# Homaloidal nets and ideals of fat points II: subhomaloidal nets

## TL;DR

This paper investigates the algebraic and homological properties of certain ideals of plane fat points, focusing on their linear systems and birational maps, extending previous work with detailed analysis of initial degrees and transformations.

## Contribution

It introduces new insights into the ideal theoretic and homological properties of subhomaloidal ideals of fat points, utilizing classical quadratic transformations and birationality concepts.

## Key findings

- Analysis of initial linear systems of subhomaloidal ideals
- Connection between ideals and birational maps to higher-dimensional spaces
- Retrieval of known birational maps including Bordiga--White parameterizations

## Abstract

This paper is a natural sequel to [22] in that it tackles problems of the same nature. Here one aims at the ideal theoretic and homological properties of a class of ideals of general plane fat points whose second symbolic powers hold virtual multiplicities of proper homaloidal types. For this purpose one carries a detailed examination of their linear systems at the initial degree, a good deal of the results depending on the method of applying the classical arithmetic quadratic transformations of Hudson--Nagata. A subsidiary guide to understand these ideals through their initial linear systems has been supplied by questions of birationality with source $\mathbb{P}^2$ and target higher dimensional spaces. This leads, in particular, to the retrieval of birational maps studied by Geramita--Gimigliano--Pitteloud, including a few of the celebrated Bordiga--White parameterizations.