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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.

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    $\begingroup$ 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. $\endgroup$
    – ali
    Feb 15, 2022 at 10:31
  • $\begingroup$ 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 ? $\endgroup$
    – Baloo
    Feb 15, 2022 at 11:55

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