About multiplicities and applications to Bezout numbers

TL;DR

This paper investigates the relationship between multiplicities in local rings, characterizes when certain multiplicity differences vanish, and applies these findings to bounds on Bezout numbers in algebraic geometry.

Contribution

It introduces new conditions for when the multiplicity difference $ppa$ equals zero and explores specific cases, with applications to Bezout number bounds.

Findings

Conditions for $ppa=0$ are identified.

Explicit calculations of $ppa$ in particular cases are provided.

Applications to bounding Bezout numbers are demonstrated.

Abstract

Let $(A,\mathfrak{m},\Bbbk)$ denote a local Noetherian ring and $\mathfrak{q}$ an ideal such that $\ell_A(M/\mathfrak{q}M) < \infty$ for a finitely generated $A$-module $M$. Let $\au = a_1,\ldots,a_d$ denote a system of parameters of $M$ such that $a_i \in \mathfrak{q}^{c_i} \setminus \mathfrak{q}^{c_i+1}$ for $i=1,\ldots,d$. It follows that $ \chi := e_0(\au;M) - c \cdot e_0(\mathfrak{q};M) \geq 0$, where $c = c_1\cdot \ldots \cdot c_d$. The main results of the report are a discussion when $\chi = 0$ resp. to describe the value of $\chi$ in some particular cases. Applications concern results on the multiplicity $e_0(\au;M)$ and applications to Bezout numbers.