Cubicderived

Cubicderived is a theoretical framework in algebraic geometry and category theory that studies objects defined by cubic relations through derived categories and homological methods. It treats cubic equations, cubic algebras, and cubic hypersurfaces as organizing principles, and records how cubic conditions influence the morphisms and extensions among coherent sheaves or modules. The term is used in expository contexts to indicate a focus on cubic-level phenomena within a derived setting.

The central construction assigns to a cubic geometric object X an enhanced derived category, such as the bounded derived category of coherent sheaves D^b(Coh X) or the derived category of a cubic algebra. In many formulations a cubic potential W is used to define a DG-category via matrix factorizations, Orlov-type equivalences, or noncommutative deformations, yielding a triangulated category that encodes cubic relations.

Key properties include semi-orthogonal decompositions separating a primitive cubic component from standard exceptional objects, potential Calabi–Yau behavior in the primitive part, and invariants that track cubic-level complexity. Researchers may define cubic-derived dimensions or cubic Calabi–Yau dimensions as measures of growth and symmetry in these categories.

Typical examples involve smooth cubic hypersurfaces in projective spaces, such as plane cubic curves, cubic surfaces in P^3, and cubic fourfolds in P^5. The primitive components of their derived categories play a central role in modern investigations, especially Kuznetsov's cubic component for cubic fourfolds, which relates to K3 surfaces and mirror symmetry.

Cubicderived intersects with broader topics in derived algebraic geometry, matrix factorizations, and Homological Mirror Symmetry, and is used to explore questions such as rationality of cubic fourfolds and the structure of noncommutative resolutions.