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Let $X\subseteq\mathbb{P}^n$ be a projective variety. The Fano scheme $F_1(X)$ of lines $L\in\mathbb{P}^n$ contained in $X$ is a subscheme of the Grassmannian $\mathbb{G}(1,n)$. We can find the class $[F_1(X)]$ in the Chow ring of the Grassmannian. Furthermore, we can write this class uniquely as a sum of the Schubert cycles $\sigma_{a,b}\in A^{a+b}(\mathbb{G}(1,n))$.

On the other hand, we can also look at the space $|F_1(X)|\subseteq X$, consisting of the union of all lines $L\in F_1(X),$ presumably with an appropriate reduced structure corresponding to lines appearing with multiplicity. We can write the class $[|F_1(X)|]$ in the Chow ring of $\mathbb{P}^n$ uniquely as a sum of planes of various dimensions.

My question is, what is the relationship between these two expressions? Here is an example. On a quadric surface $Q\subseteq\mathbb{P}^3$, the class $[F_1(X)]$ is (if I calculated it correctly) $4\sigma_{2,1}$, i.e. four times the class of lines in a plane $H$ containing a point $p\in H.$ Now it is apparently true (quadric surfaces are just the Segré surface up to change of coordinates, no?) that generically, each point of $Q$ is contained in two lines in $Q$. In other words, $|F_1(X)|$ is "two copies of $Q$", a subscheme of degree 4 in codimension 1 in $\mathbb{P}^3$. It seems like perhaps not a coincidence, then, that $[F_1(X)]$ is "degree 4 in codimension 1," whatever that should mean. Is there a natural notion of "degree" in the Chow groups of the Grassmannian, that is preserved by this map to the Chow groups of projective space?

In particular, if I calculated everything correctly, the class of lines on a general quartic surface in $\mathbb{P}^4$ is $544\sigma_{3,2}.$ Does anyone happen to know if it's true that through each point there are 544/4=136 lines?

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