Non-positive and negative at infinity divisorial valuations of Hirzebruch surfaces

Abstract

We consider rational surfaces Z defined by divisorial valuations ν of Hirzebruch surfaces. We introduce concepts of non-positivity and negativity at infinity for these valuations and prove that these concepts admit nice local and global equivalent conditions. In particular we prove that, when ν is non-positive at infinity, the extremal rays of the cone of curves of Z can be explicitly given.

Discrete Equivalence of Non-positive at Infinity Plane Valuations

“…The next theorem states the above mentioned geometrical properties. Proofs can be found in [12,14]. Theorem 3.3.…”

“…Now assume that t is the discrete class of a special divisorial valuation of F δ . With notation as in Theorem 3.3, by [14,Theorem 3.6], the inequality…”

“…This proves that 2β 1 β 0 + δβ 2 1 ≥ β g+1 because F 1 and M 0 go both through p but at most one of their transforms passes through p 2 . Conversely, let t ∈ T be (β g+1 ∈ Z >0 ) such that the inequality 2β 1 β 0 + δβ 2 1 ≥ β g+1 is satisfied, then considering the special divisorial valuation ν of F δ with discrete class t whose first free points in C ν coincide with those through which F 1 goes, by [14,Theorem 3.6] one gets an NPI special divisorial valuation on F δ with discrete class t.…”

“…The concept of non-positive at infinity (NPI) valuation was introduced in [12] (see also [20]) and afterwards extended in [14]. Valuations of quotient fields of local regular rings centered at them and their graded algebras are important tools for studying local uniformization of singularities [1,[24][25][26][27].…”

“…Finally, NPI divisorial valuations characterize those rational surfaces Z (given by simple sequences of point blowing-ups) with polyhedral cone of curves and minimum number of extremal rays. Even, in the projective case, they allow us to decide when the Cox ring of Z is finitely generated [12,14].…”