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Let $f:X \to T$ be a flat, projective morphism of noetherian schemes with $T$ an irreducible curve. Suppose that there exists a point $0 \in T$ such that the fiber $f^{-1}(0)$ is Fano.

Q. Is it true that in this case, for all $t$ near $0$, the fiber $f^{-1}(t)$ is Fano?

N.B. If necessary, one can assume that $f$ is smooth, but if such a statement holds without this assumption, it would be more interesting.

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    $\begingroup$ It is definitely asking too much for this to hold for all $t \in C$. Even if $f$ is smooth, this will typically only hold for $t$ near 0. $\endgroup$ Apr 7 '17 at 12:56
  • $\begingroup$ @potentiallydense Thanks. Could you give a reference for your second statement. $\endgroup$
    – Chen
    Apr 7 '17 at 13:02
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    $\begingroup$ To add to what @potentially dense was saying: Take $X_t$ to be a family of blow-ups of $P^2$ in three points for instance. For a general choice of three points, this is Del Pezzo, but it is not when the points are colinear. In general you get that there is a nbh where $X_t$ is Fano. This follows because ampleness of -K is an open property in families. $\endgroup$
    – byu
    Apr 7 '17 at 13:02
  • $\begingroup$ @gbp , potentiallydense, thank you. I have edited the question. $\endgroup$
    – Chen
    Apr 7 '17 at 13:05
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The answer is yes, in fact the following result holds.

Theorem. Let $f \colon X \to T$ be a flat deformation of a Fano variety $X_0:=f^{-1}(0)$ having at most terminal, $\mathbb{Q}$-factorial singularities.

Then $X_t:=f^{-1}(t)$ is a Fano variety with at most terminal, $\mathbb{Q}$-factorial singularities, for all $t$ in a neighborhood of $0$ in $T$.

For a proof, see Corollary 3.2 and Proposition 3.8 in

De Fernex, Tommaso; Hacon, Christopher D., Deformations of canonical pairs and Fano varieties, J. Reine Angew. Math. 651, 97-126 (2011). ZBL1220.14026.

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