Lemma 44.3.4. Let X \to S be a smooth morphism of schemes of relative dimension 1 such that the functors \mathrm{Hilb}^ d_{X/S} are representable. The morphism \underline{\mathrm{Hilb}}^ d_{X/S} \times _ S X \to \underline{\mathrm{Hilb}}^{d + 1}_{X/S} is finite locally free of degree d + 1.
Proof. Let D_{univ} \subset X \times _ S \underline{\mathrm{Hilb}}^{d + 1}_{X/S} be the universal object. There is a commutative diagram
\xymatrix{ \underline{\mathrm{Hilb}}^ d_{X/S} \times _ S X \ar[rr] \ar[rd] & & D_{univ} \ar[ld] \ar@{^{(}->}[r] & \underline{\mathrm{Hilb}}^{d + 1}_{X/S} \times _ S X \\ & \underline{\mathrm{Hilb}}^{d + 1}_{X/S} }
where the top horizontal arrow maps (D', x) to (D' + x, x). We claim this morphism is an isomorphism which certainly proves the lemma. Namely, given a scheme T over S, a T-valued point \xi of D_{univ} is given by a pair \xi = (D, x) where D \subset X_ T is a closed subscheme finite locally free of degree d + 1 over T and x : T \to X is a morphism whose graph x : T \to X_ T factors through D. Then by Lemma 44.3.3 we can write D = D' + x for some D' \subset X_ T finite locally free of degree d over T. Sending \xi = (D, x) to the pair (D', x) is the desired inverse. \square
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