In a family of smooth projective curves over a reduced complex scheme of finite type the list of isomorphism classes of automorphism groups of the fibers is finite. This follows from the Hurwitz's bound and the constancy of the genus on each connected component of the base.

Can one prove a similar result for families of higher-dimensional varieties?

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    $\begingroup$ For every elliptic curve $E$, for every endomorphism $\phi:E\to E$, the following self-map of $E\times E$ is an automorphism, $\widetilde{\phi}(e,f) = (e,f+\phi(e))$. Now consider the countably many points in the $j$-line that parameterize elliptic curves with complex multiplication. $\endgroup$ – Jason Starr Dec 9 '19 at 16:23

In the case of a family of minimal smooth projective varieties of general type, we can repeat exactly the same argument as in the case of curves of genus $\geq 2$.

In fact, by a result of Hacon, McKernan and Xu [HMcKX13] we know that, if $X$ is any such a variety, we have $$|\mathrm{Aut}(X)| \leq c \cdot \mathrm{vol}(X, \, K_X), $$ where $c$ is a constant that only depends on $d:=\mathrm{dim}(X)$.

Now, it sufficies to observe that the canonical volume $$\mathrm{vol}(X, \, K_X):=\limsup_{m \to + \infty} \frac{d !\; h^0(X, \, mK_X)}{m^d}$$ is deformation invariant, by the celebrated invariance of plurigenera in smooth families proved by Siu [S98].


[HMcKX13] Hacon, Christopher D.; McKernan, James; Xu, Chenyang, On the birational automorphisms of varieties of general type, Ann. Math. (2) 177, No. 3, 1077-1111 (2013). ZBL1281.14036.

[S98] Siu, Yum-Tong, Invariance of plurigenera, Invent. Math. 134, No. 3, 661-673 (1998). ZBL0955.32017.


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