On the coupled stability thresholds of graded linear series

TL;DR

This paper explores properties of graded linear series, introduces coupled stability thresholds, and generalizes existing formulas and results in algebraic geometry related to stability and invariants.

Contribution

It defines coupled stability thresholds for graded linear series and extends several formulas and properties, generalizing prior work by Zhang and Zhuang.

Findings

Coupled stability threshold function extends continuously over the interior support.

Product-type formula for coupled stability thresholds is established.

Abban--Zhuang type formulas for local stability thresholds are derived.

Abstract

In this paper, we see several basic properties of graded linear series. We firstly see that, if a graded linear series contains an ample series, then so are the pullbacks of the system under birational morphisms. Using this proposition, we define the refinements of graded linear series with respects to primitive flags. Moreover, we give several formulas to compute the $S$-invariant of those refinements. Secondly, we introduce the notion of coupled stability thresholds for graded linear series, which is a generalization of the notion introduced by Rubinstein--Tian--Zhang. We see that, over the interior of the support for finite numbers of graded linear series containing an ample series, the coupled stability threshold function can be uniquely extended continuously, which generalizes the work by Kewei Zhang. Thirdly, we get a product-type formula for coupled stability thresholds, which generalizes the work of Zhuang. Fourthly, we see Abban--Zhuang's type formulas for estimating local coupled stability thresholds.