Fiber product

Definition

The fiber product is a construction in category theory and algebraic geometry that combines two schemes over a common base scheme into a new scheme, reflecting the relationships between them. It allows for a geometric understanding of how different schemes relate when restricted to a specific base. This construction is especially useful when dealing with morphisms and studying properties of varieties under different projections.

5 Must Know Facts For Your Next Test

1. In a fiber product, given two schemes $X$ and $Y$ over a base scheme $S$, the fiber product $X \times_S Y$ consists of pairs $(x, y)$ where the images of $x$ and $y$ under their respective morphisms to $S$ coincide.
2. The fiber product is particularly important in the context of blowing up, where it helps to analyze the pre-image of a point or subvariety under blow-up morphisms.
3. Fiber products can also be used to construct moduli spaces, where they help define families of geometric objects parametrized by schemes.
4. The universal property of fiber products states that for any scheme mapping to both $X$ and $Y$, there exists a unique morphism from that scheme to the fiber product.
5. When dealing with varieties, fiber products can provide insight into how intersections and various geometric properties behave under changes in base schemes.

Review Questions

• How does the fiber product help in understanding morphisms between schemes when focusing on a specific base?
  • The fiber product allows us to analyze the relationships between two schemes over a common base by creating a new scheme that captures pairs of points from each scheme whose images match under their respective morphisms to the base. This construction makes it easier to understand how these morphisms interact, particularly when studying properties like continuity, intersection, or smoothness, as it consolidates information from both schemes into one framework focused on their behavior over the base.
• Discuss the role of fiber products in the context of blowing up singularities and how they facilitate resolution processes.
  • In the blowing up process, fiber products are used to study how points in the original variety are transformed when we blow up along a subvariety. The fiber product enables us to understand the local structure around singular points by providing a new space that represents all possible points that can be 'lifted' from the original variety while maintaining consistency with their images. This is crucial for resolving singularities, as it helps visualize and analyze how local geometry changes during this transformation.
• Evaluate the implications of fiber products in constructing moduli spaces of curves and stable curves, particularly regarding families of geometric objects.
  • Fiber products are fundamental in constructing moduli spaces as they allow for the examination of families of geometric objects parametrized by schemes. When creating moduli spaces of curves, fiber products can express families of curves that satisfy certain conditions relative to base points, revealing insights into their stability and deformation. By analyzing these constructions through fiber products, we gain essential understanding regarding how various families interact and relate within broader geometric contexts, leading to deeper insights into stability criteria and deformation theory.

Related terms

Base change: A process of changing the base scheme in a fiber product, allowing for the analysis of how morphisms behave under various conditions.

Product scheme: The categorical product of schemes, which serves as a generalization of the notion of Cartesian products in set theory.

Morphisms of schemes: The structure-preserving maps between schemes that capture the relationships and transformations among them.