Nagata type statements
Nagata solved Hilbert’s 14-th problem in 1958 in the negative. The solution naturally led him to a tantalizing conjecture that remains widely open after more than half a century of intense efforts. Using Nagata’s theorem as starting point, and the conjecture, with its multiple variations, as motivation, we explore the important questions of finite generation for invariant rings, for support semigroups of multigraded algebras, and for Mori cones of divisors on blown up surfaces, and the rationality of Waldschmidt constants. Finally we suggest a connection between the Mori cone of the Zariski–Riemann space and the continuity of the Waldschmidt constant as a function on the space of valuations.
These notes correspond to the course of the same title given by the first author in the workshop Asymptotic invariants attached to linear series held in the Pedagogical University of Cracow from May 16 to 20, 2016.