Section 4: Dense CubicsΒΆ

In Section 4 we look at cubic surfaces defined by a dense polynomial

\[\begin{split}\begin{matrix} & c_0 w^3 + c_1w^2z + c_2wz^2 + c_3z^3 + c_4w^2y + c_5wyz + c_6yz^2 \\ & + \, c_7wy^2 + c_8y^2z + c_9y^3 + c_{10}w^2x + c_{11}wxz + c_{12}xz^2 + c_{13}wxy \\ & + \, c_{14}xyz + c_{15}xy^2 + c_{16}wx^2 + c_{17}x^2z + c_{18}x^2y + c_{19}x^3. \end{matrix}\end{split}\]

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

\[({\rm val}(c_0),\ldots,{\rm val}(c_{19})) \,\,=\,\, (16,7,3,11,9,2,5,3,0,0,5,0,4,3,0,2,6,5,8,15).\]

We fix \(p=5\) and choose the canonical lifts

\[c_0 = 5^{16}, \,c_1 = 5^7\! ,\,c_2 = 5^3\!,\, \ldots, \,c_{19} = 5^{15}.\]

The six nearby points over \(\mathbb{Q}\) of the blow-up points are

\[\begin{split}\begin{array}{rcl} q_1 &=& ( 1 \,\,:\,\, 0\,\,:\,\, 0),\\ q_2 &=& (0\,\,:\,\, 1 \,\,:\,\,0 ),\\ q_3 &=& (0\,\,:\,\, 0\,\,:\,\,1 ),\\ q_4 &=& (1\,\,:\,\, 1\,\,:\,\,1 ),\\ q_5 &=& (2473616049/5 \,\,:\,\, -425393750 \,\,: \,\, 1 ), \\ q_6 &=& (1331718750 \,\,:\,\, -2324221875 \,\,:\, \, 1 ). \end{array}\end{split}\]

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.