Timeline for symplectic classes on rational surfaces.

9 events

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May 22, 2011 at 23:32	comment	added	Dmitri Panov		Paul, yes, indeed $C\cdot X$ must be positive, the point is that if you we blow up $\mathbb CP^2$ in $n\ge 9$ generic points the number of rational $-1$-curves is infinite, and the structure of this set infinite is quite involved. If you blow up $\mathbb CP^2$ in $8$ points the number of $-1$ curves is around 250 (I am lazy now to make the precise calculation)
May 22, 2011 at 20:05	comment	added	Paul		OK, I get it, its the converse that is tricky.
May 22, 2011 at 16:44	comment	added	Paul		@Dmitri: I didn't think the question was that involved: Isn't it true that if $C$ is a symplectic class, then $C^2>0$ (the symplectic class is a volume form) and $C\cdot X>0$ for any complex curve $X$ (since every complex curve is a symplectic submanifold). So $C$ has to satisfy $C^2=a^2-\sum b_i^2 >0$ and $aA-\sum b_iB_i>0$ when $X=AH+\sum_i B_iE_i$ is represented by any complex curve. Maybe I'm missing something obvious.
May 22, 2011 at 13:36	comment	added	Dmitri Panov		You are very welcome :) !
May 22, 2011 at 13:22	vote	accept	Yunhyung Cho
May 22, 2011 at 9:51	history	edited	Dmitri Panov	CC BY-SA 3.0	added 6 characters in body
May 22, 2011 at 9:35	history	edited	Dmitri Panov	CC BY-SA 3.0	improved exposition
May 22, 2011 at 0:22	history	edited	Dmitri Panov	CC BY-SA 3.0	added 202 characters in body
May 22, 2011 at 0:06	history	answered	Dmitri Panov	CC BY-SA 3.0