User philiph - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T01:59:18Z http://mathoverflow.net/feeds/user/17768 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/106476/blowing-down-1-curves Blowing down -1 curves philiph 2012-09-06T04:58:41Z 2012-09-09T07:53:27Z <p>After a good look for anything in the way of an explicit example, and harassing the algebraic geometers in the department, I still am unable to find an answer to my question, so any light will be very much appreciated.</p> <p>Suppose we have blown up $\mathbb{P}^2$ at some singular points (of a mapping), and we find that in this new space we have 3 curves which have self-intersection -1, which we would like to blow down. In my particular case I have two degree 1 curves (which pass through 2 base points in $\mathbb{P}^2$) and a degree 2 curve which passes through 5 base points in $\mathbb{P}^2$. Given that I have these curves explicitly, is it possible to find the blow down of each curve, and if so how? </p> <p>Also, what happens to the Picard group in this blown down space? If one of my curves is $H-E_1-E_2$ in my blown up space, where $H$ is the representative of a generic degree 1 curve in $\mathbb{P}^2$ and $E_1$ and $E_2$ are the total transforms of two base points, then what is the Picard group in this space where we have blown down the line $H-E_1-E_2$?</p> <p>My experience has been that all these things are theoretically calculable but nobody seems to apply the theory in specific cases. </p> <p>Many thanks!</p> http://mathoverflow.net/questions/77672/expressing-a-element-of-a-matrix-subgroup-in-terms-of-subgroup-generators Expressing a element of a Matrix subgroup in terms of subgroup generators philiph 2011-10-10T07:53:55Z 2011-10-10T08:33:31Z <p>I'm no (computational) algebraist, and my searches have been pretty unyielding (probably due to the vast amounts written on the key words), but perhaps someone may know if this is possible, and if so, lead me to some solution.</p> <p>Consider the subgroup $S$ of $GL(n,\mathbb{Z})$ which is generated by elements $s_1,...,s_k$. If $x\in S$, then $x$ has a representation as a word in the form $x=\Pi_{i}s_i$. Is it possible to find such a representation? </p> <p>If this is possible, and computable, is there any efficient software out there? My current problem is a 10x10 matrix and I'm trying to fit this in to a subgroup generated by 9 10x10 matrices.</p> http://mathoverflow.net/questions/75810/self-intersection-of-blown-up-points-and-the-lines-which-they-lie-on Self intersection of blown up points and the lines which they lie on philiph 2011-09-19T02:03:56Z 2011-09-19T03:13:49Z <p>I'm currently trying to understand the process of blowing-up, and a few things strike me as a little difficult to get an intuitive understanding of what's happening.</p> <p>The current problem is on self intersection.</p> <p>Why does the exceptional divisor I get by blowing up at a point $\mathbb{P}^1(\mathbb{C})\times\mathbb{P}^1(\mathbb{C})$ have self intersection -1? A mathematically abstract answer would be nice, but an included intuitive explanation of negative self-intersection would really take the cake.</p> http://mathoverflow.net/questions/75284/anticanonical-divisor-of-a-blow-up (Anti)Canonical divisor of a blow up philiph 2011-09-13T07:47:46Z 2011-09-13T08:33:11Z <p>This question may be utterly trivial, or not, but as someone with hardly any knowledge of algebraic geometry I thought there could be a chance I get lucky.</p> <p>Let X be a rational surface obtained by n blows up of $\mathbb{P}^1(\mathbb{C})\times\mathbb{P}^1(\mathbb{C})$. I write $H_x$ and $H_y$ for the lines x=constant and y=constant, and $E_i$ for the total transform of the point of the $i-th$ blow up. Is there some way to find what the canonical divisor on X is? Take for example the surface X blown up at the point $x=\infty$ , $y=\infty$, and then again on that divisor at some point. </p> <p>(This example above occured when I cooked up an example to try and understand the resolution of singularities for the simple mapping x(n+2)+ax(n+1)+bx(n)=0).</p> <p>I have seen for example, that the canonical divisor of $\mathbb{P}^2(\mathbb{C})$ blown up at 9 points is given by $K_X=-3H+E_1+E_2+...+E_9$. Is this true in general or only for a special class of surfaces?</p> http://mathoverflow.net/questions/106476/blowing-down-1-curves/106486#106486 Comment by philiph philiph 2012-09-09T00:19:25Z 2012-09-09T00:19:25Z Yes this seems to work. As for the Picard group, I know when we blow up we just add another element, the line which is the blowup of a point. But in the case of blow down, we must remove an element, and in this case, what are we removing? Some combination of the lines blown down? http://mathoverflow.net/questions/106476/blowing-down-1-curves Comment by philiph philiph 2012-09-06T07:04:02Z 2012-09-06T07:04:02Z Yes, $\mathbb{P}^2$ blown up in 5 points, the other curves are $H-E_1-E_3$ and $2H-E_1-E_2-E_3-E_4-E_5$. http://mathoverflow.net/questions/75810/self-intersection-of-blown-up-points-and-the-lines-which-they-lie-on Comment by philiph philiph 2011-09-19T06:25:52Z 2011-09-19T06:25:52Z Ah cheers Emerton, I did a search but obviously my keywords weren't exhaustive enough. http://mathoverflow.net/questions/75284/anticanonical-divisor-of-a-blow-up/75289#75289 Comment by philiph philiph 2011-09-13T08:42:48Z 2011-09-13T08:42:48Z Yes I've seen that before, but was unsure. Thanks for the clarification.