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 |
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
???
-Lp
???
-p
???
-D
???
-P
???
-t
???
-c*
???
-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 *);
