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This is a question from an online note. Let $A$ be a two-dimensional $\mathbb C$-torus. And there is an involution on $A$: $A\to A, x\mapsto -x$. The action has 16 fixed points. Let $Y:=A/\{\pm1\}$, then $Y$ is a complex surface with 16 ordinary double points. Let $X$ be the blow up of $Y$ at all 16 singular points. After some calculations, we can see $X$ is a $K3$ surface. Then it's claimed that if $A$ is not projective then $X$ is not projective and we get an example of a non-projective $K3$ surface. But I feel confused why "if $A$ is not projective then $X$ is not projective".

I know since $A\to X$ is finite, thus if $Y$ is projective, then we can pullback an ample line bundle to an ample line to $A$. But I can't see why $X$ is projective implies $Y$ is projective? Do we have contracting a rational curve on a complex surface preserves the projectivity?

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    $\begingroup$ Contracting a curve with negative square on a surface preserves projectivity. See this question of MO and its answer. $\endgroup$
    – abx
    Commented Sep 4, 2020 at 4:00
  • $\begingroup$ Crossposted from MSE. When crossposting, it is important to link all versions together to prevent duplicating work. $\endgroup$
    – KReiser
    Commented Sep 4, 2020 at 9:52

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In fact, $X$ is projective if and only if $A$ is projective.

If $A$ is projective, then $Y$ is so, being the quotient of a projective variety by a finite group (this is a toy model of GIT, see this MO question). Then $X$ is projective, too, being the blow-up of the projective variety $Y$ at a finite number of points.

Conversely, assume $X$ projective. Then there is a double cover $\tilde{A} \to X$, where $\tilde{A}$ is the blow-up of $A$ at its $16$ points of order $2$. This shows that $\tilde{A}$ is projective, so the blow-down $A$ is projective as well (an alternative argument is noticing that $X$ projective implies $Y$ projective and so $A$ projective, since contracting a $(-2)$ curve on a projective surface preserves projectivity, as explained in abx's comment).

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  • $\begingroup$ Thanks for the nice answer, but why is the blow down also projective? Can you explain or tell me a reference? $\endgroup$
    – 6666
    Commented Sep 4, 2020 at 6:30
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    $\begingroup$ For surfaces, this is usually called "Castelnuovo contractibility theorem". Look for it in Griffiths-Harris, p. 476 (where it is called "Castelnuovo-Enriques criterion"), or in Beauville's book on surfaces (II.17) $\endgroup$
    – Francesco Polizzi
    Commented Sep 4, 2020 at 7:49
  • $\begingroup$ Sorry, when we apply Castelnuovo contractibility theorem, how can I know the blow down (projective) surface is exactly $A$? $\endgroup$
    – 6666
    Commented Sep 4, 2020 at 17:20
  • $\begingroup$ The contracting map must factor through the blow-down by the universal property, and from this it is an immediate exercise to show that it coincides with the blow-down (in fact, this is part of the statement of Castelnuovo's criterion). $\endgroup$
    – Francesco Polizzi
    Commented Sep 4, 2020 at 18:35

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