Cox rings of blow-ups of multiprojective spaces

This thesis deals with birational geometry, which is a subfield of algebraic geometry. In particular, we study Mori dream spaces, which are varieties that are strictly related to the theory of Mori’s minimal model program. Mori dream spaces were introduced by Y. Hu and S. Keel in the beginning of the 21th century. Roughly speaking, a Mori dream space is a projective variety, whose cone of effective divisors admits a well-behaved decomposition into convex sets, called Mori chambers. These chambers are the nef cones of the birational models of X. Several geometric objects are of fundamental importance if we want to proceed in the study of the birational geometry of a normal projective variety. In particular its cones of curves and of divisors. Another notion of great importance to establish whether a variety is a Mori dream space or not is the property of being weak or log Fano. Weak Fano varieties are log Fano and log Fano varieties are Mori dream spaces. In 2021, T. Grange, E. Postinghel and A. Prendergast-Smith focussed on blow-ups of P 1 × P 2 and of P 1 × P 3 in sets of up to six points in very general position. Their main result is the explicit descriptions of the cones of effective divisors on these varieties and the description of the geometry of the generating classes. More explicitly, they proved that the blow-up of P 1×P 2 is weak Fano if and only if the number of blown up points is less or equal than six and that if the number of blown up points is less or equal than six, the variety P 1 × P 3 blown-up in those points is log Fano. Hence, these varieties are also Mori dream spaces. In chapter 2 of this thesis we give an overview on the theory of Cox rings, Mori dream spaces and log Fano varieties. In the first sections we give the definitions of the various cones inside N1(X) and inside N1 (X) and their inclusion relations. We then give a description of the Cox ring of a variety equipped with an algebraic torus action. We conclude the chapter with the result that permits to find generators for the moving cone of a variety from the generators of its Cox ring. It will follow an explanation on the main results concerning Mori dream spaces and log Fano varieties, and many examples. Finally, we introduce the main object of study of this thesis: the variety X1,n r , which is the blow-up of P 1 ×P n in r points in general position. In particular, we focus on X 1,n n+1, X 1,n n+2 and on X 1,n n+3 when n ≤ 4. In chapter 3 we compute the cone of effective curves of X1,n r for r = n + 1, n + 2 and r = n + 3 when n ≤ 4. We then prove that X1,n r is log Fano for r ≤ n + 1. In chapter 4, we compute generators and relations of the Cox ring of X 1,n n+1. We then use these generators to compute generators of the moving cone of X 1,n n+1. In order to do the computation, we wrote some scripts on Maple and Magma, some of which are provided in chapter 6. At the end if chapter 4 we compute the nef cones of X1,n r for r = n + 1, n + 2 and for r = n + 3 when n ≤ 4. Then, in chapter 5 we also give a Mori chamber decomposition of X 1,n n+1 in Magma and we display the case n = 2.