Timeline for A criterion for isotriviality of families of varieties of general type
Current License: CC BY-SA 4.0
7 events
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| Jun 18, 2020 at 17:14 | comment | added | Hacon | I agree with Piotr; in fact the "corrected conjecture" seems to be a theorem in great generality (log case; any dimension fiber/base) see Theorem 1.3 arxiv.org/pdf/1503.02952.pdf. For surfaces, I would expect $\nu =5$ or $4$ should suffice (as $5K_X$ recovers the canonical model and $4K_X$ gives a birational map if I remember correctly). | |
| Jun 17, 2020 at 9:51 | comment | added | Piotr Achinger | A stupid counterexample would be to blow up a moving family of curves inside a product of two fake projective planes, in which case $f_* \omega_{X/Y} = 0$. I guess that a corrected conjecture would be that if $\deg f_* \omega_{X/Y}^\nu = 0$ for all $\nu>0$ then the family is birationally isotrivial. | |
| Jun 17, 2020 at 9:28 | comment | added | Piotr Achinger | Couldn't this sheaf be zero? | |
| Jun 17, 2020 at 7:42 | comment | added | Mobius | Thank you for your remind! The assumption of the generic fiber of $f$ being of general type is necessary for my question. | |
| Jun 17, 2020 at 7:21 | history | edited | Mobius | CC BY-SA 4.0 |
added 49 characters in body; edited title
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| Jun 16, 2020 at 16:19 | comment | added | Evgeny Shinder | The title of the question has general type in it, but not the question itself! Without general type assumption, e.g. for del Pezzo surfaces, the statement will be definitely false because the canonical class of the fibers will be negative, so the pushforward will be zero, but del Pezzo surfaces have moduli. | |
| Jun 16, 2020 at 13:42 | history | asked | Mobius | CC BY-SA 4.0 |