The regularity condition refers to a specific requirement that a toric variety satisfies in relation to its associated fan and combinatorial structure. It ensures that the toric variety is well-behaved, particularly regarding its geometric properties such as smoothness and dimensionality, by establishing necessary conditions on the cones in the fan.
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The regularity condition ensures that each cone in the fan corresponds to a well-defined affine piece of the toric variety.
If a toric variety satisfies the regularity condition, it can be constructed using the associated fan without ambiguity in its geometric properties.
The condition plays a crucial role in determining whether the resulting toric variety is smooth or has singularities.
Regularity is closely linked to the concept of ample line bundles on toric varieties, influencing their intersection theory.
One common formulation of the regularity condition involves checking that for every cone, the dimension of the corresponding affine space matches the dimension of the toric variety.
Review Questions
How does the regularity condition influence the properties of a toric variety?
The regularity condition directly impacts whether a toric variety exhibits desirable properties such as smoothness and proper dimensionality. When satisfied, this condition indicates that each cone in the associated fan accurately represents an affine piece of the variety. This means that we can expect consistent behavior across different regions of the toric variety, making it easier to study its geometric features.
Discuss how the regularity condition relates to the concept of fans and cones within toric geometry.
The regularity condition is intimately tied to how fans and cones are constructed in toric geometry. Each cone corresponds to an affine piece of a toric variety, and if the regularity condition holds, it ensures that these cones are appropriately defined. This relationship allows mathematicians to utilize combinatorial techniques for understanding complex geometric structures, enabling them to analyze properties like smoothness and singularities within these varieties.
Evaluate the implications of violating the regularity condition on the geometric structure of a toric variety.
If the regularity condition is violated, it can lead to significant issues in understanding the geometric structure of a toric variety. For instance, one might encounter unexpected singularities or inconsistencies in dimensionality across different affine pieces. This lack of coherence complicates not only the geometric interpretation but also impacts algebraic properties such as intersection theory and line bundles, making it challenging to apply classical results from algebraic geometry to these varieties.
A toric variety is a special type of algebraic variety that is defined by combinatorial data from a fan, allowing for a connection between geometry and algebraic structures.
A fan is a collection of cones that describes the combinatorial structure of a toric variety, capturing the relationships between different coordinate patches of the variety.
Cone: In the context of fans, a cone is a geometric object formed by rays emanating from a single point, which represent the combinatorial data associated with the corresponding toric variety.