Let $X\subsetneq\mathbb{P}^{N}$ be a smooth projective variety, and let $$ S_{X}=\overline{\{(x,y,z)\in X\times X\times \mathbb{P}^{N}:x\neq y, z\in\langle x,y\rangle\}}. $$ The secant variety of $X$ is defined to be $$ SX:=\varphi(S_{X}), $$ where $\varphi:S_{X}\rightarrow \mathbb{P}^{N}$ is the projection to the third factor.

If $\dim X=n$, then it is easy to prove that $S_{X}$ is irreducible and $\dim S_{X}=2n+1$. Since $$ \varphi^{-1}(X)\neq S_{X} $$ and given $x\in X$ $$ \varphi^{-1}(x)\supseteq \{(x,y,x)\in S_{X}:y\in X\}\simeq X, $$ we deduce that $\dim \varphi^{-1}(x)=n$ for general $x\in X$ and $\dim \varphi^{-1}(X)=2n$.

Accordingly, $\varphi^{-1}(X)$ is a hypersurface in $S_{X}$. Which is the equation defining $\varphi^{-1}(X)$ in $S_{X}$?

Let $X\subsetneq\mathbb{P}^{N}$ be a smooth projective variety, and let $$ S_{X}=\overline{\{(x,y,z)\in X\times X\times \mathbb{P}^{N}:x\neq y, z\in\langle x,y\rangle\}}. $$ The secant variety of $X$ is defined to be $$ SX:=\varphi(S_{X}), $$ where $\varphi:S_{X}\rightarrow \mathbb{P}^{N}$ is the projection to the third factor.

If $\dim X=n$, then it is easy to prove that $S_{X}$ is irreducible and $\dim S_{X}=2n+1$. Since $$ \varphi^{-1}(X)\neq S_{X} $$ and given $x\in X$ $$ \varphi^{-1}(x)\supseteq \{(x,y,x)\in S_{X}:y\in X\}\simeq X, $$ we deduce that $\dim \varphi^{-1}(x)=n$ for general $x\in X$ and $\dim \varphi^{-1}(X)=2n$.

Accordingly, $\varphi^{-1}(X)$ is a hypersurface in $S_{X}$. Which is the equation defining $\varphi^{-1}(X)$ in $S_{X}$ in a neigbourhood of a point $p\in \varphi^{-1}(X)$?