Blowing up

Definition

Blowing up is a technique used in algebraic geometry to replace a point (or a subvariety) on a variety with a more complex structure, often a projective space, to resolve singularities or better understand the geometric properties of the variety. This process helps to analyze and visualize singular points on plane curves, allowing for clearer insights into their local behavior and interactions.

5 Must Know Facts For Your Next Test

1. Blowing up is primarily used to handle singularities in varieties by replacing the singular point with an entire projective space.
2. This technique can be thought of as 'zooming in' on a point to better understand the local geometry around it.
3. When blowing up a point on a curve, new curves called exceptional divisors are created, which help to represent the original curve's behavior near the singularity.
4. The blow-up process maintains important topological properties, ensuring that the overall structure of the variety is preserved while enhancing its geometric detail.
5. Blowing up is reversible; after applying the technique, one can return to the original variety through a blow-down process.

Review Questions

• How does blowing up improve our understanding of singularities in plane curves?
  • Blowing up enhances our understanding of singularities by transforming them into more manageable structures, such as projective spaces. By focusing on these singular points, we can analyze their local behavior and interactions with other points or curves. This helps mathematicians visualize complex relationships and provides insight into how singular points affect the overall geometry of the variety.
• Discuss how exceptional divisors arise from the blowing up process and their significance in studying plane curves.
  • Exceptional divisors emerge when a point is blown up in a plane curve, acting as new curves that replace the singular point. These divisors are significant because they offer a way to represent the local geometry around the original singularity. Analyzing these exceptional divisors enables mathematicians to explore how the curve behaves near these problematic points and helps in resolving the singularities while retaining essential characteristics of the original curve.
• Evaluate the importance of blowing up in the context of resolving singularities and its implications for algebraic geometry as a whole.
  • The importance of blowing up lies in its ability to effectively resolve singularities, which are critical challenges in algebraic geometry. By transforming singular varieties into non-singular ones, this technique allows mathematicians to apply classical geometric methods to study more complex shapes. This has broad implications for algebraic geometry since it enables deeper investigations into curves and surfaces, enriching our understanding of their structures and leading to advancements in various mathematical fields.

Related terms

Singularities:

Points on a curve or surface where the function fails to be well-defined or differentiable, often resulting in 'sharp' or 'cuspidal' features.

Resolution of singularities: A process aimed at transforming a singular variety into a non-singular variety, often through techniques like blowing up.

Projective space: A geometric space that allows for the study of properties of geometric figures that are invariant under projection, providing a framework for blowing up points in varieties.