Fano Schemes of Intersections of Quadrics

Question

Let $g\geqslant 2$, and denote by $\mathrm{X}=\mathrm{Q}_1\cap\mathrm{Q}_2\subset\mathbf{P}^{2g+1}$ a smooth intersection of quadrics. By considering the pencil generated by $\mathrm{Q}_1,\mathrm{Q}_2$ we obtain an associated hyperelliptic curve $\mathrm{C}$ of genus $g$. The following results on the Fano scheme $\mathrm{F}_n(\mathrm{X})$ of $n$-planes on $\mathrm{X}$ are well-known:

(i) $\mathrm{F}_n(\mathrm{X})=\emptyset$ for $n>g-1$;

(ii) $\mathrm{F}_{g-1}(\mathrm{X})\simeq\mathrm{J}(\mathrm{C})$, the Jacobian of $\mathrm{C}$ (theorem of A. Weil);

(iii) $\mathrm{F}_{g-2}(\mathrm{X})\simeq\mathrm{U}_{\mathrm{C}}(2,\mathscr{L})$, the moduli space of rank $2$ bundles on $\mathrm{C}$ with fixed determinant $\mathscr{L}$ of odd degree (theorem of Desale-Ramanan).

Is there an interpretation of the $\mathrm{F}_n(\mathrm{X})$ for $n<g-2$ in terms of the geometry of (moduli spaces on) $\mathrm{C}$?

Answer 1

It turns out that this question was resolved by Ramanan in Orthogonal and spin bundles over hyperelliptic curves. He proves that $\mathrm{F}_{g-n}(\mathrm{X})$ is isomorphic to the following moduli space $\mathscr{M}$. Let $\iota$ be the hyperelliptic involution of $\mathrm{C}$, and $\mathscr{L}$ a $\iota$-invariant line bundle of degree $2g-1$. Then $\mathscr{M}$ is the moduli space of $\iota$-invariant orthogonal bundles $\mathscr{E}$ of rank $2n$ on $\mathrm{C}$ with $\Gamma^{+}(2n)$-structure such that for every Weierstrass point $x$ the $-1$-eigenspace of $\iota$ on the fibre $\mathscr{E}\otimes\mathcal{L}(x)$ has dimension $1$.

Here $\Gamma^{+}(2n)$ is a certain subgroup of the group of units of the Clifford algebra.