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For questions related to divisors in the sense of algebraic geometry (Cartier divisors, Weil divisors and so on). For question on divisors in the number theoretic sense please use the tag divisors-multiples.
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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 …