Problems on the Pseudoeffective Cone

The study of divisors is a very powerful tool to achieve the understanding of the geometry of a projective variety. This book deals with some questions concerning the pseudoeffective divisors. In particular we give an account, with some original proofs, of recent results about cones associated to projective algebraic varieties. In the first part, we present a generalization of the so-called SHGH Conjectures to a surface different from the projective plane. We study the influence of this conjecture on the Mori cone of a blown-up surface at many general points and we show that this conjecture implies that a slice of this cone is, in a certain sense, circular. In the second part of this book, we give a positive answer to the existence of a Weak Zariski Decomposition for the elements of the pseudoeffective cone in the case of some projectivized vector bundles.