blow up of segre primal and $\mathcal{M}_{0,6}$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T13:13:30Z http://mathoverflow.net/feeds/question/120025 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/120025/blow-up-of-segre-primal-and-mathcalm-0-6 blow up of segre primal and $\mathcal{M}_{0,6}$ IMeasy 2013-01-27T13:52:12Z 2013-01-28T04:35:26Z <p>The segre cubic primal $X\subset P^4$ is the GIT quotient of 6 points on $P^1$. Let $M_{0,6}$ the DM compactification of the moduli of 6-pointed rational curves. The Segre primal $X$ is a cubic 3-fold with ten double points and there exists a natural map $M_{0,6}\to X$ that contracts 10 boundary divisors (each iso to $P^1 \times P^1$) to the singular points. Now if I blow up the singular points of $X$, since they are ordinary double points, I get a $P^1 \times P^1$ exceptional divisor over each of them. Call $\tilde{X}$ the blown up variety. Maybe it is a silly question, but it is clear that there exists an iso $\tilde{X}\cong M_{0,6}$? Why?</p> http://mathoverflow.net/questions/120025/blow-up-of-segre-primal-and-mathcalm-0-6/120059#120059 Answer by Steven Sam for blow up of segre primal and $\mathcal{M}_{0,6}$ Steven Sam 2013-01-27T22:09:37Z 2013-01-27T22:09:37Z <p>From what you've written you'll get a map $M_{0,6} \to \tilde{X}$ via the universal property of blowups (see Proposition II.7.14 of Hartshorne's Algebraic Geometry). In fact this map is an isomorphism.</p> <p>See the paragraph after Theorem 4.2 of Harvey-Lloyd Philipps, Symmetry and moduli spaces for Riemann surfaces for a short argument: <a href="http://books.google.com/books?id=pvKdD74g6UEC&amp;pg=PA164&amp;lpg=PA164" rel="nofollow">http://books.google.com/books?id=pvKdD74g6UEC&amp;pg=PA164&amp;lpg=PA164</a></p> http://mathoverflow.net/questions/120025/blow-up-of-segre-primal-and-mathcalm-0-6/120078#120078 Answer by Moon for blow up of segre primal and $\mathcal{M}_{0,6}$ Moon 2013-01-28T04:35:26Z 2013-01-28T04:35:26Z <p>As Steven said, there is a morphism $\overline M_{0,6} \to \tilde{X}$ because the inverse image sheaf of the ideal of double points generates a Cartier divisor. Now $\tilde{X}$ is nonsingular, so to check the fact $\overline{M}_{0,6} \to \tilde{X}$ is an isomorphism it suffices to show injectivity of the map, or show the equality of Picard numbers. </p> <p>In general, the birational morphism $\overline M_{0,n} \to (\mathbb{P}^1)^n//SL_2$ is a composition of smooth blow-ups and Kirwan's desingularization. Consult <a href="http://arxiv.org/abs/1002.2461" rel="nofollow">http://arxiv.org/abs/1002.2461</a>.</p>