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?



blow up of segre primal and $\mathcal{M}_{0,6}$
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.


See the paragraph after Theorem 4.2 of 
Harvey-Lloyd Philipps, Symmetry and moduli spaces for Riemann surfaces for a short argument:
<http://books.google.com/books?id=pvKdD74g6UEC&pg=PA164&lpg=PA164>


