# Space of initial conditions for a cubic Hamiltonian system

## TL;DR

This paper constructs the space of initial conditions for a specific cubic Hamiltonian system by compactifying its phase space and performing blow-ups to regularize singular points, following Okamoto's method.

## Contribution

It explicitly derives the Okamoto space of initial conditions for a particular cubic Hamiltonian system through geometric blow-up procedures.

## Key findings

- Successfully compactified the phase space to  and identified base points.
- Performed blow-ups to regularize singularities in the phase space.
- Established a space where the system admits regular initial value problems.

## Abstract

In this paper we perform the analysis that leads to the space of initial conditions for the Hamiltonian system $q' = p^2 + zq + \alpha$, $p' = -q^2 - zp - \beta$, studied by the author in an earlier article. By compactifying the phase space of the system from $\mathbb{C}^2$ to $\mathbb{CP}^2$ three base points arise in the standard coordinate charts covering the complex projective space. Each of these is removed by a sequence of three blow-ups, a construction to regularise the system at these points. The resulting space, where the exceptional curves introduced after the first and second blow-up are removed, is the so-called Okamoto's space of initial conditions for this system which, at every point, defines a regular initial value problem in some coordinate chart of the space.