ups are performed, as it dictates which points or subvarieties are targeted for resolution. By leveraging this combinatorial perspective, mathematicians can systematically address singularities in toric varieties, leading to clearer geometric interpretations and facilitating the analysis of their structures.
• Evaluate the significance of blowing up within the broader context of algebraic geometry, particularly in relation to resolution of singularities and its implications for understanding variety properties.
  • Blowing up is pivotal in algebraic geometry as it directly addresses the challenges posed by singularities, offering a pathway to resolution. By transforming singular varieties into non-singular ones, this process enriches our understanding of their geometric properties, intersection theory, and cohomological aspects. The implications are profound: not only does it allow mathematicians to classify varieties more effectively, but it also opens avenues for deeper investigations into their intrinsic structures and relationships with other mathematical entities.

Related terms

Resolution of Singularities: A method in algebraic geometry aimed at modifying a singular variety into a non-singular one, typically involving the blowing up process.

Toric Varieties: A class of algebraic varieties that are defined by combinatorial data, often used in conjunction with blowing up to understand their singularities and resolutions.

Exceptional Divisor: The divisor that appears in the blow-up of a variety, representing the new structure introduced at the blown-up point or subvariety.