Blowing up

Definition

Blowing up is a technique in algebraic geometry used to resolve singularities of a variety by replacing a point with an entire projective space. This method introduces new geometric structures that help to smooth out the singularities, making the variety easier to analyze and understand. By performing a blow-up, one can often gain insights into the properties of the original variety and its underlying structure.

5 Must Know Facts For Your Next Test

1. The blow-up process replaces a singular point with an entire projective space, creating new points called exceptional divisors.
2. This technique can be repeated multiple times to resolve more complex singularities, leading to a smoother variety.
3. Blowing up has applications in various areas of mathematics, including algebraic geometry, number theory, and complex geometry.
4. The blow-up map is typically constructed using local coordinates around the singular point, allowing for a detailed analysis of the local structure.
5. After blowing up, one can apply further techniques to obtain a minimal model, which is often more manageable for studying the variety's properties.

Review Questions

• How does blowing up help in resolving singularities in algebraic varieties?
  • Blowing up assists in resolving singularities by replacing a problematic point with a projective space, which allows for a smoother local structure. This replacement creates exceptional divisors that help eliminate singular behavior. As a result, mathematicians can analyze the new structure more effectively, leading to insights about the original variety's geometric properties.
• Discuss the relationship between blowing up and the resolution of singularities in the context of algebraic geometry.
  • Blowing up is a key technique within the broader framework of resolution of singularities in algebraic geometry. While resolution refers to the overall process of transforming a singular variety into a non-singular one, blowing up serves as one of the primary methods employed during this transformation. The introduction of projective spaces via blowing up allows for systematic handling of different types of singularities and helps facilitate the overall goal of achieving a smooth variety.
• Evaluate how blowing up contributes to our understanding of algebraic varieties and their geometric structures.
  • Blowing up plays a significant role in deepening our understanding of algebraic varieties by providing tools to systematically address and resolve their singularities. By transforming these varieties through the introduction of exceptional divisors and projective spaces, mathematicians gain insights into their local and global properties. This process not only clarifies complex structures but also aids in the classification and study of varieties, revealing connections between algebraic geometry and other mathematical disciplines.

Related terms

Singularity: A point at which a mathematical object is not well-behaved, such as a point where a function does not have a derivative or where a curve intersects itself.

Resolution of singularities: A process in algebraic geometry aimed at modifying a singular variety into a non-singular one through techniques like blowing up.

Projective space: A mathematical space that extends the concept of ordinary geometric spaces, allowing for the representation of points at infinity and providing a more comprehensive framework for geometry.