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) $\mathrm{C}$?


1 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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.