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Let $X$ be a quasi-projective variety over $\mathbb{C}$, we say it is "nearly proper" if $X=Y-Z$ for some projective variety $Y$ and a closed subset $Z\subset Y$ of codimension at least two.

(1) For every morphism $f\colon X\to\mathbb{P}^n$, is the image $f(X)$ always locally closed?

(2) For every morphism $f\colon X\to\mathbb{P}^n$, is the map $d_f\colon x\mapsto \dim f^{-1}(x)$ always upper semi-continuous on $f(X)$?

(3) If $H^0(X,\mathcal{O}_X)=\mathbb{C}$, is $X$ necessarily "nearly proper"?

[If (1) holds, then $f(X)$ can be equipped with the structure of a variety. (1) is not true for $f\colon\mathbb{A}^2\to\mathbb{P}^2,(x,y)\mapsto [x:xy:1]$. (2) Is true for points in smooth surfaces: we can resolve $Y\dashrightarrow \overline{f(X)}$ by blow ups, then deleting points and curves in fibers do not change the dimension or make the fiber empty. (3) Being "nearly proper" implies $H^0(X,\mathcal{O}_X)=\mathbb{C}$, as any two points in $X$ can be connected by proper irreducible curves.]

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    $\begingroup$ The answer to (1) is no. Let $Y\to\mathbf{P}^3$ be the blowup of $\mathbf{P}^3$ at a point $P$ and let $Z\subseteq Y$ be the strict transform of a line $L$ through $P$. Then $X=Y-Z$ is nearly proper, but the image of $X\to \mathbf{P}^3$ is $(\mathbf{P}^3 - L)\cup \{P\}$ which is not locally closed. $\endgroup$
    – Piotr Achinger
    Commented Mar 20, 2023 at 21:36
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    $\begingroup$ For (3) you might consider the total space of a line bundle on an elliptic curve that (a) has c1=0 and (b) is not a torsion line bundle. $\endgroup$
    – Yosemite Stan
    Commented Mar 21, 2023 at 3:53
  • $\begingroup$ And for (2) the answer is again no. Let $X' = X\times \mathbf{P}^1 \sqcup L$ where $X$ and $L$ are as in the previous example. Again, $X'$ is nearly proper. We have the map $X'\to \mathbf{P}^3$ which on $X\times\mathbf{P}^1$ is the projection onto $X$ followed by the map to $\mathbf{P}^3$ as before, and on $L$ is the inclusion (we add $L$ so that the fibers are at least nonempty). Then the generic fiber has dimension one while the fiber over a point $Q\in L-\{P\}$ is zero-dimensional. $\endgroup$
    – Piotr Achinger
    Commented Mar 21, 2023 at 11:47
  • $\begingroup$ @YosemiteStan Thanks for the comment, I tried the following: take $\overline{X}=\mathbb{P}(L\oplus 1)$, then $\mathrm{Pic}(\overline{X})=\mathrm{Pic}(E)[h]/(h^2+c_1(L)h)$. Let $H$ be the curve at infinity, then $H^2=0$. Not sure if we can contract the curve at infinity $H$, so as to get a codim-2 compactification? $\endgroup$
    – chord_213
    Commented Mar 21, 2023 at 11:53
  • $\begingroup$ @PiotrAchinger Thanks for the nice examples! (It would be great if $X$ is irreducible. The motivation for these questions was an attempt to relax the properness condition in the rigidity lemma. For the classical rigidity lemma, one may relax properness to "fiberwise admitting a compactification in codim 2", using the trick that any two point can be connected by proper curves. But for the generalized versions (mathoverflow.net/questions/230466/…) seems the trick does not work..and semi-continuity is wondered) $\endgroup$
    – chord_213
    Commented Mar 21, 2023 at 12:13

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