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. 



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



Deformation invariance of Fano varieties
The answer is *yes*, in fact the following result holds.



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> **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.
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> 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 $T$.
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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**](http://dx.doi.org/10.1515/CRELLE.2011.010), J. Reine Angew. Math. **651**, 97-126 (2011). [ZBL1220.14026](https://zbmath.org/?q=an:1220.14026).


