Let $X_0$ be a smooth projective variety over the complex numbers and let $X$ be an infinitesimal deformation of $X_0$ over the ring of dual numbers.

If the canonical bundle of $X_0$ is ample (resp. anti-ample), does it follow that the canonical bundle of $X$ over $\mathbb C[\epsilon]$ is ample (resp. anti-ample)?


This is true in a much more general setting:

Fact Let $X$ be a proper variety over a noetherian ring $A$ and let $\mathscr L$ be a line bundle on $X$. Then $\mathscr L$ is ample if and only if $\mathscr L_{\mathrm{red}}\simeq \mathscr L\otimes \mathscr O_{X_{\mathrm{red}}}\ $ is ample on $X_{\mathrm{red}}$.

See Exc.III.5.7(b) in [Hartshorne] for the proper case and EGA II, 4.5.13 without the proper assumption (thanks to @Laurent Moret-Bailly for the EGA reference).

In your case $X_0\simeq X_{\mathrm{red}}$ and $\omega_{X_0}\simeq \left(\omega_{X} \right)_{\mathrm{red}}\simeq \left(\omega_{X/\mathbb C[\varepsilon]}\ \ \right)_{\mathrm{red}}$ .

  • 2
    $\begingroup$ In fact $X$ need not be proper; or even of finite type (EGA II, 4.5.13). $\endgroup$ – Laurent Moret-Bailly Jul 1 '15 at 5:56
  • $\begingroup$ Right, of course. I guess I got stock in the proper world by the wording of the question... :) $\endgroup$ – Sándor Kovács Jul 1 '15 at 6:58

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