# Reference quest: variety of lines and variety of planes

Let $X\subset \mathbb P_{\mathbb C}^n$ be a smooth projective variety, $F(X)\subset G(2,n+1)$ its Fano variety of lines and $$I_F=\left\{([l],[l'])\in F(X)\times F(X), l\cap l'\neq \emptyset\right\}$$ the incidence correspondence. There is a natural application $\alpha:I_F\backslash \Delta_{F(X)}\rightarrow G(3, n+1)$. Are there some references were this application (or something close to) is "studied", to compute, for example, the class (in the Chow group of the variety resolving the indeterminacies) of the secant lines which span a $\mathbb P^2$ contained in $X$?

• Denote by $\text{Flag}(1,2,n+1)$ the flag variety parameterizing pointed lines, and denote by $F_{0,1}(X) \subset \text{Flag}(1,2,n+1)$ the variety of pointed lines in $X$. Denote by $\text{ev}:F_{0,1}(X)\to X$ the morphism forgetting the line. If the fiber of $\text{ev}$ over a general point of $X$ is nonempty, then it is smooth of the expected dimension $\langle c_1(T_X), [\ell] \rangle - 2$, and its first Chern class can be computed (e.g., in my papers with de Jong). Lines in fibers of $\text{ev}$ are 2-planes. You can use this to compute your cycle class for $X$ a general hypersurface. – Jason Starr Jun 27 '16 at 15:10
• The idea in my last comment is close to what is in my note, "Fano Varieties and Linear Sections of Hypersurfaces". – Jason Starr Jun 27 '16 at 18:14