Understanding Biplicity in Algebraic Geometry and Commutative Algebra

In mathematics, particularly in the context of algebraic geometry and commutative algebra, biplicity refers to a specific type of singularity that can occur at a point of a variety.

A point $P$ on a variety $X$ is said to have biplicity if there exist two distinct branches of the variety passing through $P$, such that each branch has a tangent line at $P$ that is not contained in the other branch. In other words, the tangent space at $P$ decomposes into two non-rivial subspaces, one associated with each branch.

Biplicity is a stronger condition than singularity, as it implies that the variety has a non-trivial tangent space at the point, and that the singularity is not just a simple point of inflection or a cusp. Biplicity is also a necessary condition for the existence of certain types of singularities, such as nodal singularities.

In the context of algebraic curves, biplicity can be used to study the geometry and topology of the curve near a singular point. For example, if a curve has a bipoint, then it must have at least one inflection point nearby, and the tangent line at the bipoint must be horizontal.

Overall, biplicity is an important concept in algebraic geometry and commutative algebra, and it has applications in various areas of mathematics and physics, such as the study of algebraic surfaces, the geometry of moduli spaces, and the study of quantum field theories.