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On the valuative Nagata conjecture

Carlos Galindo, Francisco Monserrat, Carlos-Jesús Moreno-Ávila, Julio José Moyano-Fernández

TL;DR

This work extends the concept of minimality from plane valuations to divisorial valuations on smooth projective surfaces, by introducing and relating valuative Seshadri constants $\varepsilon(D,\nu_r)$ and Seshadri-type constants $\hat{\mu}_D(\nu_r)$ to the geometry of the Newton-Okounkov body. It proves a suite of equivalent conditions for a divisorial valuation to be minimal with respect to an ample divisor, including equality cases in area bounds of Newton-Okounkov bodies, nefness with vanishing self-intersection, and confinement of the Newton-Okounkov body to a canonical triangle. The results yield several equivalent statements of the valuative Nagata conjecture (in particular for $S=\mathbb{P}^2$ and a general line $L$) and offer quantitative controls via Zariski chambers and submaximal curves. The framework unifies several tools from intersection theory, valuation theory, and Newton-Okounkov theory to produce Nagata-type insights on a broad class of surfaces.

Abstract

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 valuative Seshadri constant and other global tools of the surface defined by the divisorial valuation. As a consequence, we derive several equivalent statements for the valuative Nagata conjecture and some related results.

On the valuative Nagata conjecture

TL;DR

This work extends the concept of minimality from plane valuations to divisorial valuations on smooth projective surfaces, by introducing and relating valuative Seshadri constants and Seshadri-type constants to the geometry of the Newton-Okounkov body. It proves a suite of equivalent conditions for a divisorial valuation to be minimal with respect to an ample divisor, including equality cases in area bounds of Newton-Okounkov bodies, nefness with vanishing self-intersection, and confinement of the Newton-Okounkov body to a canonical triangle. The results yield several equivalent statements of the valuative Nagata conjecture (in particular for and a general line ) and offer quantitative controls via Zariski chambers and submaximal curves. The framework unifies several tools from intersection theory, valuation theory, and Newton-Okounkov theory to produce Nagata-type insights on a broad class of surfaces.

Abstract

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 valuative Seshadri constant and other global tools of the surface defined by the divisorial valuation. As a consequence, we derive several equivalent statements for the valuative Nagata conjecture and some related results.
Paper Structure (10 sections, 23 theorems, 76 equations, 1 figure)

This paper contains 10 sections, 23 theorems, 76 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Convex cones associated to surfaces
  4. Plane valuations
  5. Some invariants of plane divisorial valuations
  6. Seshadri-type constants for divisorial valuations.
  7. Newton-Okounkov bodies of exceptional curve valuations
  8. The valuative Nagata conjecture for plane divisorial valuations
  9. Seshadri constant for a divisorial valuation of a smooth projective surface
  10. Minimal plane divisorial valuations with respect to an ample divisor

Key Result

Theorem 1

Let $D$ be an ample divisor on a smooth projective surface $S$ and $\nu_r$ a divisorial valuation of $S$. Set $\mathcal{C}_{\nu_r}=\{p_i\}_{i=1}^r$ the configuration of centers of $\nu_r,$$\tilde{S}$ the surface defined by the divisorial valuation $\nu_r$ and $\nu$ the valuation defined by the flag

Figures (1)

  • Figure 3.1: The triangle $\blacktriangle$ in Proposition \ref{['pro:triangle']}.

Theorems & Definitions (53)

  • Theorem 1
  • Theorem 2
  • Conjecture 3
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • Theorem 2.3
  • Definition 2.4
  • Remark 2.5
  • Lemma 2.6
  • ...and 43 more