# How to understand the geometric concept of tangent variety?

Introduction

I am a physicist struggling with some basic definitions and concepts from algebraic geometry. Therefore I apologize if I make mistakes, I'm just learning! By the way, English is not my mother tongue, sorry if I made grammar or vocabulary mistakes!

Tangent space to a point

Let $$X$$ be a projective variety ($$X\subset \mathbb{P}^n$$). The intrinsic definition of the tangent space to $$X$$ at $$p\in X$$ is $$T_p(X)=(m_p\backslash m_p^2)^\star$$, where $$m_p$$ is the maximum ideal of functions vanishing at p. Equivalently, one can define $$T_p(X)= Z(\{d_pf ~|~ f\in I(X)\})$$.

Tangent variety

The definition of the tangent variety $$\tau(X)$$ to X follows : $$\tau(X)= \underset{p\in X}{\cup} T_p(X).$$ An equivalent way to define it is by joints. Let $$Y\subset \mathbb{P}^n$$. One defines the joint of X and Y by $$J(X,Y)=\overline{\underset{x\in X, y\in Y, x\neq y}{\cup} \mathbb{P}^1_{x,y}}$$ where $$\mathbb{P}^1_{x,y}$$ is the projective line containing x and y. Let $$y_o\in Y$$ and define $$T^\star_{x,y,y_o}=\overline{\underset{x\in X, y\in Y, x,y\rightarrow y_0}{\cup} \mathbb{P}^1_{x,y}}$$. Then define $$T(X,Y)=\underset{y_0\in Y}{\cup}\mathbb{P}^\star_{x,y,y_0}$$. The tangent variety to X can therefore be defined as $$\tau(X)=T(X,X).$$

Question

Link between algebraic and geometric concept of tangent variety: What is the link between the algebraic concept of tangent variety $$\tau(X)= \underset{p\in X}{\cup} T_p(X)$$ and the geometric concept of tangent variety $$\tau(X)=T(X,X)$$ ? To me, these two definitions look totally different.

• what is the meaning of $x,y\to y_0$ in the second definition? the usual definition of a tangent line at $y_0$ is limit of lines $xy$ where $x,y$ goes to $y_0$ which should be related to the second definition, equivalently you can associate to a tangent line a partial deriverate along this line and this is the first deinition.
– ali
Feb 15, 2022 at 10:31
• The meaning of $x,y \rightarrow y_0$ is not perfectly clear in my mind. I think that it means "with x and y going to $y_0$". Reading your comment, I understand that I don't have any insight about limit of lines xy where x,y goes to $y_0$. I mean, the limit of lines xy where x and y go to $y_0$ is not the same thing as the line xy itself ? Feb 15, 2022 at 11:55