Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Limit $F$-signature functions of two-variable binomial hypersurfaces

Anna Brosowsky, Izzet Coskun, Suchitra Pande, Kevin Tucker

TL;DR

The paper determines the limiting $F$-signature function for two-variable binomial hypersurfaces as the characteristic $p$ grows, showing that the limit is a piecewise polynomial and aligns with Li’s normalized volume for the associated KLT data. It introduces a detailed Gröbner-basis method for computing lengths in the special case $f=x+y$, solving recurrences to obtain explicit polynomials that generate a Gröbner basis for $I_{k,m,n}=(x^m,y^n,(x+y)^k)$ under suitable bounds on $p$. The results provide explicit formulas for the limit $F$-signature function $oldsymbol{\psi}(t)$ in terms of the data $(a,b,u,v,c)$ and connect these invariants to singularity theory notions like ADE types and normalized volumes. The techniques yield avenues for precise length calculations and illuminate the relationship between limiting $F$-signature and geometric stability invariants under reduction mod $p$, with broader implications for understanding strong $F$-regularity via asymptotic characteristic behavior.

Abstract

The $F$-signature is a fundamental numerical invariant of singularities in positive characteristic. Its positivity detects strong $F$-regularity, an important class of singularities related to KLT singularities in characteristic zero. In this paper, we compute the limiting $F$-signature function of binomial and other related hypersurfaces in two variables as the characteristic $p \to \infty$. In particular, we show it is a piecewise polynomial function, and relate it to the normalized volume.

Limit $F$-signature functions of two-variable binomial hypersurfaces

TL;DR

The paper determines the limiting -signature function for two-variable binomial hypersurfaces as the characteristic grows, showing that the limit is a piecewise polynomial and aligns with Li’s normalized volume for the associated KLT data. It introduces a detailed Gröbner-basis method for computing lengths in the special case , solving recurrences to obtain explicit polynomials that generate a Gröbner basis for under suitable bounds on . The results provide explicit formulas for the limit -signature function in terms of the data and connect these invariants to singularity theory notions like ADE types and normalized volumes. The techniques yield avenues for precise length calculations and illuminate the relationship between limiting -signature and geometric stability invariants under reduction mod , with broader implications for understanding strong -regularity via asymptotic characteristic behavior.

Abstract

The -signature is a fundamental numerical invariant of singularities in positive characteristic. Its positivity detects strong -regularity, an important class of singularities related to KLT singularities in characteristic zero. In this paper, we compute the limiting -signature function of binomial and other related hypersurfaces in two variables as the characteristic . In particular, we show it is a piecewise polynomial function, and relate it to the normalized volume.

Paper Structure

This paper contains 4 sections, 5 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lengths and Groebner bases when f=x+y
  4. Case m at most k

Key Result

Proposition 1

Let $f\in R=\mathbb Z[x_1,\ldots, x_n]$ be a non-zero polynomial contained in $\mathfrak m=(x_1,\ldots, x_n)$, so that by abuse of notation we can view $f\in \mathbb Z/p\mathbb Z[x_1,\ldots, x_n]_{\mathfrak{m}}$ or in $\mathbb Q[x_1,\ldots, x_n]_{\mathfrak{m}}$. Then for all $p\gg 0$, we have $\math

Theorems & Definitions (14)

  • Definition 1: $F$-signature function of hypersurface pairs
  • Definition 2: $F$-pure threshold
  • Proposition 1: MustataTakagiWatanabeFThresholdsAndBernsteinSato
  • Definition 3: Limit $F$-signature function
  • Lemma 1
  • proof
  • Definition 4
  • Proposition 2
  • proof
  • Lemma 2
  • ...and 4 more