Blow-up

Definition

A blow-up is a fundamental operation in algebraic geometry that allows us to resolve singularities of a variety by replacing it with a new variety. This process transforms a point on the original variety into a higher-dimensional space, effectively 'spreading out' the structure around that point, which can lead to improved properties like smoothness. Blow-ups are closely connected to birational equivalence and rational maps as they provide a way to create new varieties that can often be related through these concepts.

5 Must Know Facts For Your Next Test

1. The blow-up of a variety at a point introduces new exceptional divisors that encode information about the original singularity.
2. The blow-up operation is reversible, meaning one can blow down from the blow-up back to the original variety under certain conditions.
3. Blow-ups can be performed successively, allowing for multiple singularities to be resolved step-by-step.
4. The composition of blow-ups leads to new varieties that may have different properties than the original, often creating smooth structures.
5. In birational geometry, blow-ups are crucial for understanding how different varieties can relate to one another via rational maps.

Review Questions

• How does the blow-up operation impact the singularities of a variety and what implications does this have for its geometric structure?
  • The blow-up operation specifically targets singularities on a variety, transforming them into smooth structures by introducing exceptional divisors. By replacing points of singularity with higher-dimensional spaces, the blow-up helps create a new variety that lacks those singular points, thereby enhancing its overall geometric properties. This smoothing out allows for better behavior under various operations and facilitates further analysis using tools like rational maps.
• Discuss the relationship between blow-ups and birational equivalence. How does this operation help in establishing connections between different varieties?
  • Blow-ups are integral to establishing birational equivalence between varieties because they allow for transformations that can simplify or clarify the relationships between complex structures. When two varieties are related by a birational map, performing blow-ups at key points can create new varieties that are easier to compare and understand. This connection reveals how seemingly different varieties can share common traits through their blown-up counterparts, facilitating deeper insights into their geometry.
• Evaluate how the resolution of singularities through blow-ups contributes to advancements in algebraic geometry and its applications in other mathematical fields.
  • The resolution of singularities through blow-ups represents a critical advancement in algebraic geometry as it provides methods to transform complex and pathological varieties into more manageable forms. This process not only aids in understanding the intrinsic properties of varieties but also connects algebraic geometry with other areas such as number theory and complex geometry. By creating smooth models from singular ones, mathematicians can apply techniques from these various fields to solve problems, explore relationships, and develop further theories in mathematics.

Related terms

Singularity: A point on a variety where the variety fails to be well-behaved, often marked by non-smoothness or other pathological features.

Birational Map: A rational map between two varieties that is an isomorphism outside of a lower-dimensional subset, often used in the context of blow-ups.

Resolution of Singularities: A process that replaces a singular variety with a non-singular one through techniques such as blow-ups, aiming to improve geometric properties.