Timeline for Reference needed for negative curves on blowup of the projective plane at generic points
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| Sep 30, 2012 at 19:19 | comment | added | user5117 | Hi Jérémy, yes, I glossed over some subtleties in the answer. One should choose the points to be in Cremona general position, meaning that they are in linear general position, and remain so after any sequence of Cremona transformations. Also, as you say, the Cremona transformations are not automorphisms of a fixed blowup. | |
| Sep 30, 2012 at 18:06 | comment | added | Jérémy Blanc | This is a nice description, but one has to take care: in general there is no automorphism on the blow-up of $r\ge 9$ pts. The application of the quadratic maps is good, but it could move the points in a bad way. In fact, the application of an element of the Weyl group to a $(-1)$-curve gives an element $C$ of the Picard group, which satisfies $C^2=-1$ and $CK=-1$. In most case, it should yield another $(-1)$-curve, but it could also happen that the divisor is not irreducible. Anyway, for most elements it works, by the result of Nagata. | |
| Sep 30, 2012 at 16:52 | history | answered | user5117 | CC BY-SA 3.0 |