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Generalization of Gurjar's Hyperplane section theorem to arbitrary analytic varieties and A$\mathbb{m}$AC classes

A. J. Parameswaran, Mohit Upmanyu

Abstract

The aim of this paper is to generalize the hyperplane section theorem of Gurjar to arbitrary (local) analytic varieties even if the intersection with of hyperplanes is not necessarily isolated. In case of formal varieties, we generalize the statement to work for different classes of functions than just hyperplanes. We call these classes (which are subsets of formal power series ring) to be algebraic $\mathbb{m}$-adicaly closed (A$\mathbb{m}$AC).

Generalization of Gurjar's Hyperplane section theorem to arbitrary analytic varieties and A$\mathbb{m}$AC classes

Abstract

The aim of this paper is to generalize the hyperplane section theorem of Gurjar to arbitrary (local) analytic varieties even if the intersection with of hyperplanes is not necessarily isolated. In case of formal varieties, we generalize the statement to work for different classes of functions than just hyperplanes. We call these classes (which are subsets of formal power series ring) to be algebraic -adicaly closed (AAC).
Paper Structure (5 sections, 29 theorems, 58 equations)

This paper contains 5 sections, 29 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Algebraic $\mathfrak{m}$-adically closed class
  4. Another result for singular surfaces
  5. The analytic case

Key Result

Theorem 1.1

(GurjarHST)Let $A = \mathbb{C}[\![x_1,x_2,\ldots,x_n]\!]$ and $\mathfrak{m}$ denote its maximal ideal. Let $f \in \mathfrak{m}^2$ such that the subscheme $X \subset \mathop{\mathrm{Spec}}\nolimits(A)$ defined by $f$ has a curve singularity. Assume $h \in \mathfrak{m}-\mathfrak{m}^2$ such that $X \ca

Theorems & Definitions (69)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 1: \ref{['AmAC defn']}
  • Theorem 1: \ref{['AmAC result']}
  • Corollary 1: \ref{['Hyperplane section theorem']}
  • Remark 1.3
  • Theorem 2: \ref{['surface result']}
  • Theorem 3: \ref{['milnor tjurina']}
  • Definition 2.1
  • Lemma 2.2
  • ...and 59 more