Suppose I have a one parameter flat family of complex surfaces (regular, of general type) whose general fibre is smooth. Is it possible for the central fibre to have singularities which are not canonical? If so, how bad can they be?
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$\begingroup$ Take a 1-parameter family of curves $(X_t)_t$ whose central fibre is "very bad". Now consider $(X_t\times C)_t$ with $C$ a fixed smooth curve. $\endgroup$– Ariyan JavanpeykarAug 18, 2018 at 8:41
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1$\begingroup$ On the positive side, if the general fiber is a canonically polarized surface with canonical singularities (which you can achieve by running an MMP), you can then assume that (after an appropriate base change) the special fiber is a canonically polarized SLC surface (if the special fiber happens to be irreducible, then it will actually have canonical sings). $\endgroup$– HaconAug 20, 2018 at 15:30
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$\begingroup$ That sounds quite well @Hacon. Do you have a reference in which I can find that statement? Many thanks in advance. $\endgroup$– R. JahvelAug 24, 2018 at 15:24
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$\begingroup$ arxiv.org/pdf/1008.0621.pdf is a good introduction; properness of the functor in full generality is proven in arxiv.org/abs/1105.1169 $\endgroup$– HaconAug 25, 2018 at 1:22
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2 Answers
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The cone over a plane curve of degree $d$ deforms to a smooth surface in $\mathbb P^3$ of degree $d$. Take $d\ge 5$ to see that things can be arbitrarily bad.
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The central fiber can even be everywhere non-reduced. This can happen when you take the central fiber of the global image of a global map whose general fiber map is the canonical embedding of a surface of general type with very ample canonical map but its central fiber map is the canonical map of a surface of general type whose canonical map is a double covering onto some rational surface.
answered Aug 18, 2018 at 8:03