Complete intersection

Definition

A complete intersection refers to a type of algebraic variety that can be defined as the common zero set of a specific number of homogeneous polynomials whose degrees are such that their total number matches the dimension of the variety. This concept connects various properties like depth, regular sequences, and Cohen-Macaulay rings, highlighting how varieties can be constructed in a structured manner. Complete intersections are also crucial in the study of toric varieties, where they represent certain combinatorial and geometric configurations within projective spaces.

5 Must Know Facts For Your Next Test

1. Complete intersections can be classified by the number of defining equations and their degrees, which relate directly to their geometric properties.
2. In a complete intersection, the dimension can be calculated as the dimension of the ambient space minus the number of defining equations.
3. Cohen-Macaulay complete intersections have well-behaved cohomological properties, making them easier to study within algebraic geometry.
4. Complete intersections play a significant role in intersection theory, particularly when analyzing how different varieties meet within a projective space.
5. The Hilbert polynomial of a complete intersection provides important information about its geometric characteristics and can be used to deduce its dimensions and degrees.

Review Questions

• How does the concept of depth relate to complete intersections and their defining equations?
  • Depth is a crucial property in understanding complete intersections because it measures how 'regular' a sequence of elements is within the ideal. In a complete intersection defined by homogeneous polynomials, the depth corresponds to how many of these equations can be considered independent without leading to redundancy. This independence indicates that the complete intersection has desirable geometric properties, linking back to concepts like Cohen-Macaulay rings where depth equates to the height of ideals.
• Discuss how complete intersections serve as examples of Cohen-Macaulay rings and their implications on algebraic geometry.
  • Complete intersections are prominent examples of Cohen-Macaulay rings because they often possess uniform depth across their ideals, allowing them to maintain structural integrity. This relationship signifies that for a complete intersection, each ideal’s depth corresponds with its height, leading to more manageable cohomological properties. Understanding this connection enriches our knowledge about how these varieties behave under various algebraic operations, influencing both theoretical research and practical applications in algebraic geometry.
• Evaluate the significance of complete intersections in toric varieties and how they enhance our understanding of combinatorial geometry.
  • Complete intersections play a vital role in toric varieties as they exemplify how algebraic geometry intertwines with combinatorial data derived from polyhedral geometry. By analyzing complete intersections within toric varieties, we gain insights into how geometric configurations can manifest from polynomial relationships. This evaluation extends our understanding not only of the algebraic structures but also provides tools for visualizing complex relationships between different geometric objects, thereby enriching both the theoretical framework and practical methodologies used in contemporary algebraic geometry.

Related terms

Cohen-Macaulay ring: A type of ring where every ideal has a depth equal to its height, indicating a certain level of regularity and structural integrity in its algebraic properties.

Regular sequence: A sequence of elements in a ring such that each element is not a zero divisor modulo the ideal generated by the previous elements, playing a key role in defining depth.

Toric variety: A type of algebraic variety that is defined by combinatorial data associated with polyhedra and their faces, often linked to complete intersections in projective spaces.