I'm trying to understand the claim that the moduli space of stable rank 2 vector bundles on a (general?) cubic 3-fold, say $X$, with $deg c_2=6$ and $det=O_X(2)$ is of dimension 9.

I belive what I can do, is to compute the tangent space at a general point, says $E$, and try to compute its dimension, and I seem to recall that this can be done by computing the $H^1$ of a certain bundle ($End(E)?$) but I haven't managed to find any specific reference for that fact.

Is it correct? And if so, where does that come from?


The Zariski tangent space of the moduli space of stable sheaves at a point $[F]$ for a stable sheaf $F$ can be canonically identified with $Ext^1(F,F)$. Now if $F$ is locally free, then this space is just $H^1(X,\mathcal{E}nd(F))$.

You can read about all that in Huybrechts-Lehn: The Geometry of Moduli Spaces of Sheaves.

Where did you read that the moduli space you investigate is 9-dimensional?

  • $\begingroup$ Thank you! I read it here math.polytechnique.fr/~voisin/Articlesweb/fibresaj.pdf on page 11. $\endgroup$ – PTV May 20 '15 at 16:03
  • $\begingroup$ Do you see an easy way prove that such a vector bundle $E$ in the moduli space satisfies $H^1(X, End(E))$ is 9-dimensionnal. I gave this some thought, and tried to use Riemann-Roch, but i don't have the right result. $\endgroup$ – PTV May 22 '15 at 14:15
  • $\begingroup$ I thought about that myself. I mean there is a fairly explicit description of the bundle, but i couldn't do it either. Maybe one should try and write this bundle as an extension using the Serre correspondence. Maybe you should ask this in a new question. $\endgroup$ – Bernie May 22 '15 at 16:50

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