November 12, 2016

Tangent space to moduli space of sheaves

Let $X$ be a projective variety. Suppose that the moduli space $\mathcal M$ of nice sheaves with fixed rank and Hilbert polynomial exists. Then
$$T_{[E]}\mathcal M=\mathrm{Ext}^1(E, E).$$

For a partial proof see
Hartshorne Lectures on Deformation Theory, theorem 2.6

From MO:253578.

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