Let $V$ be a vector space over a field with discrete valuation and let $R$ be its ring of integers. Which varieties can be reached as the special fiber of a flat degeneration over $R$, when the generic fiber is $\mathbb{P}(V)$?
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1$\begingroup$ If you don't put any restriction on your special fiber there is no hope to get a reasonable answer, already for $\mathbb{P}(V)=\mathbb{P}^1$: for instance you can get as special fiber plane curves of arbitrarily high degree (with some embedded points). $\endgroup$– abxCommented Apr 29, 2016 at 6:20
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1$\begingroup$ One famous theorem is that, if the special fiber is smooth, then the base change to the algebraic closure of the ground field is isomorphic to projective space, at least in characteristic 0 (I believe it is now known in all characteristics, but I am forgetting a reference). Siu first proved this over the complex numbers using differential geometry, but now it is a corollary of an algebraic geometry theorem of Cho, Miyaoka and Shepherd-Barron. $\endgroup$– Jason StarrCommented Apr 29, 2016 at 10:01
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$\begingroup$ A related article : "Flat deformations of $\mathbb{P}^n$" link.springer.com/article/10.1007%2Fs00574-014-0054-x $\endgroup$– Pedro MonteroCommented May 12, 2016 at 15:14
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1 Answer
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In the case of the projective plane $\mathbb{P}^2$, this problem is studied in
Marco Manetti, MR 1116920 Normal degenerations of the complex projective plane, J. Reine Angew. Math. 419 (1991), 89--118.
answered Apr 29, 2016 at 11:48