User fellow - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T07:02:46Z http://mathoverflow.net/feeds/user/4327 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/16788/blowing-up-1-curves-effective-and-ample-divisors blowing up, -1 curves, effective and ample divisors fellow 2010-03-01T21:45:38Z 2010-05-24T02:20:58Z <p>Lets say we're on a smooth surface, and we blow up at a point. Is there a simple explicit computation that shows to me the fact that the exceptional divisor E has self intersection -1 ? I don't consider the canonical divisor explicit (but am open to it). I do consider power series hacking to be explicit.</p> <p>I'm quite unnerved by this -1. Is E effective (seems to be, by definition?). Is E ample (seems to not be, by Nakai-Mozeishon type things)? More generally, I used to think of effective and ampleness as both being measures of "positivity"; but perhaps this is wrong - what do effectiveness and ampleness have to do with each other.</p> <p>What happens locally at a point of -1 intersection? I thought two irreducible curves on a surface should intersect either in 0 points, or in a positive number of points. To find E.E I would have tried to move E to some other divisor, and then I would get E.E = 0 or nonnegative.</p> <p>Sorry for the multiple questions, but I'm really distressed :(</p>