Pushforward

Definition

The pushforward is a process in mathematics that allows us to transfer geometric or algebraic structures from one space to another via a continuous map. In the context of tropical geometry, it specifically refers to how tropical cycles and divisors can be transformed when passing through a morphism, thereby allowing us to study their properties in different settings.

5 Must Know Facts For Your Next Test

1. The pushforward of a tropical cycle involves taking a tropical cycle defined on one space and mapping it onto another space using a continuous morphism.
2. The pushforward operation preserves the combinatorial structure of the tropical cycles, meaning important features like intersection numbers remain intact.
3. In tropical geometry, the pushforward can be used to define the degree of a morphism, which measures how many points in the domain map to each point in the target space.
4. The pushforward helps in defining pullbacks and understanding how structures transform under various morphisms, contributing to the study of duality and cohomology in tropical settings.
5. An important aspect of pushforward is its role in understanding how divisors behave under changes of coordinates or more general transformations in tropical geometry.

Review Questions

• How does the pushforward relate to the concept of tropical cycles and their properties when mapped through a morphism?
  • The pushforward allows us to take tropical cycles defined on one variety and map them to another while preserving their combinatorial structure. This means that properties such as intersection numbers and multiplicities are maintained through this transformation. The pushforward thus provides a way to study the behavior of tropical cycles under continuous maps, enabling deeper insights into their geometric properties.
• Discuss how the pushforward operation impacts the understanding of degrees of morphisms in tropical geometry.
  • The pushforward operation plays a crucial role in defining the degree of a morphism in tropical geometry. The degree measures how many points in the domain correspond to each point in the codomain, reflecting the local behavior of the morphism. By utilizing pushforward, we can compute these degrees systematically and explore how they influence the topology and geometry of the resulting structures.
• Evaluate the implications of the pushforward operation on the interaction between tropical cycles and divisors within different geometric contexts.
  • The implications of the pushforward operation are significant for understanding how tropical cycles and divisors interact within various geometric contexts. By transferring these structures through morphisms, we can analyze their behavior under different geometric transformations. This evaluation reveals crucial insights about duality principles, cohomological aspects, and how geometric properties evolve as we move between spaces in tropical geometry.

Related terms

Tropical Cycle: A tropical cycle is a formal sum of tropical subvarieties, typically representing a notion of 'divisor' in tropical geometry, capturing important geometric information.

Morphism: A morphism is a structure-preserving map between two mathematical objects, allowing us to relate different spaces and their properties.

Tropical Divisor:

A tropical divisor is a formal sum of points on a tropical variety, used to capture intersection data and other geometric properties in tropical geometry.