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On the additivity of Newton-Okounkov bodies

Robert Wilms

TL;DR

The paper proves that Newton–Okounkov bodies Δ_{Y_⋅} become additive on 2‑dimensional subcones of Amp(X) when the flag Y_⋅ corresponds to a fixed L, using a slice-formula inductive argument. It constructs a linear map Δ from a 2D subspace of N^1(X)_ℚ into the Grothendieck group of convex bodies that remains compatible with intersection numbers, yielding a tight link between mixed volumes and intersection theory. The additivity is shown to be optimal in general, with a precise description of when it can occur, and the results yield a nef-intersection inequality L^d·(M·N^{d−1}) ≤ d(M·L^{d−1})(L·N^{d−1}). These contributions connect convex-geometric behavior of Newton–Okounkov bodies with algebraic intersection theory beyond polyhedral pseudo-effective cones.

Abstract

We study the additivity of Newton-Okounkov bodies. Our main result states that on two-dimensional subcones of the ample cone the Newto-Okounkov body associated to an appropriate flag acts additively. We prove this by induction relying on the slice formula for Newton-Okounkov bodies. Moreover, we discuss a necessary condition for the additivity showing that our result is optimal in general situations. As an application, we deduce an inequality between intersection numbers of nef line bundles.

On the additivity of Newton-Okounkov bodies

TL;DR

The paper proves that Newton–Okounkov bodies Δ_{Y_⋅} become additive on 2‑dimensional subcones of Amp(X) when the flag Y_⋅ corresponds to a fixed L, using a slice-formula inductive argument. It constructs a linear map Δ from a 2D subspace of N^1(X)_ℚ into the Grothendieck group of convex bodies that remains compatible with intersection numbers, yielding a tight link between mixed volumes and intersection theory. The additivity is shown to be optimal in general, with a precise description of when it can occur, and the results yield a nef-intersection inequality L^d·(M·N^{d−1}) ≤ d(M·L^{d−1})(L·N^{d−1}). These contributions connect convex-geometric behavior of Newton–Okounkov bodies with algebraic intersection theory beyond polyhedral pseudo-effective cones.

Abstract

We study the additivity of Newton-Okounkov bodies. Our main result states that on two-dimensional subcones of the ample cone the Newto-Okounkov body associated to an appropriate flag acts additively. We prove this by induction relying on the slice formula for Newton-Okounkov bodies. Moreover, we discuss a necessary condition for the additivity showing that our result is optimal in general situations. As an application, we deduce an inequality between intersection numbers of nef line bundles.
Paper Structure (6 sections, 5 theorems, 73 equations)

This paper contains 6 sections, 5 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Newton--Okounkov Bodies
  3. The Additivity of $\Delta_{Y_\bullet}$
  4. A Linear Map Compatible with Intersection Products
  5. The Limits of the Additivity of $\Delta_{Y_\bullet}$
  6. An Inequality of Intersection Numbers

Key Result

Theorem 1.2

Let $X$ be any irreducible projective variety, $L$ and $M$ any $\mathbb{Q}$-line bundles on $X$, and $Y_\bullet$ an admissible flag on $X$ corresponding to $L$. Then for all $N_1,N_2\in C_L(M)$ we have

Theorems & Definitions (8)

  • Definition 1.1
  • Theorem 1.2
  • Example 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Lemma 6.1
  • proof