Effective behavior of multiple linear systems

TL;DR

This paper establishes optimal effective bounds for classical theorems on complex algebraic surfaces, improving understanding of linear systems' behavior and providing effective versions of key theorems like Matsusaka's.

Contribution

It provides the first optimal effective bounds for several foundational theorems on complex algebraic surfaces, including an effective Matsusaka's big theorem.

Findings

Derived optimal bounds for Riemann-Roch and vanishing theorems.

Established effective criteria for base-point freeness and k-very ampleness.

Provided examples demonstrating the bounds are sharp.

Abstract

We give optimal effective bounds for some well-known theorems on complex algebraic surfaces, which are respectively due to Serre, Zariski (1962), Castelnuovo (1897), Artin (1962, 1966), Benveniste (1984), Cutkosky and Srinivas (1993). These theorems are about Riemann-Roch problem (on the behavior of the function dim |nD| of n), vanishing theorems, base-point freeness and k-very ampleness of the linear systems |nD| and |nA+L|, where D is effective, A is nef and big and L is arbitrary. As a consequence, we obtain an effective version of Matsusaka's big theorem, and we give also examples to show that our bound is the best possible one.