Projective scheme

Definition

A projective scheme is a type of geometric object that is defined as a closed subscheme of projective space, which itself is constructed from the spectrum of a graded ring. These schemes allow for the study of properties of algebraic varieties in a more generalized setting, emphasizing their relationships to projective geometry and enabling the application of tools from modern algebraic geometry.

5 Must Know Facts For Your Next Test

1. Projective schemes can be constructed from projective varieties, which are defined as zero sets of homogeneous polynomials in projective space.
2. The structure sheaf on a projective scheme is derived from the homogeneous coordinate ring, linking algebraic geometry to commutative algebra.
3. One important aspect of projective schemes is their ability to compactify affine schemes, making them crucial for studying limits and continuity in algebraic geometry.
4. In projective schemes, points can correspond to linear subspaces of a certain dimension, providing a richer geometric structure compared to affine schemes.
5. The concept of duality plays a significant role in projective schemes, where one can associate dual projective spaces leading to various geometric interpretations.

Review Questions

• How do projective schemes enhance our understanding of algebraic varieties compared to affine schemes?
  • Projective schemes enhance our understanding by introducing compactification to affine varieties, allowing for the inclusion of points at infinity. This leads to a richer geometric perspective and provides tools for analyzing limits and continuity that are not readily apparent in affine spaces. The closed nature of projective schemes also facilitates connections with classical geometry, enabling more comprehensive study of their properties.
• Discuss the relationship between projective schemes and graded rings, particularly in terms of their construction and geometric implications.
  • Projective schemes are closely linked to graded rings as they are often constructed from the spectrum of a graded ring associated with homogeneous coordinates. This connection allows for the translation between algebraic properties of graded rings and geometric properties of the resulting projective scheme. The structure sheaf derived from these rings plays an essential role in defining functions on the scheme, thereby intertwining algebraic and geometric perspectives.
• Evaluate the significance of duality in projective schemes and how it relates to the broader framework of modern algebraic geometry.
  • Duality in projective schemes is significant because it enables the association between points in one projective space and hyperplanes in its dual space. This relationship provides critical insights into geometric properties such as intersection theory and dimension counting. Moreover, this duality extends into broader contexts within modern algebraic geometry, where it helps unify various concepts across different types of geometrical objects, thus enriching our understanding and methods within the field.

Related terms

Projective space: An extension of the concept of affine space that incorporates points at infinity, allowing for the treatment of lines and planes in a unified manner.

Graded ring: A ring that is decomposed into a direct sum of abelian groups, where each group consists of elements of a specific degree, facilitating operations on polynomial functions.

Closed subscheme: A subscheme that corresponds to a closed subset of a given scheme, typically defined by vanishing of a collection of functions.