July 2 – The programs have been updated.
This page will soon be updated accordingly
The program PModChecker will enable you to determine if a class in the Néron-Severi group of your
We could demonstrate PModChecker by using CGS to generate datasets of classes of square
That would be OK, but the result wouldn’t be worth much.
What would be the point of generating tons of results devoid of geometric interpretation?
Let’s follow an approach developed by Xavier Roulleau in his atlas of K3 surfaces and this paper On the geometry of K3 surfaces with finite automorphism group and no elliptic fibrations.
To study projective models of a
where
Morrison’s theorem tells us that the morphism
In this context, the above mentioned configuration has a very precise geometric interpretation :
can be interpreted as an hyperplane section of a quartic in
can be seen as a conic section of a quartic which decomposes as the union of smooth rational curves
Simple computations involving dimensions of linear systems make it possible to determine whether a quartic in
By Saint-Donat and Morrison, the base point freeness and non-hyperellipticity of an ample class
With this approach in mind, we have produced the programs SysFinder and SysDisplay : These programs enable us to first determine a configuration
with ll
We present these programs on an example and will then use PModChecker to determine whether the class
Please download the following input data file: INPUT_DATA_PICARD_THREE_XT_t_=_7.sobj
We will study projective models of the
Open a Sage console from the folder into which you stored the file. Enter these commands
INPUT_DATA=load('INITIAL_INPUT_DATA_xt_picard_three_case_7.sobj');
GramMatS = INPUT_DATA[0];
amp0 = INPUT_DATA[2];
and load PModChecker.sage :

The program asks us to compute
ListSRC = CGS(MaxDegree=30000,GramMatrix=GramMatS,Ample=amp0,SelfInt=-2)
That is, we use CGS in order to compute the set of classes of smooth rational curves
We then define
ListSRCred=ListSRC[:25]
which is the list of the first 25 elements of ListSRC, where you can replace
We then reload PModChecker.sage, and this time you should not see any warning displayed.

The function SysFinder takes the three following mandatory parameters as input data :
- GramMatrix : Gram matrix of
, - ample : ample class,
- selfint : integer,

and returns the raw data of all linear combinations of the form
SysFinder will save the data it computes in a dedicated file

You can personalize the name of this file by defining global string variables text1 and text2. For example,
text1 = str('my_surface')
text2 = str('picard_3')
Otherwise, text1 and text2 will be set by default as

Let’s compute all the possible linear combinations
Define
SYSTEMS = SysFinder(GramMatrix=GramMatS,ample=amp0,selfint=4)
Then, run
SysDisplay(SYSTEMS)
The program SysDisplay will then use the data from SYSTEMS to form configurations
associated to an ample class

That is, the program returned a configuration
associated to the ample class
with
Let’s use PModChecker to determine if a projective model of the surface under study can be exhibited from the data of
PModChecker(Matrix([1,-1,-2]))
