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 $p\in \varphi^{-1}(X)$?


If you do not make any hypothesis on $X$, then your statement $\phi^{-1}(X) \neq S_X$ is wrong. Take $X$ by a linear space, then $S(X) = X$.

Even if you assume that $X \neq S(X)$, it's not clear to me that $\phi^{-1}(X)$ is irreducible and let alone that it is pure dimensional. Certainly $X \times X$ (seen as the zero section of the $\mathbb{P}^1$-bundle (over $X \times X$) defining $S_X$) is an irreducible component of $\phi^{-1}(X)$, but I am pretty sure you may have extra components, not necessarily hypersurfaces, and these components certainly strongly depend on the geometry of $X$. I guess it would be hard to find a simple set of equations for them.

For instance, let $X \subset \mathbb{P}^n$ be a ruled surface. Then, for all $x \in X$, $\phi^{-1}(x)$ contains $\{(x,y,x), y \in X\}$, but also contains $\{(z_1,z_2,x), (z_1,z_2) \in L_x \times L_x \}$, where $L_x$ is a line of $X$ passing trough $x$. And this last set is an irreducible component of $\phi^{-1}(x)$ disjoint from $\{(x,y,x), y \in X\}$.

In this particular case, one can certainly also guess the equation of this extra component. But you see that this extra component strongly depends on the geometry of lines included in $X$. Hence I guess there is no hope to find a simple general set of equations to describe $\phi^{-1}(X)$.


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