Geometric realizations of birational maps

In this thesis we study the relation between algebraic torus actions on complex projective varieties and the birational geometry of their geometric quotients. Given a C*-action on a normal projective variety X, there exist two unique connected components of the fixed point locus, called the sink Y− and the source Y+, containing the limit at ∞ and 0 of the general orbit. Let GX− (resp. GX+) be the variety parametrizing the orbits converging to the sink (resp. the source). Since there exists an open subset of points converging to Y±, we obtain a birational map ψ: GX->GX+. By choosing different linearizations of ample line bundles on X, we obtain a factorization of the birational map ψ among inner geometric quotient, parametrizing different open subsets of stable points. In this setting, we investigate the local analytic geometry of the birational map ψ. On one hand we link certain birational transformations, called rooftop flips, with varieties with two projective bundles structures. On the other we study when the birational map ψ can be locally described by a toric flip of Atiyah type. If on one side a C*-action naturally induces a birational map among geometric quotients, it is meaningful to study the opposite direction: more precisely, given a birational map φ: Z+->Z− among normal projective varieties, how can we construct a normal projective variety X, endowed with a C*-action, such that Z− is the sink, Z+ is the source, and the natural birational map ψ constructed above coincide with φ? Such an X is called a geometric realization of the birational map φ. We propose a construction of a geometric realization of φ, whose geometry reflects the factorization of the map as a composition of flips, blow-ups and blow-downs. We describe in particular the case in which φ is a small modification of dream type, namely a birational map which is an isomorphism in codimension 1 associated to a finitely generated multisection ring. Moreover, we show that the cone of divisors associated to such multisection rings admits a chamber decomposition where the models are the geometric quotients of the C*-action. If in addition Z± are assumed to be toric varieties, we construct a function in SageMath to compute the polytope of the associated toric geometric realization.