Gasubvarieties

Gasubvarieties are a class of algebraic subvarieties defined within an ambient variety that satisfy a guiding compatibility condition with a chosen stratification. The term refers to subvarieties that locally decompose in a way that combines a smooth, transverse component with a constrained “gas” factor along specified strata. In this sense, gasubvarieties generalize the notion of smooth subvarieties and encompass many mildly singular strata that arise in stratification theories.

Construction and basic properties can be described without requiring heavy formalism. A gasubvariety is typically realized as a zero locus of a regular sequence that is adapted to a fixed stratification of the ambient space, ensuring transversal behavior to each stratum. They are stable under finite intersections and, in generic situations, obey a dimension formula consistent with standard expectations: the dimension of the subvariety equals the ambient dimension minus the codimension defined by the chosen defining equations. Gasubvarieties admit a natural filtration by gas strata, which provides a framework for inductive arguments and the application of cohomological methods.

Examples and scope help situate the concept. Every smooth subvariety of a fixed ambient variety is a gasubvariety. In two dimensions, certain nodal curves can be viewed as gasubvarieties with a simple two-stratum decomposition (smooth points and singular points), illustrating how the gas condition accommodates mild singularities while retaining structural control.

See also: Subvariety, Algebraic geometry, Stratification, Singularity theory.