Documentation on nef.x
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.
The first line of the output takes the following form:
M:# # N:# # codim=# #part=#
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 as specifiec by the option -c. The number # in #part=# is the number n of all the nef-partitions that nef.x has found. It specifies the number of lines that follow the first output line. If the polytope has been entered as a combined weight system, the line starts with the combined weight system repeated before the symbol M.
The next lines contain the information about the various nef-partitions. 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 full Hodge diamond is displayed with the option -H.
The numbers # in the square brackets [#] is the Euler number of X.
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 number # before sec indicates the time that was needed to compute this partition.
The number # before cpu indicates the number of CPU seconds that was needed to compute the Hodge numbers. For determining the nef-partitions without computing the Hodge numbers see the option -p.
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 -Lv -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
6 7 Vertices in N-lattice:
-1 0 0 0 0 0 1
-1 0 1 0 0 0 0
-1 0 0 1 0 0 0
-1 0 0 0 1 0 0
-1 0 0 0 0 1 0
-1 1 0 0 0 0 0
-----------------------------------
1 1 1 1 1 1 1 d=7 codim=0
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!
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
???
-t
???
-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
???
-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 *);
