Milnor Number

Definition

The Milnor number is an important invariant in algebraic geometry that measures the complexity of singularities of a smooth function or a variety. It is defined as the dimension of the local ring of the singularity divided by the dimension of the ambient space, providing insight into how singular the point is. This number helps classify different types of singularities and plays a critical role in understanding their geometric and topological properties.

5 Must Know Facts For Your Next Test

1. The Milnor number is calculated using the formula: $$ u = ext{dim}( ext{local ring}) - ext{dim}( ext{ambient space})$$.
2. For an isolated singularity, the Milnor number can help determine whether the singularity is non-degenerate or degenerate.
3. The Milnor number can also provide information about the topology of the singularity, such as its link or the number of holes.
4. It is particularly useful when studying hypersurface singularities, where it can indicate how many branches intersect at a singular point.
5. The Milnor number is zero for non-singular points, indicating that there are no complications in the structure at that point.

Review Questions

• How does the Milnor number help in classifying singularities in algebraic geometry?
  • The Milnor number provides a way to classify singularities by quantifying their complexity. By measuring how many dimensions are lost at a singularity compared to a smooth point, it allows us to distinguish between different types of singularities. A higher Milnor number often indicates a more complex or intricate structure around the singularity, which can help mathematicians understand the local behavior of varieties and functions.
• Discuss the significance of the Milnor number when analyzing isolated singularities and their geometric implications.
  • When analyzing isolated singularities, the Milnor number becomes crucial for understanding their geometric properties. A non-zero Milnor number suggests that there are multiple branches or features emerging from the singularity, while a zero Milnor number indicates smoothness. This distinction helps in visualizing and categorizing how these singular points affect the overall structure of a variety, influencing both its topology and its potential applications in algebraic geometry.
• Evaluate how changes in the Milnor number can influence our understanding of deformation theory in algebraic geometry.
  • Changes in the Milnor number can significantly impact our understanding of deformation theory as they provide insights into how families of varieties behave near singular points. For instance, if the Milnor number increases, it may suggest that new features or complexities are arising as we deform a variety. This knowledge allows researchers to predict how variations affect singularities and enables them to formulate theories on stability and change within families of algebraic structures. Thus, analyzing shifts in the Milnor number informs broader conjectures regarding algebraic varieties and their moduli spaces.

Related terms

Singularity: A point at which a mathematical object is not well-defined or fails to be smooth.

Local Ring:

A ring that allows for the study of functions defined near a specific point, particularly useful in analyzing singularities.

Jacobian: A matrix of all first-order partial derivatives of a vector-valued function, used to study the behavior of functions near critical points.