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mrw
mrw
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If I understand your question, the answer is no.

Catanese produces examples of surfaces with $K_{Y_0}$ big and nef but not ample and such that if $Y_0\to X_0$ is the canonical model, thenYou can consider any surface $X_0$ admits only RDP singularities which dodoes not admit anya global smoothing. Now if Let $Y\to \Delta$ is$Y_0\to X_0$ be the minimal resolution and consider a smooth family $Y\to \Delta$ such that $K_{Y_t}$ is ample for the general $t$, then I guess what you are asking isfiber does not possibleadmit any curve with negative self-intersection.

If I understand your question, the answer is no.

Catanese produces examples of surfaces with $K_{Y_0}$ big and nef but not ample and such that if $Y_0\to X_0$ is the canonical model, then $X_0$ admits only RDP singularities which do not admit any global smoothing. Now if $Y\to \Delta$ is a smooth family such that $K_{Y_t}$ is ample for the general $t$, then I guess what you are asking is not possible.

If I understand your question, the answer is no.

You can consider any surface $X_0$ which does not admit a global smoothing. Let $Y_0\to X_0$ be the minimal resolution and consider a family $Y\to \Delta$ such that the general fiber does not admit any curve with negative self-intersection.

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mrw
mrw
  • 31
  • 2

If I understand your question, the answer is no.

Catanese produces examples of surfaces with $K_{Y_0}$ big and nef but not ample and such that if $Y_0\to X_0$ is the canonical model, then $X_0$ admits only RDP singularities which do not admit any global smoothing. Now if $Y\to \Delta$ is a smooth family such that $K_{Y_t}$ is ample for the general $t$, then I guess what you are asking is not possible.