Timeline for Finite orbits on an elliptic curve with two generic involutions
Current License: CC BY-SA 3.0
8 events
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| Oct 6, 2015 at 3:26 | comment | added | user47305 | @Joe Silverman Yep, that's right. In fact being an isomorphism in codim 1 is even stronger than being algebraically stable (e.g. take any algebraically stable map in dim 2 that isn't an isomorphism -- it contracts a curve). These "pseudoautomorphisms" are fairly common on CY varieties in higher dimension, and the "cone conjecture" they talk about says something about the group of these maps. | |
| Oct 6, 2015 at 1:15 | comment | added | Joe Silverman | @Mark Ah, the map is what the dynamics crowd calls algebraically stable. That's definitely of interest. | |
| Oct 5, 2015 at 23:14 | comment | added | user47305 | Whoops, sorry. The point is that not only is the indeterminacy locus codim 2 (duh), but that the map is an isomorphism in codimension 1 -- no divisors are contracted in either direction. This means in particular that there is still a functorial pullback on $N^1(X)$ (i.e. $(f^\ast)^n = (f^{\circ n})^\ast$), which you won't get for a general rational map. Thanks! | |
| Oct 5, 2015 at 17:08 | comment | added | Joe Silverman | @Mark Thanks for the reference. But is your point that the indeterminacy locus in that setting is exactly codimension 2. In general, the indeterminacy locus of any rational map between smooth varieties has codimension at least 2. | |
| Oct 5, 2015 at 15:01 | comment | added | user47305 | A reference for this construction in higher-dimensional settings is Cantat--Oguiso, " Birational automorphism groups and the movable cone theorem for Calabi-Yau manifolds of Wehler type via universal Coxeter groups". In general you only get rational maps, but they have codimension 2 indeterminacy -- such maps are sometimes called "pseudoautomorphisms". | |
| Oct 5, 2015 at 2:50 | history | edited | Joe Silverman | CC BY-SA 3.0 |
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| Oct 4, 2015 at 23:28 | comment | added | gsvr | Thank you! I'm mostly interested the question for K3s and higher dimensional Calabi-Yaus, so these references look useful! | |
| Oct 4, 2015 at 23:16 | history | answered | Joe Silverman | CC BY-SA 3.0 |