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Grassmannians are algebraic varieties whose points corresponds to vector subspaces of a fixed dimension in a fixed vector space.
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Differential of a specific morphism to a Grassmannian
The differential of $\Phi$ at a point $(D,\theta)$ in the fiber product is given by the induced map on tangent spaces:
$$T(\mathrm{Div} _{\ge 0}(X)) \times T(\mathrm{Pic} (X)) \times T(\mathrm{Aut} ^0 …