Kapranov's realization of the moduli spaces of n-pointed stable curves of genus zero and generalizations

This work presents an introduction to the geometry of the moduli space M_{0,n} of smooth algebraic curves of genus zero and presents two ways of compactifying it, yielding the space \overline{M}_{0,n}. In his first construction Kapranov, starting from a result by Castelnuovo, compactifies M_{0,n} by means of Veronese curves and their limit positions, yielding boundary divisors. On the other hand, Kapranov compactifies M_{0,n} by iterated blow-up of the projective space P^{n-3} at n-1 points in general position and at all linear spaces spanned by them, yielding exceptional divisors and the hyperplane class. Focusing on the correspondence between boundary divisors and Veronese curves and on the role of the hyperplane class, we show that there is a dictionary between boundary divisors, exceptional divisors and the hyperplane class. In particular, we present a realization of \overline{M}_{0,5} as a del Pezzo surface of degree 5 and we compare the space \overline{M}_{0,6} with the Segre cubic. We conclude by presenting some generalizations of Kapranov’s construction by Hassett and Losev Manin in order to appreciate the modular structure of the rounds of blow-up.