Section 4: Dense Cubics
In Section 4 we look at cubic surfaces defined by a dense polynomial
The magma code to transform such a surface into octanomial form and to study the tree arrangement of its line can be found in the github repository https://github.com/emresertoz/pAdicCubicSurface.
Computations are explicitly illustrated in Example 4.4 where we start from the following tropical coefficients vector
We fix \(p=5\) and choose the canonical lifts
The six nearby points over \(\mathbb{Q}\) of the blow-up points are
We choose the point \(q_7=( -4: 10: -1) \in \mathbb{Q}\) and proceed by finding a cuspidal cubic over \(\mathbb{Q}_p\) passing throught the \(7\) points. Finally we compute the linear forms \(\ell_0, \ell_1, \ell_2\) determining the automorphism of \(\mathbb{P}^2\) which brings the cuspidal cubic in the standard form \(\{X^2Z=Y^3\}\) and allows us to read the moduli paramaters \(d_1, d_2, \ldots, d_6\).
All the details and outputs of this computation can be found in the following magma file
.