Non-singular variety

Definition

A non-singular variety is a geometric object in algebraic geometry that has no 'sharp points' or 'edges', meaning it is smooth and well-behaved everywhere. This quality implies that the variety has a well-defined tangent space at every point, which leads to important implications for its local and global properties, including the ability to apply various mathematical tools effectively.

5 Must Know Facts For Your Next Test

1. In algebraic geometry, a non-singular variety is one where all points have a well-defined tangent space, meaning the derivatives of defining polynomials do not vanish simultaneously.
2. The Jacobian criterion helps establish whether a variety is non-singular by analyzing the rank of the Jacobian matrix; if the rank is maximal, then the variety is non-singular at that point.
3. Non-singular varieties can be thought of as generalizations of smooth manifolds in differential geometry, allowing for similar techniques and intuitions to be applied.
4. Many important results in algebraic geometry, like theorems concerning intersection theory and cohomology, rely heavily on working within non-singular varieties.
5. If a variety is not non-singular (i.e., it is singular), it may exhibit complications such as undefined tangent spaces or issues with local properties that complicate analysis.

Review Questions

• How does the concept of non-singular varieties relate to the definition and properties of tangent spaces?
  • Non-singular varieties are defined by their smoothness at all points, which directly relates to tangent spaces since each point on such a variety has a well-defined tangent space. This means that at every point, the variety behaves like a smooth surface in the local vicinity. The ability to define these tangent spaces allows mathematicians to apply calculus-like techniques to study local behavior and properties of non-singular varieties.
• Discuss how the Jacobian criterion can be used to verify whether a given point on a variety is part of a non-singular variety.
  • The Jacobian criterion involves evaluating the Jacobian matrix associated with the defining equations of a variety at a specific point. To determine if this point lies in a non-singular variety, one checks if the rank of this matrix is maximal. If it is, then that point is non-singular, indicating that nearby points also have well-defined tangent spaces and confirming the smooth nature of the variety at that location.
• Evaluate the significance of studying non-singular varieties in algebraic geometry and their impact on broader mathematical theories.
  • Studying non-singular varieties is crucial because they serve as foundational structures in algebraic geometry. The absence of singularities simplifies many calculations and theoretical results, making them essential for understanding more complex geometrical configurations. Non-singular varieties allow for the application of advanced mathematical concepts such as intersection theory and cohomology, which are pivotal in various areas of mathematics and its applications across fields like physics and engineering.

Related terms

Singular variety: A singular variety is a geometric object that contains points where it fails to be smooth, often characterized by cusp or node points where the tangent space is not well-defined.

Tangent space: The tangent space at a point on a variety represents the best linear approximation of the variety at that point, capturing its local behavior and geometry.

Jacobian criterion: The Jacobian criterion is a method used to determine whether a point on a variety is singular or non-singular by examining the rank of the Jacobian matrix associated with a set of polynomial equations.