The cone of curves and the Cox ring of rational surfaces given by divisorial valuations

Abstract

Abstract. We consider surfaces X defined by plane divisorial valuations ν of the quotient field of the local ring R at a closed point p of the projective plane P 2 over an arbitrary algebraically closed field k and centered at R. We prove that the regularity of the cone of curves of X is equivalent to the fact that ν is non positive on O P 2 (P 2 \ L), where L is a certain line containing p. Under these conditions, we characterize when the characteristic cone of X is closed and its Cox ring finitely generated. Equivalent conditions to the fact that ν is negative on O P 2 (P 2 \ L) \ k are also given.

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

“…A remarkable property of non-positive at infinity valuations is that they determine (in fact, are equivalent to) the surfaces Z as above such that its cone of curves N E(Z) is finite polyhedral and generated either by the classes of the strict transforms of the fiber F 1 , the special section M 0 and the exceptional divisors (special valuations) or by the mentioned generators plus the class of the section M 1 (non-special valuations). Since the Hirzebruch surface F 1 can be obtained by blowing-up a point in P 2 , our results recover those in [17] concerning the characterization of non-positive and negative at infinity divisorial valuations of P 2 , and provide a very simple characterization of the rational surfaces (obtained from a classical minimal model by a finite simple sequence of point blowing-ups) whose cone of curves has the above mentioned generators.…”

“…As a consequence, Theorem 3.6 allows us to provide equivalent conditions to the non-positivity of a valuation of P 2 (see Section 2.3). In fact, our Theorem 3.6 recovers Theorem 1 in [17] considering δ = 1 and p 1 a special point, and the above proof is an adaptation and extension to our more general situation of that of [17, Theorem 1].…”

“…Recently, in [17], it was considered a class N of divisorial valuations ν (of K(O P 2 ,p ) centered at O P 2 ,p ) which have a similar behaviour as that of curves with only one place at infinity [3]. They were named non-positive at infinity because satisfy ν(f ) ≤ 0 for every f ∈ k[x, y] \ {0}, {x, y} being affine coordinates in the chart of points which are not in the line at infinity (which contains p).…”

“…For surfaces defined by non-positive at infinity valuations of P 2 , we are able to decide whether their Cox rings are finitely generated [17], and, as mentioned, for these same valuations, we know how to compute their Seshadri-like constants and to explicitly obtain their corresponding Newton-Okounkov bodies. In a forthcoming paper, we will prove that similar properties can be deduced when considering non-positive at infinity valuations of F δ .…”

“…The concepts of non-positivity (and negativity) at infinity of valuations of P 2 and F δ are different. For instance, by[17, Theorem 1] there is no non-positive at infinity divisorial valuation of P 2 with maximal contact values 3, 11 and 122; however Theorem 3.6 proves the existence of non-positive at infinity valuations of F δ with those maximal contact values, when δ ≥ 2.…”