Remark 43.25.3. Lemma 43.24.3 and Theorem 43.25.2 also hold for nonsingular quasi-projective varieties with the same proof. The only change is that one needs to prove the following version of the moving Lemma 43.24.1: Let $X \subset \mathbf{P}^ N$ be a closed subvariety. Let $n = \dim (X)$ and $0 \leq d, d' < n$. Let $X^{reg} \subset X$ be the open subset of nonsingular points. Let $Z \subset X^{reg}$ be a closed subvariety of dimension $d$ and $T_ i \subset X^{reg}$, $i \in I$ be a finite collection of closed subvarieties of dimension $d'$. Then there exists a subvariety $C \subset \mathbf{P}^ N$ such that $C$ intersects $X$ properly and such that

$(C \cdot X)|_{X^{reg}} = Z + \sum \nolimits _{j \in J} m_ j Z_ j$

where $Z_ j \subset X^{reg}$ are irreducible of dimension $d$, distinct from $Z$, and

$\dim (Z_ j \cap T_ i) \leq \dim (Z \cap T_ i)$

with strict inequality if $Z$ does not intersect $T_ i$ properly in $X^{reg}$.

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