# Deformation invariance of Fano varieties

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.

• 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. Apr 7 '17 at 12:56
• @potentiallydense Thanks. Could you give a reference for your second statement.
– Chen
Apr 7 '17 at 13:02
• 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.
– byu
Apr 7 '17 at 13:02
• @gbp , potentiallydense, thank you. I have edited the question.
– Chen
Apr 7 '17 at 13:05

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 $$T$$.