Timeline for Intersection multiplicity in abelian varieties

5 events

when toggle format	what		by	license	comment
Jul 23, 2013 at 3:18	comment	added	Will Sawin		$X \cap Y + T$ is a closed subset of $A \times A$, where the first $A$ is where the intersection happens and the second $A$ parameterizes $T$. So it has a proper projection morphism to the second $A$. Etale-locally, we can lift the decomposition into connected components of a special fiber of this morphism to the whole morphism, by lifting the idempotents. So following the connected components to sets of points in a well-defined way is no problem, although it's not immediately obvious to me that this preserves intersections numbers.
Jul 22, 2013 at 20:22	comment	added	David E Speyer		@OlegEroshkin The conclusion is still right in characteristic $p$. You can't always make $X \cap (Y+T)$ reduced, but you can always arrange that it is zero dimensional.
Jul 22, 2013 at 14:48	comment	added	Vivek Shende		It's certainly true you can move to make the intersections isolated, which is why I believed the statement in the first place. But in trying to actually make a proof out of this: how do you know that each connected component before you moved can really be followed to a bunch of points along the move?
Jul 22, 2013 at 14:35	comment	added	Oleg Eroshkin		Indeed, for field of characteristic zero this is proved in Kleiman, 1974
Jul 22, 2013 at 14:18	history	answered	Joe Silverman	CC BY-SA 3.0