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$\begingroup$ Thank you! It seems that the same should hold for smooth isotrivial families of varieties of non-negative Kodaira dimension? $\endgroup$

Vesselin Dimitrov
– Vesselin Dimitrov

2014-07-24 20:53:55 +00:00
Commented Jul 24, 2014 at 20:53
$\begingroup$ It seems likely, but I am not completely sure: you might get into trouble with automorphisms acting trivially in cohomology, for instance. $\endgroup$

abx
– abx

2014-07-25 06:54:26 +00:00
Commented Jul 25, 2014 at 6:54
2

$\begingroup$ At least in characteristic 0, I think it should be fine for all non-uniruled varieties. Of course everything is fine when the automorphism group is finite (or can be "naturally" reduced to a finite group). That leaves varieties whose automorphism group contains a positive-dimensional (connected) linear group. But all such groups are (geometrically) rational. Thus the orbits are unirational. So the variety is uniruled. $\endgroup$

Jason Starr
– Jason Starr

2014-07-25 12:44:43 +00:00
Commented Jul 25, 2014 at 12:44

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