This code is part of the publication “Mordell-Weil Torsion in the Mirror of Multi-Sections” (MWTMM)
by Paul Oehlmann, Jonas Reuter and Thorsten Schimannek, http://arxiv.org/abs/1604.00011.

The code has been tested with Sage 6.10.

The code is provided as a set of Jupyter notebooks for a Sage kernel.
In case you do not have Jupyter installed we also provide html dumps.
The individual notebooks serve the following purposes:

Code:

	generateDualIds.ipynb
	generateDualIds.html

	 This notebook contains the code to calculate the dictionary of
	 mirror nef ids. The dictionary is also provided in dualids.txt and this code is
	 mainly provided for documentation.

	generateIntersections.ipynb
	generateIntersections.html

	 This notebook calculates the dictionary of intersections of toric
	 divisors with the complete intersection curves. The dictionary is also provided in intersection.txt and this code
	 is mainly provided for documentation

	loadData1411.2615.ipynb
	loadData1411.2615.html

	 This notebook loads the data about the toric Mordell-Weil group and codimension one singularities,
	 calculated by Volker Braun, Thomas W. Grimm and Jan Keitel in http://arxiv.org/abs/1411.2615.
	 The data has to be downloaded from Jan Keitel website first and instructions are provided in the notebook.
	 You should follow the instructions before proceeding with worksheet.ipynb.

	worksheet.ipynb
	worksheet.html

	 This notebook loads the dictionaries of mirror ids, intersections, equivalence classes and the above data to check the Mirror conjecture for
	 complete intersections in three-dimensional toric ambient spaces.
	 Note that using pythons list comprehension, the four datasets provide powerful means to answer other questions about 
	 the complete intersection fibres as well.

The dictionaries in the data files

	dualids.txt
	intersection.txt
	equivalence.txt

can also be loaded independently, using

“
with open(‘filename‘, ‘r‘) as f:

	dict = eval(f.read())
“

All of the dictionaries are indexed by tuples of the form (polyid, nefid).
The entries are the following:

	dualids.txt		-	A tuple (dualPolyId, dualNefId)

	intersections.txt	-	A list with the intersections of the toric divisors associated to the points in poly.points()
					when for example “poly = ReflexivePolytope(3, polyid)”.
					One of the points will be the origin and does not correspond to a divisor.
					The corresponding entry is zero.
					The indices of the list correspond to the indices of the z variables in the equations p1, p2
					obtained in http://arxiv.org/abs/1411.2615.

	equivalence.txt		-	A list of tuples (polyid, nefid) of the nef partitions that are equivalent - in the sense outlined
					in MWTMM - to the key tuple. 
					The code to obtain this list is somewhat messy and we do not provide it here.
					If you are interested in seeing it feel free to contact us.