On Nakayama's theorem

TL;DR

This paper provides an accessible proof of Nakayama's theorem applicable to algebraic varieties over any algebraically closed field, emphasizing the negative definiteness of intersection matrices of exceptional curves.

Contribution

It offers a new proof of Nakayama's theorem that extends its applicability beyond complex analytic spaces to arbitrary characteristic fields.

Findings

Proof based on negative definiteness of intersection matrices

Applicable to algebraic varieties over any algebraically closed field

Extends Nakayama's theorem beyond complex analytic spaces

Abstract

The main purpose of this paper is to make Nakayama's theorem more accessible. We give a proof of Nakayama's theorem based on the negative definiteness of intersection matrices of exceptional curves. In this paper, we treat Nakayama's theorem on algebraic varieties over any algebraically closed field of arbitrary characteristic although Nakayama's original statement is formulated for complex analytic spaces.