Blow-ups at points in non-general position - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T12:00:29Z http://mathoverflow.net/feeds/question/58926 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/58926/blow-ups-at-points-in-non-general-position Blow-ups at points in non-general position math_donk 2011-03-19T16:20:47Z 2011-03-20T14:01:01Z <p>Is it well known what happens if one blows-up $\mathbb{P}^2$ at points in non-general position (ie. 3 points on a line, 6 on a conic etc)? Are these objects isomorphic to something nice? </p> http://mathoverflow.net/questions/58926/blow-ups-at-points-in-non-general-position/58928#58928 Answer by Francesco Polizzi for Blow-ups at points in non-general position Francesco Polizzi 2011-03-19T16:48:31Z 2011-03-19T17:09:22Z <p>In both examples you are considering, the anticanonical model is a <em>singular</em> del Pezzo surface.</p> <p>In fact, let $X$ be the blow-up of $\mathbb{P}^2$ at three points lying on a line $L$. By Bezout's theorem, the birational map associated with the linear system of cubics through the three points contracts $L$. Since $L^2=1$, the blow-up of $L$ at three points gives a $(-2)$-curve. Therefore, the anticanonical model of $X$ is a Del Pezzo of degree $6$ in $\mathbb{P}^6$ with an ordinary double point (i.e., a node).</p> <p>Analogously, let $Y$ be the blow-up of $\mathbb{P}^2$ at six points lying on a conic $C$. By Bezout's theorem, the birational map associated with the linear system of cubics through the six points contracts $C$. Since $C^2=4$ and we are blowing-up six points over $C$, we obtain again a $(-2)$-curve. Therefore, the anticanonical model of $Y$ is a Del Pezzo surface of degree $3$ in $\mathbb{P}^3$ with an ordinary double point, i.e. a cubic surface with a node.</p> <p>Of course, if you blow-up more than $8$ points then the result is not a Del Pezzo anymore. For instance, the blow-up of $\mathbb{P}^2$ at nine points which are the base locus of a pencil of cubics is an elliptic fibration $X \to \mathbb{P}^1$ with nine sections; in general, such fibration has exactly $12$ nodal fibres, corresponding to the singular elements of the pencil.</p> <p>When the number of points increases the situation becomes more and more complicated, and I guess that a satisfatory description is out of reach.</p> http://mathoverflow.net/questions/58926/blow-ups-at-points-in-non-general-position/58973#58973 Answer by known google for Blow-ups at points in non-general position known google 2011-03-20T14:01:01Z 2011-03-20T14:01:01Z <p>I'll just add to Francesco's answer by saying that general position of the points on the plane is equivalent to ampleness of the anticanonical sheaf $\omega_X^{\otimes -1}$.</p> <p>The key observation is that on a del Pezzo surface, an irreducible negative curve ($C^2 &lt; 0$) must be an exceptional curve (i.e. $C^2 = C\cdot K_X = -1$). This follow from the adjunction formula and the Nakai-Moishezon criterion. </p> <p>If you blow up 3 colinear points, then the strict transform of the line containing these points will have self-intersection $-2$, which is not allowed by the key observation. Similiarly for the strict transform of a conic through 6 blown-up points. There is one more condition you have to impose: if you blow up 8 points, they cannot lie on a singular cubic with one of the points at the singularity.</p> <p>If you relax the requirement that $\omega_X^{\otimes -1}$ be ample to just big and nef, then you can have some degenerate point configurations: this time $C^2 = -2$ is allowed, so 3 colinear points ar OK. However, 4 colinear points would not be OK.</p>