New appendix: An effective expression for the decomposition of Fano schemes of intersections of two quadrics

This post concerns the appendix (in its standalone version, written by me) to Lena Ji and Fumiaki Suzuki's paper on motivic classes of Fano schemes of lines. A few years ago we wrote a paper on decompositions of Fano schemes of linear subspaces on intersections of two quadrics. For $g\ge 2$ and $0\le k\le g-2$, consider a smooth intersection $Q_1\cap Q_2\subset\mathbb{P}^{2g+1}$ with associated hyperelliptic curve $C$. One of our conjectures predicts the following identity in $\mathrm{K}_0(\mathrm{Var})$: \begin{equation} [\mathrm{F}_k(Q_1\cap Q_2)] = \sum_{i=0}^{k+1}\mathrm{M}_{g,k,i}(\mathbb{L})[\operatorname{Sym}^i C], \end{equation} where $\mathbb{L}=[\mathbb{A}^1]$, $\operatorname{Sym}^i C$ is the $i$th symmetric power of $C$, and for $0\le i\le k+1$, the coefficient is \begin{equation} \begin{aligned} \mathrm{M}_{g,k,i}(\mathbb{L}) ={}&\mathbb{L}^{i(g-k-1)}\Biggl( \binom{2g-k-i}{k+1-i}_{\mathbb{L}}\\ &-\bigl(\mathbb{L}^{g-k-1}+\mathbb{L}^{g+2k-3i}\bigr)…