Proof.
The functors \mathrm{Hilb}^ d_{X/k} are representable by Proposition 44.2.6 (see also Remark 44.2.7) and the fact that X is projective (Varieties, Lemma 33.43.4). The schemes \underline{\mathrm{Hilb}}^ d_{X/k} are separated over k by Lemma 44.2.4. The schemes \underline{\mathrm{Hilb}}^ d_{X/k} are smooth over k by Lemma 44.3.5. Starting with X = \underline{\mathrm{Hilb}}^1_{X/k}, the morphisms of Lemma 44.3.4, and induction we find a morphism
X^ d = X \times _ k X \times _ k \ldots \times _ k X \longrightarrow \underline{\mathrm{Hilb}}^ d_{X/k},\quad (x_1, \ldots , x_ d) \longrightarrow x_1 + \ldots + x_ d
which is finite locally free of degree d!. Since X is proper over k, so is X^ d, hence \underline{\mathrm{Hilb}}^ d_{X/k} is proper over k by Morphisms, Lemma 29.41.9. Since X is geometrically irreducible over k, the product X^ d is irreducible (Varieties, Lemma 33.8.4) hence the image is irreducible (in fact geometrically irreducible). This proves (1). Part (2) follows from the definitions. Part (3) follows from the commutative diagram
\xymatrix{ X^{d_1} \times _ k X^{d_2} \ar[d] \ar@{=}[r] & X^{d_1 + d_2} \ar[d] \\ \underline{\mathrm{Hilb}}^{d_1}_{X/k} \times _ k \underline{\mathrm{Hilb}}^{d_2}_{X/k} \ar[r] & \underline{\mathrm{Hilb}}^{d_1 + d_2}_{X/k} }
and multiplicativity of degrees of finite locally free morphisms.
\square
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