# Why is $[l_1]+[l_2]+[l_3]$ a constant in $CH_0(F(X))$, where $F(X)$ is the Fano variety of lines of a cubic fourfold?

Let $X\subset \mathbb P^5$ be a cubic fourfold and $F(X)$ be its Fano varieties of lines. Let $\mathbb P^2$ be a plane such that the intersection $\mathbb P^2\cap X$ consists of three lines $L_1,L_2,$ and $L_3$. It is clear that $[L_1]+[L_2]+[L_3]$ is constant in the Chow group $CH_1(X)$ of $X$ (i.e., it does not depend on the choice of $\mathbb P^2$).

Why is $[L_1]+[L_2]+[L_3]$ a constant in $CH_0(F(X))$ of $F(X)$?

• The family of $2$-planes in $\mathbb{P}^5$ is parameterized by the Grassmannian $\text{Grass}(\mathbb{P}^2,\mathbb{P}^5)$. This is a rational variety. Rational equivalence of cycles is the weakest equivalence relation stable for addition and that forms quotients by families of cycles parameterized by a rational variety. So all of the cycles obtained by intersection with $\mathbb{P}^2$s (that are not contained in $X$) give the same cycle class in the Chow group. – Jason Starr May 2 '17 at 10:32
• Typo correction: "... and that forms quotients by families ..." --> "... and that factors through the quotient by all families ..." – Jason Starr May 2 '17 at 11:11
• @Jason Starr: what you say proves that the sum is constant in $CH_1(X)$, but why in $CH_0(F(X))$? – abx May 2 '17 at 14:23
• @abx. Good point -- I missed that the OP is asking about $\text{CH}_0(F(X))$. – Jason Starr May 2 '17 at 14:55
• Sorry again for my mistake above, and thanks to abx for noticing this. I do not know the answer, but here is something interesting: the parameter space of triples of concurrent lines in $X$ is a conic bundle over a double cover of $X$. If that double cover is a Fano manifold (probably a long shot), that might give a strategy to prove this. – Jason Starr May 2 '17 at 16:04