Documentation on nef.x
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Revision as of 07:34, 30 December 2011
nef.x is the application designed to address the following problems.
...
The corresponding routines are listed in the header file Nef.h.
Contents |
General output
The general output slightly depends on whether the polytope is input as a combined weight system or by a list of points. If the polytope was input as a list of points, the first line of the output takes the following form:
M:# # N:# # codim=# #part=#
Note that by default the input polytope is in the lattice M, while the output polytope is its dual in the lattice N. To change the lattice of the input polytope see the option -N.
If the input was a combined weight system, the line starts with the combined weight system repeated before the symbol M.
# M:# # N:# # codim=# #part=#
where # is the sequence of numbers describing the combined weight system. Note that the combined weight system describes the output polytope in the lattice N.
The two numbers # after M correspond to the number of points and vertices of the input polytope in the lattice M, respectively.
The two numbers # after N correspond to the number of points and vertices of the output polytope in the lattice N, respectively.
The number # in codim=# is the codimension of the nef-partition. The default is 2, otherwise it is specified by the option -c*.
The number # in #part=# is the number n of all the nef-partitions that nef.x has found, up to symmetries of the underlying lattice. It specifies the number of lines that follow the first output line. If the symmetries of the underlying lattice should not be taken into account, use the option -s.
The next lines contain the information about the various nef-partitions. If the codimension is 2 they take the following form:
H:# [#] P:# V:# # #sec #cpu
The numbers # after H: are the Hodge numbers 
, where 
is the dimension of the Calabi-Yau manifold 
that can be obtained from this nef-partition.
The number # in the square brackets [#] is the Euler number of . Under the assumption that 
, the remaining Hodge numbers can be computed from the ones given after H: and the Euler number. If this assumption is not satified, the Calabi-Yau manifold contains 
factors and the Hodge numbers 
are nonvanishing. See the option -D for this case. In any case, the full Hodge diamond is displayed with the option -H.
The number # after P: is a counter specfiying the nef-partition. It runs from 0 to n - 1, where n is the number of nef-partitions. Note that nef-partitions corresponding to direct products and projections to nef-partitions of lower codimensions are omitted by default. To display them use the options -D and -P, respectively.
The sequence of numbers # separated by a single space after V: corresponds to the vertices that belong to the first part of the nef-partition. Note that the vertices are counted starting from 0. These numbers only make sense if the options -Lv or -Lp are used. The vertices that belong to the second part of the nef-partition are not displayed, since they are simply the remaining ones. If the polytope entered also has points that are not vertices, then the second sequence of numbers # that is separated from the first sequence by two spaces corresponds to the non-vertex points that belong to the first partition. Again, the non-vertex points that belong to the second part of the nef-partition are not displayed.
The number # before sec indicates the time that was needed to compute this partition.
The number # before cpu indicates the number of CPU seconds that were needed to compute the Hodge numbers. For determining the nef-partitions without computing the Hodge numbers see the option -p.
If the codimension c is bigger than 2 the lines containing the information about the various nef-partitions take the following form:
H:# [#] P:# V0:# # V1:# # ... V(c-2):# # #sec #cpu
Now, there are c - 1 expressions of the form Vi:# #, where i runs from 0 to c - 2. Each expression consists of two sequences of numbers # separated by two spaces from each other. The first sequence of numbers # separated by a single space corresponds to the vertices that belong to the i-th part of the nef-partition. The second sequence of numbers # separated by a single space corresponds to the non-vertex points that belong to the i-th part of the nef-partition. The second sequence of numbers # is omitted if the polytope has no points that are not vertices or if the option -Lv is used.
The final line always takes the following form
np=# d:# p:# #sec #cpu
The number # in np=# is the number of nef-partitions which are neither direct products nor projections.
The number # after d: is the number of nef-partitions that are direct products.
The number # after p: is the number of nef-partitions that are projections. The total of the three numbers adds up to the total number of nef-partitions as indicated in the first line after #part=.
The number # before sec indicates the time that was needed to compute all the partitions.
The number # before cpu indicates the number of CPU seconds that were needed to compute the Hodge numbers of all the nef-partitions.
The following examples illustrate the general output of nef.x. We consider complete intersections of codimension 2 in 
discussed in arXiv:0704.0449. First, we enter the polytope by giving the combined weight system (which is essentially the same as a linearly independent set of linear relations) of this product of projective spaces
palp$ nef.x Degrees and weights `d1 w11 w12 ... d2 w21 w22 ...' or `#lines #colums' (= `PolyDim #Points' or `#Points PolyDim'): 3 1 1 1 0 0 0 0 0 2 0 0 0 1 1 0 0 0 3 0 0 0 0 0 1 1 1 3 1 1 1 0 0 0 0 0 2 0 0 0 1 1 0 0 0 3 0 0 0 0 0 1 1 1 M:300 18 N:9 8 codim=2 #part=15 H:19 19 [0] P:0 V:2 4 6 7 1sec 0cpu H:9 27 [-36] P:2 V:3 4 6 7 1sec 0cpu H:3 51 [-96] P:3 V:3 5 6 7 1sec 1cpu H:3 75 [-144] P:4 V:3 6 7 1sec 0cpu H:3 51 [-96] P:6 V:4 5 6 7 2sec 1cpu H:3 51 [-96] P:7 V:4 5 6 1sec 1cpu H:6 51 [-90] P:8 V:4 6 7 1sec 1cpu H:3 75 [-144] P:9 V:4 6 1sec 1cpu H:3 60 [-114] P:10 V:5 6 7 2sec 1cpu H:3 69 [-132] P:11 V:5 6 1sec 1cpu H:3 75 [-144] P:12 V:6 7 1sec 0cpu np=11 d:2 p:2 0sec 0cpu
Equivalently, we can use the option -N and enter the points of the polytope of the normal fan of
esche$ nef.x -N Degrees and weights `d1 w11 w12 ... d2 w21 w22 ...' or `#lines #colums' (= `PolyDim #Points' or `#Points PolyDim'): 5 8 Type the 40 coordinates as dim=5 lines with #pts=8 colums: 1 0 -1 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 1 0 -1 0 0 0 0 0 0 1 -1 M:300 18 N:9 8 codim=2 #part=15 H:3 51 [-96] P:0 V:2 3 4 7 1sec 1cpu H:3 51 [-96] P:1 V:2 4 6 7 2sec 1cpu H:3 60 [-114] P:2 V:2 4 7 2sec 1cpu H:3 51 [-96] P:3 V:2 6 7 1sec 1cpu H:3 69 [-132] P:4 V:2 7 1sec 1cpu H:9 27 [-36] P:5 V:3 4 6 7 1sec 0cpu H:3 75 [-144] P:6 V:3 4 7 0sec 0cpu H:19 19 [0] P:8 V:4 5 6 7 1sec 0cpu H:6 51 [-90] P:9 V:4 6 7 1sec 1cpu H:3 75 [-144] P:10 V:4 7 1sec 0cpu H:3 75 [-144] P:13 V:6 7 1sec 1cpu np=11 d:2 p:2 0sec 0cpu
Note that the nef-partitions are given in different orders. The first lines of the outputs respectively read
3 1 1 1 0 0 0 0 0 2 0 0 0 1 1 0 0 0 3 0 0 0 0 0 1 1 1 M:300 18 N:9 8 codim=2 #part=15 and M:300 18 N:9 8 codim=2 #part=15
Hence the polytope in the lattice N has 9 points, 8 vertices and the interior point, while the polytope in the lattice M has 300 points, 18 of which are vertices. The codimension is 2 and there are 15 nef-partitions. These are listed as follows:
H:3 51 [-96] P:0 V:2 3 4 7 1sec 1cpu H:3 51 [-96] P:1 V:2 4 6 7 2sec 1cpu H:3 60 [-114] P:2 V:2 4 7 2sec 1cpu H:3 51 [-96] P:3 V:2 6 7 1sec 1cpu H:3 69 [-132] P:4 V:2 7 1sec 1cpu H:9 27 [-36] P:5 V:3 4 6 7 1sec 0cpu H:3 75 [-144] P:6 V:3 4 7 0sec 0cpu H:19 19 [0] P:8 V:4 5 6 7 1sec 0cpu H:6 51 [-90] P:9 V:4 6 7 1sec 1cpu H:3 75 [-144] P:10 V:4 7 1sec 0cpu H:3 75 [-144] P:13 V:6 7 1sec 1cpu np=11 d:2 p:2 0sec 0cpu
There are 11 nef-partitions listed, furthermore there are 2 nef-partitions which are direct products, and 2 which are projections. Let 
denote the vertices of the polytope in the lattice 
. Let 
be the standard basis of 
. The according to the input, we have
The nef-partitions are then as follows (with the Hodge numbers and the Euler number of the corresponding Calabi-Yau 3-fold X):
Global parameters and limitations
Help screens
The help screens for nef.x are
nef.x -h
This is ``nef.x'': calculate hodge numbers of nef-partitions
Usage: cws.x -<options>
Options: -h print this information
-f or - use as filter; otherwise parameters denote I/O files
-N starting-poly is in N-lattice (detault is M)
-H gives full list of hodge numbers
-Lv prints L vector of Vertices (in N-lattice)
-Lp prints L vector of Points (in N-lattice)
-p prints only Partitions, no Hodge numbers
-D calculates also direct products
-P calculates also projections
-t full time info
-cCODIM codimension (default = 2)
-Fcodim FIBRATIONS up to codim (default = 2)
(note that 'cws.x' should become 'nef.x'!!!) and
nef.x -x
This is extended help for ``nef.x'':
-y print poly/CWS in M lattice if it has nef-partitions
-S information about #points calculated in S-Poly
-T checks Serre-duality
-s don't remove symmetric nef-partitions
-n prints Poly only if it has nef-partitions
-v prints Vertices and #points of starting-poly in one
line. with the following option the output is limited
by #points:
-uPOINTS ... upper limit of #points (default = POINT_Nmax)
-lPOINTS ... lower limit of #points (default = 0)
-m starts with [d w1 w2 ... wk d=d_1 d_2 (Minkowski sum)
-R prints Vertices of starting-poly if it is
not reflexive
-V prints Vertices of poly (in N-lattice)
-Q only direct products (up to lattice Quotient)
-gNUMBER prints Points of Gorenstein Poly in N-lattice
-dNUMBER prints Points of Gorenstein Poly in M-lattice
if NUMBER = 0 ... no 0/1 info
if NUMBER = 1 ... no redundant 0/1 info (=default)
if NUMBER = 2 ... full 0/1 info
The options in detail
-N
This option interprets the input polytope in the lattice N. The default lattice for the input polytope, however, is the lattice M. The default lattice for the output polytope is the lattice N.
The following example of a complete intersection of degree (2,2) in 
illustrates the difference:
palp$ nef.x -Lv
Degrees and weights `d1 w11 w12 ... d2 w21 w22 ...'
or `#lines #colums' (= `PolyDim #Points' or `#Points PolyDim'):
3 4
Type the 12 coordinates as dim=3 lines with #pts=4 colums:
-1 0 0 1
-1 0 1 0
-1 1 0 0
M:5 4 N:35 4 codim=2 #part=0
3 4 Vertices in N-lattice:
-1 -1 -1 3
-1 -1 3 -1
-1 3 -1 -1
--------------------
1 1 1 1 d=4 codim=0
np=0 d:0 p:0 0sec 0cpu
Without the option -N, the output polytope is the dual of the input polytope with 35 points and no nef partition.
palp$ nef.x -Lv -N
Degrees and weights `d1 w11 w12 ... d2 w21 w22 ...'
or `#lines #colums' (= `PolyDim #Points' or `#Points PolyDim'):
3 4
Type the 12 coordinates as dim=3 lines with #pts=4 colums:
-1 0 0 1
-1 0 1 0
-1 1 0 0
M:35 4 N:5 4 codim=2 #part=2
3 4 Vertices in N-lattice:
-1 0 0 1
-1 0 1 0
-1 1 0 0
--------------------
1 1 1 1 d=4 codim=0
H:[0] P:0 V:2 3 (2 2) 0sec 0cpu
np=1 d:0 p:1 0sec 0cpu
With the option -N, the output polytope is the same as input polytope with 4 points and the expected nef partition corresponding to the complete intersection of degree (2,2) in .
-H
The option -H replaces the output lines starting with H: with the full Hodge diamond of the corresponding partition. Note that the information about the nef partition is omitted. The following example of codimension 2 complete intersections in 
illustrates this option
esche$ nef.x -H
Degrees and weights `d1 w11 w12 ... d2 w21 w22 ...'
or `#lines #colums' (= `PolyDim #Points' or `#Points PolyDim'):
7 1 1 1 1 1 1 1
7 1 1 1 1 1 1 1 M:1716 7 N:8 7 codim=2 #part=3
h 0 0
h 1 0 h 0 1
h 2 0 h 1 1 h 0 2
h 3 0 h 2 1 h 1 2 h 0 3
h 4 0 h 3 1 h 2 2 h 1 3 h 0 4
h 4 1 h 3 2 h 2 3 h 1 4
h 4 2 h 3 3 h 2 4
h 4 3 h 3 4
h 4 4
1
0 0
0 1 0
0 0 0 0
1 237 996 237 1
0 0 0 0
0 1 0
0 0
1
16sec 15cpu
h 0 0
h 1 0 h 0 1
h 2 0 h 1 1 h 0 2
h 3 0 h 2 1 h 1 2 h 0 3
h 4 0 h 3 1 h 2 2 h 1 3 h 0 4
h 4 1 h 3 2 h 2 3 h 1 4
h 4 2 h 3 3 h 2 4
h 4 3 h 3 4
h 4 4
1
0 0
0 1 0
0 0 0 0
1 356 1472 356 1
0 0 0 0
0 1 0
0 0
1
42sec 41cpu
np=2 d:0 p:1 0sec 0cpu
-Lv
The option -Lv prints the vertices of the output polytope and the relations among them in addition to the standard output. The output takes the following form: The first line is
D n Vertices in N-lattice:
This line says that the vertices of the polytope in the lattice are given by the subsequent 
lines with 
entries. The means that the polytope has dimension and is given by vertices which are the columns of the subsequent 
array of numbers. Note that an arbitrary basis of 
will be chosen.
Below the dashed line the linear relations among these vertices are indicated as follows: Let 
denote the vertices corresponding to the columns above the dashed line. For each linear relation among the vertices 
given by
,
denote its degree by
.
The vertices with non-zero 
span a subpolytope of codimension 
.
For each linearly independent linear relation there is a line in the output of the following form:
l_0 l_1 ... l_{n-1} d=l codim=c
In other words, these lines give a basis of the vector space of linear relations among the vertices. The basis is completely fixed by the order of the vertices, and the conditions that each vector, i.e. each linear relation is positive and primitive.
The following example of a codimension 2 complete intersection in 
illustrates this option
palp$ nef.x -Lv
Degrees and weights `d1 w11 w12 ... d2 w21 w22 ...'
or `#lines #colums' (= `PolyDim #Points' or `#Points PolyDim'):
4 1 1 1 1 0 0 0 0 4 0 0 0 0 1 1 1 1
4 1 1 1 1 0 0 0 0 4 0 0 0 0 1 1 1 1 M:1225 16 N:9 8 codim=2 #part=8
6 8 Vertices in N-lattice:
0 0 0 0 1 0 0 -1
0 0 1 0 0 0 0 -1
0 0 0 1 0 0 0 -1
-1 0 0 0 0 0 1 0
-1 0 0 0 0 1 0 0
-1 1 0 0 0 0 0 0
----------------------------------------
1 1 0 0 0 1 1 0 d=4 codim=3
0 0 1 1 1 0 0 1 d=4 codim=3
H:2 0 128 [828] P:1 V:3 4 6 7 (1 3) (3 1) 19sec 18cpu
H:8 0 128 [864] P:2 V:3 4 7 (0 4) (3 1) 13sec 13cpu
H:2 0 166 [1056] P:3 V:4 5 6 7 (2 2) (2 2) 8sec 8cpu
H:2 0 198 [1248] P:4 V:5 6 7 (2 2) (1 3) 9sec 8cpu
H:3 0 293 [1824] P:5 V:5 6 (2 2) (0 4) 6sec 6cpu
H:2 0 284 [1764] P:6 V:6 7 (1 3) (1 3) 10sec 9cpu
np=6 d:1 p:1 0sec 0cpu
The line 6 8 Vertices in N-lattice: says that the polytope has dimension 6 and is given by 8 vertices 
.
Let 
be the standard basis of 
. From the columns of the subsequent 6 by 8 array of numbers, we read off that the 8 vertices are
.
Note that an arbitrary basis has been chosen. There are two linearly independent linear relations:
.
Each of these linear relations has degree 4. The corresponding subpolytopes have codimension 3 each and, of course, correspond to the two factors of , respectively.
-Lp
???
-p
Computes the nef partitions without the (time-consuming) calculation of Hodge numbers.
Example: Complete intersection Calabi-Yau fourfold of codimension two discussed in arXiv:0912.3524. Important: in Global.h set POLY_Dmax=7 or higher and recompile!
Input with -p:
palp$ nef.x -p Degrees and weights `d1 w11 w12 ... d2 w21 w22 ...' or `#lines #columns' (= `PolyDim #Points' or `#Points PolyDim'): 6 3 2 1 0 0 0 0 0 0 0 0 13 6 4 0 1 0 0 0 0 0 1 1 7 3 2 0 0 1 0 0 0 0 1 0 8 3 2 0 0 0 1 0 0 1 1 0 15 6 4 0 0 0 0 1 1 1 1 1 6 3 2 1 0 0 0 0 0 0 0 0 13 6 4 0 1 0 0 0 0 0 1 1 7 3 2 0 0 1 0 0 0 0 1 0 8 3 2 0 0 0 1 0 0 1 1 0 15 6 4 0 0 0 0 1 1 1 1 1 M:4738 39 N:15 11 codim=2 #part=11 P:0 V:0 4 7 0sec 0cpu P:1 V:0 2 3 5 9 11 12 13 0sec 0cpu P:2 V:1 5 6 8 0sec 0cpu P:3 V:1 6 7 8 10 0sec 0cpu P:4 V:0 1 7 8 0sec 0cpu P:5 V:0 1 4 7 8 0sec 0cpu P:6 V:4 5 6 0sec 0cpu P:7 V:4 6 7 10 0sec 0cpu P:9 V:5 6 0sec 0cpu P:10 V:6 8 0sec 0cpu np=10 d:0 p:1 0sec 0cpu
Input without -p (note the calculation time! (32-bit system)):
palp$ nef.x Degrees and weights `d1 w11 w12 ... d2 w21 w22 ...' or `#lines #columns' (= `PolyDim #Points' or `#Points PolyDim'): 6 3 2 1 0 0 0 0 0 0 0 0 13 6 4 0 1 0 0 0 0 0 1 1 7 3 2 0 0 1 0 0 0 0 1 0 8 3 2 0 0 0 1 0 0 1 1 0 15 6 4 0 0 0 0 1 1 1 1 1 6 3 2 1 0 0 0 0 0 0 0 0 13 6 4 0 1 0 0 0 0 0 1 1 7 3 2 0 0 1 0 0 0 0 1 0 8 3 2 0 0 0 1 0 0 1 1 0 15 6 4 0 0 0 0 1 1 1 1 1 M:4738 39 N:15 11 codim=2 #part=11 H:8 0 1113 [6774] P:0 V:0 4 7 247sec 246cpu H:5 0 1115 [6768] P:1 V:0 2 3 5 9 11 12 13 141sec 141cpu H:5 0 1115 [6768] P:2 V:1 5 6 8 136sec 136cpu H:8 0 1113 [6774] P:3 V:1 6 7 8 10 183sec 182cpu H:8 0 1113 [6774] P:4 V:0 1 7 8 203sec 202cpu H:5 0 1115 [6768] P:5 V:0 1 4 7 8 157sec 156cpu H:5 0 1115 [6768] P:6 V:4 5 6 190sec 189cpu H:8 0 1113 [6774] P:7 V:4 6 7 10 226sec 225cpu H:8 0 1113 [6774] P:9 V:5 6 246sec 246cpu H:7 0 958 [5838] P:10 V:6 8 236sec 234cpu np=10 d:0 p:1 236sec 234cpu
-D
This option keeps those nef partitions which are direct products of lower-dimensional nef partitions.
Example: Codimension 2 CICY in 
with option -D:
palp$ nef.x -D 3 1 1 1 0 0 0 3 0 0 0 1 1 1 3 1 1 1 0 0 0 3 0 0 0 1 1 1 M:100 9 N:7 6 codim=2 #part=5 H:4 [0] h1=2 P:0 V:2 3 5 D 0sec 0cpu H:20 [24] P:1 V:3 4 5 0sec 0cpu H:20 [24] P:2 V:3 5 0sec 0cpu H:20 [24] P:3 V:4 5 0sec 0cpu np=3 d:1 p:1 0sec 0cpu
The last three nef partitions describe a K3 manifold. The first one is a 
. The extra output triggered by -D is:
H:4 [0] h1=2 P:0 V:2 3 5 D 0sec 0cpu
h1 = 2 indicates that the Hodge number h1,0 = 2. Furthermore D indicates that the nef partition is a direct product.
Compare this to the output without the option -D where the first nef partition is not shown:
palp$ nef.x 3 1 1 1 0 0 0 3 0 0 0 1 1 1 3 1 1 1 0 0 0 3 0 0 0 1 1 1 M:100 9 N:7 6 codim=2 #part=5 H:20 [24] P:1 V:3 4 5 0sec 0cpu H:20 [24] P:2 V:3 5 0sec 0cpu H:20 [24] P:3 V:4 5 1sec 0cpu np=3 d:1 p:1 0sec 0cpu
-P
This option also shows nef partitions corresponding to projections. If a nef partition has k elements which only contain one vertex this corresponds to k linear equations which set k of the variables to zero. The Calabi-Yau is thus a complete intersection of codimension k less.
Example: Complete intersection of codimension 2 in :
palp$ nef.x -P 4 1 1 1 1 4 1 1 1 1 M:35 4 N:5 4 codim=2 #part=2 H:[0] P:0 V:2 3 0sec 0cpu H:[0] P:1 V:3 0sec 0cpu np=1 d:0 p:1 0sec 0cpu
Compared to the output without -P there is one additional line:
H:[0] P:1 V:3 0sec 0cpu
One element of the nef partition only contains the vertex labeled by 3. Therefore one of the equations of the complete intersections reads x[3] = 0. Thus, we are left with a hypersurface in 
, i.e. the cubic curve.
Example: A complete intersection of codimension 6 which is reduced to codimension 3 by projections. We use the option -c* to set the codimension and -p to suppress the calculation of the Hodge numbers. Furthermore we list the vertices using the option -Lv:
palp$ nef.x -P -c6 -p -Lv
Degrees and weights `d1 w11 w12 ... d2 w21 w22 ...'
or `#lines #columns' (= `PolyDim #Points' or `#Points PolyDim'):
6 1 1 1 1 1 1 0 0 0 6 1 1 1 0 0 0 1 1 1
6 1 1 1 1 1 1 0 0 0 6 1 1 1 0 0 0 1 1 1 M:5214 12 N:10 9 codim=6 #part=1
7 9 Vertices in N-lattice:
-1 0 0 1 0 0 0 0 0
-1 0 1 0 0 0 0 0 0
0 -1 0 0 0 1 0 0 0
0 -1 0 0 1 0 0 0 0
-1 1 0 0 0 0 0 0 1
-1 1 0 0 0 0 1 0 0
-1 1 0 0 0 0 0 1 0
---------------------------------------------
1 1 1 1 1 1 0 0 0 d=6 codim=2
1 0 1 1 0 0 1 1 1 d=6 codim=2
P:0 V0:0 V1:2 V2:3 V3:4 7 V4:5 8 (1 1 1 1 1 1) (1 1 1 1 1 1) 0sec 0cpu
np=0 d:0 p:1 0sec 0cpu
The output shows that three elements of the nef partition contain only one vertex:
P:0 V0:0 V1:2 V2:3 V3:4 7 V4:5 8 0sec 0cpu
Therefore the variables associated to the vertices labeled by 0,2 and 3 can be set to zero and we are left with a complete intersection of codimension 3 in .
-t
The option -t gives detailed information about the calculation times of the Hodge numbers. The Hodge numbers of a complete intersection are generated by the so called stringy E-function introduced by Batyrev and Borisov in alg-geom/9509009. The combinatorial construction of the E-function involves the construction of a B-polynomial and an S-polynomial defined in alg-geom/9509009. The option -t returns the accumulated computing times of the respective polynomials.
Example: Complete intersection Calabi-Yau fourfold discussed in arXiv:0908.1784. Important: in Global.h set POLY_Dmax=7 or higher and recompile!
palp$ nef.x -t 10 3 2 0 1 1 1 1 1 6 3 2 1 0 0 0 0 0 10 3 2 0 1 1 1 1 1 6 3 2 1 0 0 0 0 0 M:2302 15 N:12 8 codim=2 #part=4 BEGIN S-Poly 0sec 0cpu BEGIN B-Poly 61sec 57cpu BEGIN E-Poly 66sec 61cpu H:2 30 308 [1728] P:0 V:4 5 6 7 66sec 61cpu BEGIN S-Poly 0sec 0cpu BEGIN B-Poly 92sec 83cpu BEGIN E-Poly 100sec 91cpu H:5 5 448 [2736] P:1 V:5 6 7 100sec 91cpu BEGIN S-Poly 0sec 0cpu BEGIN B-Poly 152sec 138cpu BEGIN E-Poly 160sec 146cpu H:5 0 567 [3480] P:2 V:6 7 160sec 146cpu np=3 d:0 p:1 0sec 0cpu
-c*
The option -c* where * is a number > = 1 allows to specify the codimension of the Calabi-Yau. The default value is for the codimension is 2. Note that the calculation can become very slow for high codimensions and PALP may crash because the limits such as the number of vertices etc. set in Global.h may be exceeded.
Important: for codimension 2 or higher the parameter POLY_Dmax specifying the dimension of the polytope in Global.h needs to be changed accordingly and PALP has to be recompiled.
Example: Codimension 3 CICY in :
palp$ nef.x -c3 Degrees and weights `d1 w11 w12 ... d2 w21 w22 ...' or `#lines #columns' (= `PolyDim #Points' or `#Points PolyDim'): 3 1 1 1 0 0 0 3 0 0 0 1 1 1 3 1 1 1 0 0 0 3 0 0 0 1 1 1 M:100 9 N:7 6 codim=3 #part=7 H:[0] P:0 V0:1 3 V1:4 5 1sec 1cpu H:[0] P:1 V0:2 3 V1:4 5 1sec 0cpu np=1 d:1 p:5 0sec 0cpu
Also hypersurfaces are possible.
Example: Quintic in 
:
palp$ nef.x -c1 Degrees and weights `d1 w11 w12 ... d2 w21 w22 ...' or `#lines #columns' (= `PolyDim #Points' or `#Points PolyDim'): 5 1 1 1 1 1 5 1 1 1 1 1 M:126 5 N:6 5 codim=1 #part=1 H:1 101 [-200] P:0 0sec 0cpu np=1 d:0 p:0 0sec 0cpu
Compare to that the output of poly.x:
palp$ poly.x Degrees and weights `d1 w11 w12 ... d2 w21 w22 ...' or `#lines #columns' (= `PolyDim #Points' or `#Points PolyDim'): 5 1 1 1 1 1 5 1 1 1 1 1 M:126 5 N:6 5 H:1,101 [-200]
-F*
???
-y
???
-S
???
-T
???
-s
The option -s also includes nef partitions in the output which are related bt to symmetries of the weight matrix. Note that the option -s does not print all possible nef partitions as those corresponding to projections (cf. option -P) or direct products (cf. option -D) are left out.
Expample: Complete intersection of codimension 2 in . We add the option -Lv in order to print the vertices and the weight matrix.
palp$ nef.x -s -Lv
Degrees and weights `d1 w11 w12 ... d2 w21 w22 ...'
or `#lines #columns' (= `PolyDim #Points' or `#Points PolyDim'):
3 1 1 1 0 0 0 3 0 0 0 1 1 1
3 1 1 1 0 0 0 3 0 0 0 1 1 1 M:100 9 N:7 6 codim=2 #part=31
4 6 Vertices in N-lattice:
0 0 0 1 0 -1
0 0 1 0 0 -1
-1 0 0 0 1 0
-1 1 0 0 0 0
------------------------------
1 1 0 0 1 0 d=3 codim=2
0 0 1 1 0 1 d=3 codim=2
H:20 [24] P:2 V:4 5 (1 2) (1 2) 0sec 0cpu
H:20 [24] P:4 V:0 5 (1 2) (1 2) 0sec 0cpu
H:20 [24] P:5 V:0 4 (2 1) (0 3) 0sec 0cpu
H:20 [24] P:6 V:0 4 5 (2 1) (1 2) 0sec 0cpu
H:20 [24] P:8 V:1 5 (1 2) (1 2) 1sec 0cpu
H:20 [24] P:9 V:1 4 (2 1) (0 3) 0sec 0cpu
H:20 [24] P:10 V:1 4 5 (2 1) (1 2) 0sec 0cpu
H:20 [24] P:11 V:0 1 (2 1) (0 3) 0sec 0cpu
H:20 [24] P:12 V:0 1 5 (2 1) (1 2) 0sec 0cpu
H:20 [24] P:14 V:2 3 (0 3) (2 1) 0sec 0cpu
H:20 [24] P:16 V:2 5 (0 3) (2 1) 0sec 0cpu
H:20 [24] P:17 V:2 4 (1 2) (1 2) 0sec 0cpu
H:20 [24] P:18 V:2 4 5 (1 2) (2 1) 0sec 0cpu
H:20 [24] P:19 V:0 2 (1 2) (1 2) 0sec 0cpu
H:20 [24] P:20 V:0 2 5 (1 2) (2 1) 1sec 0cpu
H:20 [24] P:21 V:0 2 4 (2 1) (1 2) 0sec 0cpu
H:20 [24] P:22 V:1 3 (1 2) (1 2) 0sec 0cpu
H:20 [24] P:23 V:1 2 (1 2) (1 2) 0sec 0cpu
H:20 [24] P:24 V:1 2 5 (1 2) (2 1) 0sec 0cpu
H:20 [24] P:25 V:1 2 4 (2 1) (1 2) 0sec 0cpu
H:20 [24] P:26 V:0 3 (1 2) (1 2) 0sec 0cpu
H:20 [24] P:27 V:0 1 2 (2 1) (1 2) 0sec 0cpu
H:20 [24] P:28 V:3 4 (1 2) (1 2) 1sec 0cpu
H:20 [24] P:29 V:3 5 (0 3) (2 1) 0sec 0cpu
np=24 d:1 p:6 0sec 0cpu
Note that the weight matrix is symmetric under permutations of the vertices labeled by 0,1,4 and those labeled by 2,3,5. Furthermore there only exist three pairs of degrees of the complete intersection (up to exchange within a pair): {(1,2),(1,2)},{(0,3),(2,1)},{(1,2),(2,1)}. Therefore we conclude that there are only three inequivalent nef partitions. This is indeed confirmed by calling nef without the option -s:
palp$ nef.x -Lv
Degrees and weights `d1 w11 w12 ... d2 w21 w22 ...'
or `#lines #columns' (= `PolyDim #Points' or `#Points PolyDim'):
3 1 1 1 0 0 0 3 0 0 0 1 1 1
3 1 1 1 0 0 0 3 0 0 0 1 1 1 M:100 9 N:7 6 codim=2 #part=5
4 6 Vertices in N-lattice:
0 0 0 1 0 -1
0 0 1 0 0 -1
-1 0 0 0 1 0
-1 1 0 0 0 0
------------------------------
1 1 0 0 1 0 d=3 codim=2
0 0 1 1 0 1 d=3 codim=2
H:20 [24] P:1 V:3 4 5 (1 2) (2 1) 0sec 0cpu
H:20 [24] P:2 V:3 5 (0 3) (2 1) 0sec 0cpu
H:20 [24] P:3 V:4 5 (1 2) (1 2) 0sec 0cpu
np=3 d:1 p:1 0sec 0cpu
-n
???
-v
???
-m
???
-R
???
-V
???
-Q
???
-g*
???
-d*
???
Strategy for future versions
Should we keep both help screens, or try to get everything into 'nef.x -h'?
It would be good to have a documentation of the header file Nef.h: just type whatever information you can supply into the following listing, in the standard ('/* ... */') C comment format.
#define Nef_Max 500000
#define NP_Max 500000
#define W_Nmax (POLY_Dmax+1)
#define MAXSTRING 100
#undef WRITE_CWS
#define WRITE_CWS
#define Pos_Max (POLY_Dmax + 2)
#define FIB_Nmax 10*EQUA_Nmax
#define FIB_POINT_Nmax VERT_Nmax
typedef struct {
Long W[FIB_Nmax][FIB_POINT_Nmax];
Long VM[FIB_POINT_Nmax][POLY_Dmax];
int nw;
int nv;
int d;
int Wmax;
} LInfo;
struct Poset_Element {
int num, dim;
};
struct Interval {
int min, max;
};
typedef struct Interval Interval;
typedef struct {
struct Interval *L;
int n;
} Interval_List;
typedef struct Poset_Element Poset_Element;
typedef struct {
struct Poset_Element x, y;
} Poset;
typedef struct {
struct Poset_Element *L;
int n;
} Poset_Element_List;
typedef struct {
int nface[Pos_Max];
int dim;
INCI edge[Pos_Max][FACE_Nmax];
} Cone;
typedef struct {
Long S[2*Pos_Max];
} SPoly;
typedef struct {
Long B[Pos_Max][Pos_Max];
} BPoly;
typedef struct {
int E[4*(Pos_Max)][4*(Pos_Max)];
} EPoly;
typedef struct {
Long x[POINT_Nmax][W_Nmax];
int N, np;
} AmbiPointList;
typedef struct {
int n;
int nv;
int codim;
int S[Nef_Max][VERT_Nmax];
int DirProduct[Nef_Max];
int Proj[Nef_Max];
int DProj[Nef_Max];
} PartList;
typedef struct {
int n;
int nv;
int S[Nef_Max][VERT_Nmax];
} Part;
typedef struct {
int n, y, w, p, t, S, Lv, Lp, N, u, d, g, VP, B, T, H, dd, gd,
noconvex, Msum, Sym, V, Rv, Test, Sort, Dir, Proj, f;
} Flags;
typedef struct {
int noconvex, Sym, Test, Sort;
} NEF_Flags;
struct Vector {
Long x[POLY_Dmax];
};
typedef struct Vector Vector ;
typedef struct {
struct Vector *L;
int n;
Long np, NP_max; } DYN_PPL;
void part_nef(PolyPointList *, VertexNumList *, EqList *, PartList *,
int *, NEF_Flags *);
void Make_E_Poly(FILE *, CWS *, PolyPointList *, VertexNumList *, EqList *,
int *, Flags *, int *);
void Mink_WPCICY(AmbiPointList * _AP_1, AmbiPointList * _AP_2,
AmbiPointList * _AP);
int IsDigit(char);
int IntSqrt(int q);
void Die(char *);
void Print_CWS_Zinfo(CWS *);
