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Infinitesimal successive minima, partial jets and convex geometry

Mihai Fulger, Victor Lozovanu

TL;DR

The paper develops two invariant families—infinitesimal successive minima ε_i(ξ;x) and asymptotic partial jet separation s_i(ξ;x)—to study local positivity of line bundles via blow-ups, base loci, and jets. It then translates these invariants into convex-geometry data through infinitesimal Newton–Okounkov bodies (iNObodies), introducing the generic straightened iNObody Δ^S_{x}(ξ) and showing that, at very general points, its widths equal the ε_i and that the body is bounded by a canonical box whose side-lengths are the ε_i. A key achievement is proving ε_i(ξ;x)=s_i(ξ;x) for normal X and connecting the geometry of Δ^S_{x}(ξ) to when it is simplicial, and hence to precise tangent-cone information for divisors in |mL|. The authors further establish Borel-fixed properties for slices of Δ and derive polytopal bounds for generic iNObodies, with sharp results in several classes of examples, including products of curves and Jacobians, and provide concrete applications to Seshadri constants for curves and bounds on restricted volumes. Overall, the work unifies base-locus, jet-separation, and convex-geometric perspectives to extract quantitative local positivity data and to bound classical invariants such as Seshadri constants.

Abstract

We introduce two sets of invariants for a line bundle at a point: infinitesimal successive minima and asymptotic partial jet separation. They are inspired by the local analogue of Ambro-Ito, and by the jet-theoretic interpretation of the Seshadri constant respectively. Under mild restrictions the two sets are equal. Moving to convex geometry, we prove that the lengths of the maximal simplex inside the generic infinitesimal Newton-Okounkov body (iNObody) of the line bundle at the point are precisely the successive minima. As application we characterize when this body is simplicial, and give examples when it is not. When the point is very general the convex body has a shape that we call Borel-fixed, a property inspired by generic initial ideals. Borel-fixed convex bodies satisfy simplicial lower bounds and polytopal upper bounds determined by their widths. For the generic iNObody of the line bundle at very general points these widths are again the infinitesimal successive minima.

Infinitesimal successive minima, partial jets and convex geometry

TL;DR

The paper develops two invariant families—infinitesimal successive minima ε_i(ξ;x) and asymptotic partial jet separation s_i(ξ;x)—to study local positivity of line bundles via blow-ups, base loci, and jets. It then translates these invariants into convex-geometry data through infinitesimal Newton–Okounkov bodies (iNObodies), introducing the generic straightened iNObody Δ^S_{x}(ξ) and showing that, at very general points, its widths equal the ε_i and that the body is bounded by a canonical box whose side-lengths are the ε_i. A key achievement is proving ε_i(ξ;x)=s_i(ξ;x) for normal X and connecting the geometry of Δ^S_{x}(ξ) to when it is simplicial, and hence to precise tangent-cone information for divisors in |mL|. The authors further establish Borel-fixed properties for slices of Δ and derive polytopal bounds for generic iNObodies, with sharp results in several classes of examples, including products of curves and Jacobians, and provide concrete applications to Seshadri constants for curves and bounds on restricted volumes. Overall, the work unifies base-locus, jet-separation, and convex-geometric perspectives to extract quantitative local positivity data and to bound classical invariants such as Seshadri constants.

Abstract

We introduce two sets of invariants for a line bundle at a point: infinitesimal successive minima and asymptotic partial jet separation. They are inspired by the local analogue of Ambro-Ito, and by the jet-theoretic interpretation of the Seshadri constant respectively. Under mild restrictions the two sets are equal. Moving to convex geometry, we prove that the lengths of the maximal simplex inside the generic infinitesimal Newton-Okounkov body (iNObody) of the line bundle at the point are precisely the successive minima. As application we characterize when this body is simplicial, and give examples when it is not. When the point is very general the convex body has a shape that we call Borel-fixed, a property inspired by generic initial ideals. Borel-fixed convex bodies satisfy simplicial lower bounds and polytopal upper bounds determined by their widths. For the generic iNObody of the line bundle at very general points these widths are again the infinitesimal successive minima.

Paper Structure

This paper contains 30 sections, 44 theorems, 133 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Infinitesimal successive minima
  3. Partial jet separation
  4. Infinitesimal Newton--Okounkov bodies
  5. The shape of generic iNObodies at arbitrary smooth points
  6. The shape of generic iNObodies for a very general point
  7. Examples and applications
  8. Comparison with other work in the literature
  9. Infinitesimal successive minima and partial jets
  10. Base loci
  11. Infinitesimal successive minima
  12. Asymptotic partial jet separation
  13. Infinitesimal successive minima versus partial jet separation
  14. Historical background and the successive minima of Ambro--Ito
  15. Borel-fixed properties for generic valuations on $\mathbb{P}^{n-1}$
  16. ...and 15 more sections

Key Result

Theorem 1.1

Let $X$ be a normal projective variety of dimension $n$. Let $\xi\in N^1(X)_{\mathbb R}$ and $x\in X\setminus{\bf B}_+(\xi)$ a smooth point. Then for all $i=1,\ldots ,n$.

Figures (3)

  • Figure 1: Generic vs. special flag choices for ${\Delta}_{Y_{\bullet}}(L)$ and respectively ${\Delta}^S_{Y_{\bullet}}(L)$
  • Figure 2: Vertical slice of ${\Delta}_{x}(\xi)_{\nu_1=t}$ in dimension $3$.
  • Figure 3: Failure of upper bounds at special points even for very general flags

Theorems & Definitions (117)

  • Theorem 1.1: Infinitesimal successive minima via partial jet separation
  • Theorem 1.2: Convex geometric interpretations of the infinitesimal successive minima
  • Theorem 1.3: Slice bounds
  • Theorem 1.4: Upper bounds at very general points
  • Theorem 1.5
  • Proposition 2.1
  • Proposition 2.2
  • Lemma 2.3
  • proof
  • Remark 2.4
  • ...and 107 more