We consider rational surfaces Z defined by divisorial valuations ν ν of Hirzebruch surfaces.
We introduce concepts of non-positivity and negativity at infinity for these valuations and …

We provide several equivalent conditions for a plane divisorial valuation of a smooth
projective surface to be minimal with respect to an ample divisor. These conditions involve a …

We prove that the Newton–Okounkov body associated to the flag E•:={X= Xr⊃ Er⊃{q}},
defined by the surface X and the exceptional divisor Er given by any divisorial valuation of …

We study the class of planar polynomial vector fields admitting Darboux first integrals of the
type∏ i= 1 rfi α i, where the α i's are positive real numbers and the fi's are polynomials …

Let $ X $ be a rational surface obtained by blowing up at a configuration $\mathcal {C} $ of
infinitely near points over a Hirzebruch surface $\mathbb {F} _\delta $. We prove that there …

We introduce the concepts of non-positive and negative at infinity plane valuation of a
Hirzebruch surface and determine nice global and local geometric properties of the surfaces …

We introduce surfaces at infinity, a class of rational surfaces linked to curves with only one
place at infinity. The cone of curves of these surfaces is finite polyhedral and minimally …

We consider flags E•={X⊃ E⊃{q}}, where E is an exceptional divisor defining a non-positive
at infinity divisorial valuation νE of a Hirzebruch surface, qa point in E and X the surface …

We provide a lower bound on the degree of curves of the projective plane passing through
the centers of a divisorial valuation of with prescribed multiplicities, and an upper bound for …

Nagata solved Hilbert's 14-th problem in 1958 in the negative. The solution naturally lead
him to a tantalizing conjecture that remains widely open after more than half a century of …