# Fano Schemes of Intersections of Quadrics

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 $$n in terms of the geometry of (moduli spaces on) $$\mathrm{C}$$?

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.