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The F-signature Function on the Ample Cone

Seungsu Lee, Suchitra Pande

TL;DR

The paper introduces the F-signature function on the ample cone of a globally F-regular projective variety, defining $s_X([L])$ via the F-signature of the section ring $S(X,L)$. It proves that this function is homogeneous, extends continuously from the rational to the real ample cone, and is locally Lipschitz with respect to the Néron-Severi norm. It further demonstrates that $s_X$ extends to the nef cone with a vanishing value on nef divisors that are not big, and provides a sharp upper bound in terms of the volume function. The work connects the F-signature to the volume and develops effective techniques based on Frobenius splittings to control variations of $s_X$ across the ample cone, offering new tools for studying globally F-regular varieties and their birational geometry in positive characteristic.

Abstract

For any fixed globally F-regular projective variety X over an algebraically closed field of positive characteristic, we study the F-signature of section rings of X with respect to the ample Cartier divisors on X. In particular, we define an F-signature function on the ample cone of X and show that it is locally Lipschitz continuous. We further prove that the F-signature function extends to the boundary of the ample cone. We also establish an effective comparison between the F-signature function and the volume function on the ample cone. As a consequence, we show that for divisors that are nef but not big, the extension of the F-signature is zero.

The F-signature Function on the Ample Cone

TL;DR

The paper introduces the F-signature function on the ample cone of a globally F-regular projective variety, defining via the F-signature of the section ring . It proves that this function is homogeneous, extends continuously from the rational to the real ample cone, and is locally Lipschitz with respect to the Néron-Severi norm. It further demonstrates that extends to the nef cone with a vanishing value on nef divisors that are not big, and provides a sharp upper bound in terms of the volume function. The work connects the F-signature to the volume and develops effective techniques based on Frobenius splittings to control variations of across the ample cone, offering new tools for studying globally F-regular varieties and their birational geometry in positive characteristic.

Abstract

For any fixed globally F-regular projective variety X over an algebraically closed field of positive characteristic, we study the F-signature of section rings of X with respect to the ample Cartier divisors on X. In particular, we define an F-signature function on the ample cone of X and show that it is locally Lipschitz continuous. We further prove that the F-signature function extends to the boundary of the ample cone. We also establish an effective comparison between the F-signature function and the volume function on the ample cone. As a consequence, we show that for divisors that are nef but not big, the extension of the F-signature is zero.
Paper Structure (13 sections, 29 theorems, 97 equations, 1 figure)

This paper contains 13 sections, 29 theorems, 97 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $F$-signature
  4. Section Rings:
  5. $F$-regularity:
  6. Definition of the $F$-signature Funtion
  7. Continuity of the $F$-signature function
  8. Informal sketch of the proof of \ref{['mainthm']}:
  9. Proof of \ref{['mainthm']}
  10. Completion of the proof of \ref{['ampledifference']}:
  11. Completion of the proof of \ref{['mainthm']}:
  12. Extending the $F$-signature function to the Nef Cone.
  13. Local upper bounds for the $F$-signature function

Key Result

Theorem 1.1

Fix any globally $F$-regular projective variety $X$ over an algebraically closed field $k$ of positive characteristic $p$. Assume that the dimension of $X$ is positive. Then, the $F$-signature function $L \mapsto s_{X}(L)$, assigning to any ample Cartier divisor $L$, the $F$-signature of the section

Figures (1)

  • Figure 1: The $F$-signature function of the blow up of $\mathbb {P}^2$ at a point.

Theorems & Definitions (68)

  • Theorem 1.1
  • Theorem 1.2: \ref{['extensionthm']}
  • Definition 1: Free rank
  • Definition 2: $F$-signature
  • Definition 3: Graded free rank
  • Lemma 1
  • proof
  • Definition 4: $F$-signature of $\mathbb{N}$-graded rings
  • Definition 5: Section Rings and Modules
  • Lemma 2
  • ...and 58 more