Is there a standard reference for the fact that, in an appropriate algebraic-geometrical context, the tangent space at the point $[E]$ to the moduli space $\mathcal M$ is something like $\operatorname{Ext}^1(E, E)$? I am primary interested in two situations:

  • $\mathcal M$ is the moduli space of semi-stable coherent torsion-free sheaves on $\mathbb CP^2$ with given Hilbert polynomial,
  • or $\mathcal M$ is the moduli space of semi-stable finite-dimensional complex representations of finite acyclic quiver with given dimension vector,

and in both situation, if the coefficients of the Hilbert polynomial/dimension vector are coprime, there is a smooth projective variety that is a fine moduli space $\mathcal M$.

Unfortunately, I do not know any deformation theory, so a study book on it that also proves something like $T_{[E]} \mathcal M=\operatorname{Ext}^1(E, E)$ would be welcome.

  • 6
    $\begingroup$ Have a look at The Geometry of moduli space of sheaves by Huybrechts and Lehn -- see e.g. Corollary 4.5.2. $\endgroup$
    – abx
    Oct 31 '16 at 20:44

Theorem 2.6 (page 9) from Hartshorne Lectures on Deformation Theory (it seems that Hartshorne uses the same notation both for an affine scheme $D$ and its function algebra):

Let $X$ be a scheme over $k$, and let $\mathcal F$ be a coherent sheaf on $X$. We define a deformation of $\mathcal F$ over $D=\operatorname{Spec} (k[t]/t^2)$ to be a coherent sheaf $\mathcal F_0$ on $X_0 = X \times D$, flat over $D$, together with a homomorphism $\mathcal F_0 \to \mathcal F$ such that the induced map $\mathcal F_0 \otimes_{k[t]/t^2} k \to \mathcal F$ is an isomorphism. Then the deformations of $\mathcal F$ over $D$ are in natural one-to-one correspondence with the elements of $Ext^1 (\mathcal F, \mathcal F)$, with the zero-element corresponding to the trivial deformation.

Now by the universal property of the moduli space $\mathcal M$ of coherent torsion-free sheaves on $\mathbb{CP}^2$ its tangent space at $[\mathcal F]$ consists of the deformations of $\mathcal F$ over $D$. Moreover, I believe that the proof should work as well over "a non-commutative affine scheme" whose function algebra is the path algebra of a finite acyclic quiver.


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