# Dimension of moduli space of some stable vector bundles on a cubic 3-fold.

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?

## 1 Answer

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?

• Thank you! I read it here math.polytechnique.fr/~voisin/Articlesweb/fibresaj.pdf on page 11. – PTV May 20 '15 at 16:03
• 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. – PTV May 22 '15 at 14:15
• 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. – Bernie May 22 '15 at 16:50