Remark 108.10.4 (Numerical invariants). Let B, X, Y be as in the introduction to this section. Let I be a set and for i \in I let E_ i \in D(\mathcal{O}_{Y \times _ B X}) be perfect. Let P : I \to \mathbf{Z} be a function. Recall that
\mathit{Mor}_ B(Y, X) \subset \mathrm{Hilb}_{Y \times _ B X/B}
is an open subspace, see Quot, Lemma 99.12.2. Thus we can define
\mathit{Mor}^ P_ B(Y, X) = \mathit{Mor}_ B(Y, X) \cap \mathrm{Hilb}^ P_{Y \times _ B X/B}
where \mathrm{Hilb}^ P_{Y \times _ B X/B} is as in Remark 108.7.6. The morphism
\mathit{Mor}^ P_ B(Y, X) \longrightarrow \mathit{Mor}_ B(Y, X)
is a flat closed immersion which is an open and closed immersion for example if I is finite, or B is locally Noetherian, or I = \mathbf{Z}, E_ i = \mathcal{L}^{\otimes i} for some invertible \mathcal{O}_{Y \times _ B X}-module \mathcal{L}. In the last case we sometimes use the notation \mathit{Mor}^{P, \mathcal{L}}_ B(Y, X).
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