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Let C be a smooth projective curve, and let $L_1$ and $L_2$ be two line bundles with degree $d_1$ and $d_2$ respectively, where $gcd(r,d_i)=1$ for $i=1,2$. Suppose $M(r, L_1)$ and $M(r, L_2)$ are two isomorphic moduli spaces of rank $r$ semi-stable vector bundles over $C$ with fixed determinants $L_1$ and $L_2$. What can you say about $L_1$ and $L_2$? Are there any relations between them?

I know that there are papers,

  1. Biswas-Gomez-Munoz, Automorphisms of moduli spaces of vector bundles over a curve, arXiv:1202.2961
  2. Kouvidakis-Pantev, The automorphism group of the moduli space of semi stable vector bundles, arXiv:9306001

which talk about automorphisms of $M(r,L)$, giving a nice equivalent conditions between line bundles.

PS: In "Periods of a moduli space of bundles on curves" by Mumford and Newstead, they said that the variety $M(r,L)$ essentially depends on the residue class of $\deg(L) (\mod n)$. Also, https://mathoverflow.net/questions/347975/is-the-determinant-map-det-mathcalmr-d-rightarrow-picdx-on-moduli-spac#:~:text=Is%20the%20determinant%20map%20d,Thanks%20in%20advance!, this claims that the $\det$ map is a fibre bundle, which implies $M(r,L)$ isomorphic for all $L\in Pic^d(C)$ and $\det$ map is locally trivial.

Now, I am confused. If anyone can provide me with any insight (possibly a complete answer) on this, I would be grateful.

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  • $\begingroup$ Check out the paper "Periods of a moduli space of bundles on curves" by Mumford and Newstead. It is from 1967 and maybe(?) the first paper which studies this Torelli type theorems. $\endgroup$ Commented Oct 24 at 9:31
  • $\begingroup$ @Anubhab Pahari - "Let C be a smooth projective curve, and let L1 and L2 be two line bundles with rank r and degree d1 and d2 respectively" - A line bundle has rank 1 by definition. $\endgroup$ Commented Oct 24 at 9:41
  • $\begingroup$ @hm2020, silly mistake. Fixed it. $\endgroup$ Commented Oct 24 at 9:47
  • $\begingroup$ @Bernie There is a Torelli type theorem that says, if given two curves $C_1$ and $C_2$, we have two moduli spaces $M(r,L_1)$ and $M(r,L_2)$ respectively over $C_1$ and $C_2$ respectively, then we have $C_1 \simeq C_2$ and $r=r'$. But nothing is said about $L_1$ and $L_2$. $\endgroup$ Commented Oct 24 at 9:51
  • $\begingroup$ The determinant morphism (from the moduli space of semistable vector bundles of fixed rank and degree) to the Picard variety is an isotrivial fibration, so the choice of line bundle is invisible from the isomorphism class of the moduli space with fixed determinant. $\endgroup$ Commented Oct 24 at 10:46

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This is more of a comment/summary but it's too long.

If $L_1$ and $L_2$ have the same degree, then $M(r,L_1)$ and $M(r,L_2)$ are isomorphic, and they are fibers of the determinant map $M(r,d) \to \mathrm{Pic}^d(C)$. An isomorphism $M(r,L_1)\to M(r,L_2)$ is given by tensoring by a line bundle $N$ of degree $0$ such that $N^r \cong L_2 \otimes L_1^*$.

Similarly, twisting by line bundles of nonzero degree shows that $M(r,L_1)$ and $M(r,L_2)$ are isomorphic if $\deg L_1 \equiv \deg L_2 \pmod r$.

There is an additional isomorphism between $M(r,L_1)$ and $M(r,L_1^*)$ given by taking the dual. This then shows that $M(r,L_1)\cong M(r,L_2)$ whenever $\deg L_1 \equiv \pm \deg L_2 \pmod r$.

Thus every moduli space $M(r,L)$ is isomorphic to a space of the form $M(r,L)$ with $0\leq \deg L \leq r/2$.

"Generically" I believe these are the only isomorphisms, but for special curves there might possibly be other isomorphisms:

(1) On $\mathbb{P}^1$ the moduli spaces $M(r,d)$ are all empty unless $d\equiv 0 \pmod r$.

(2) On an elliptic curve $C$, the moduli space $M(r,d)$ is isomorphic to $C$ whenever $r$ and $d$ are coprime. (See Le Potier, Lectures on Vector Bundles, section 8.6.)

(3) I don't know what is known in higher genus. I would suspect that for a general curve of genus $g \geq 2$ there are no additional isomorphisms, but that maybe a special curve could have some additional isomorphisms.

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  • $\begingroup$ Dear Professor @jackhuizenga, Thank you so much for answering this question. I was hoping this would be true. Now I guess I have to prove this formally or find it in the literature. $\endgroup$ Commented Oct 25 at 8:46

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