Let $Q\subset \mathbb P_k^3$ be a smooth quadric. Its Fano variety of lines $F(Q)$ is the union of two disjoints lines of $G(2,4)$ and according to the answer to this question the locus $F_{osc} =\{[l]\in F(Q), \exists P\mathrm{\ plane\ s.t.\ } 2l\subseteq P\cap Q\}$ is a zero cycle in $F(Q)$. How can one compute its degree ?

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$F_{osc}$ is empty. The quadratic form defining $Q$ has rank 4, so its restriction to any hyperplane has rank 3 or 2. So for any $P$, $P\cap Q$ consists of either a smooth conic or two distinct lines.