Magma package to compute lines, blow-ups, blow-downs,... for tropically smooth cubic surfaces over p-adics.

This code performs almost everything explained in Section 4 of our joint paper with Marta Panizzut and Bernd Sturmfels (https://arxiv.org/abs/1908.06106). See example_usage.mag on how to use key functionalities.

The input cubic surface is assumed to have all its lines defined over the p-adic field (for user specified p). An error will occur if this is not the case. If the cubic surface is tropically smooth, then this assumption seems to hold. See the paper cited above for the relevant discussion.

In Magma, attach the package:

Magma> Attach("blowdown.mag");

Specify a prime and a cubic quaternary form:

Magma> p:=5;
Magma> f:=3125*w^3 + 25*w^2*x + 25*w^2*y + 5*w^2*z + 25*w*x^2 + w*x*y + w*x*z + 25*w*y^2 + w*y*z + 5*w*z^2 + 3125*x^3 + 5*x^2*y + 25*x^2*z + 5*x*y^2 + x*y*z + 25*x*z^2 + 3125*y^3 + 25*y^2*z + 25*y*z^2 + 3125*z^3;

Now the main function pAdicCubicSurface builds a record storing the lines and other data induced from them:

Magma> S:=pAdicCubicSurface(f,p);

There is a certain amount of randomness involved, in case of failure try again, or increase precision:

Magma> S:=pAdicCubicSurface(f,p : precision:=5000);

The cubics defining P2 - -> S, and the valuation of their coefficients, can be viewed as follows,

Magma> S`blowup_cubics;
Magma> [valuation_of_form(form) : form in S`blowup_cubics];

The six base points in P2 and their valuations:

Magma> pts:=blowdown_lines(S`marking); 
Magma> [[Valuation(p) : p in pt] : pt in pts];

Tree statistics are obtained as below:

Magma> splits:=[tropicalize_line_in_cubic(S,i) : i in [1..27]]; [#s : s in splits];
Magma> SetToMultiset([encode_split(split) : split in splits]);

You can also "perturb" the six base points in P2, and the cubics passing through them appropriately to get everything to be over the rationals, while preserving the combinatorial type of the cubic surface:

Magma> g,newpts,cubs,err:=perturb(S,14 : internal_precision:=20); 

Here g is the new cubic surface, newpts are the six new points in P2 rounded from the old ones while keeping 14 digits of p-adic presecision, cubs are the cubic curves through them and err is the difference in the valuation of the coefficients of g and f. If err is not zero, try again by increasing the two precisions 14 and 20; the former governs the height of the points and the latter the height of the cubic curves through them.

In case you want to compute the lines in a cubic surface which has all its lines in Qp, but the equation is not dense (e.g. octanomials) then you can do the following:

Magma> f:=4*w^2*x + w*x*y + w*x*z + 2*w*y*z + w*z^2 + 8*x^2*y + 2*x*y*z + y^2*z;
Magma> S:=pAdicCubicSurface(f,2: with_blowup:=false);

The with the blow-up computation turned off, we can handle non-dense cubics.

The lines of S are stored in a list consisting of 27 matricies of size 2x4. For each matrix, the rows span the corresponding line.

Magma> S`lines;

We also impose a labelling on the lines. The exceptional lines and all the others can be viewed by:

Magma> S`e; S`l; S`q;

Where the third entry in e is the index of the line E3 inside the list lines;