(co)tangent space (of a variety)

Last updated January 27, 2022

Tangent space

Recall the differential of a regular map.

Direct definition

Given an algebraic set $U={F_1=0,\dots,F_m=0}$, we may the define the tangent space of $U$ at the point $p$ as the solution set $$T_pU={\text{d}_pF_1=\dots=\text{d}_pF_m=0}.$$

Arguable more intrinsically, we may define the tangent space to simply be the dual of the cotangent space (see below).

Cotangent space

Given a point $p\in X$, we may define the cotangent space as $$\mathfrak{m}{p,X}/\mathfrak{m}{p,X}^2$$ where $\mathfrak{m}_{p,X}$ is the maximal ideal coresponding to the point $p$.

Examples

1. For a quadric defined by the quadratic form $Q={q=0}$, let $\tilde{q}$ denote the polarization of $q$. Then $$T_pQ={x\mid \tilde{q}(x,p)=0}.$$

Properties

1. $\text{dim } T_p X\geq\text{dim }X$ for all $p\in X$.