Cremona -- package for some computations on rational maps between projective varieties
Description
Cremona is a package to perform some basic computations on rational and birational maps between absolutely irreducible projective varieties over a field K. For instance, it provides general methods to compute degrees and projective degrees of rational maps (see degreeOfRationalMap and projectiveDegrees) and a general method to compute the push-forward to projective space of Segre classes (see SegreClass). Moreover, all the main methods are available both in version probabilistic and in version deterministic, and one can switch from one to the other with the boolean option MathMode.Let Φ:X ---> Y be a rational map from a subvariety X=V(I)⊆ℙn=Proj(K[x0,...,xn]) to a subvariety Y=V(J)⊆ℙm=Proj(K[y0,...,ym]). We assume that the map Φ can be represented, although not uniquely, by a homogeneous ring map φ:K[y0,...,ym]/J →K[x0,...,xn]/I of quotients of polynomial rings by homogeneous ideals. These kinds of ring maps, together with the objects of the RationalMap class, are the typical inputs for the methods in this package. The method toMap (resp. rationalMap) constructs such a ring map (resp. rational map) from a list of m+1 homogeneous elements of the same degree in K[x0,...,xn]/I.
Below is an example using the methods provided by this package, dealing with a birational transformation Φ:ℙ6 ---> G(2,4)⊂ℙ9 of bidegree (3,3).
i1 : ZZ/300007[t_0..t_6]; |
i2 : time phi = toMap minors(3,matrix{{t_0..t_4},{t_1..t_5},{t_2..t_6}})
-- used 0.00439695 seconds
ZZ ZZ 3 2 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 2 2
o2 = map(------[t , t , t , t , t , t , t ],------[x , x , x , x , x , x , x , x , x , x ],{- t + 2t t t - t t - t t + t t t , - t t + t t + t t t - t t t - t t + t t t , - t t + t t + t t t - t t - t t t + t t t , - t + 2t t t - t t - t t + t t t , - t t + t t t + t t t - t t t - t t + t t t , - t t t + t t + t t - t t t - t t t + t t t , - t t + t t + t t t - t t t - t t + t t t , - t t + t t t + t t t - t t - t t t + t t t , - t t + t t + t t t - t t - t t t + t t t , - t + 2t t t - t t - t t + t t t })
300007 0 1 2 3 4 5 6 300007 0 1 2 3 4 5 6 7 8 9 2 1 2 3 0 3 1 4 0 2 4 2 3 1 3 1 2 4 0 3 4 1 5 0 2 5 2 3 2 4 1 3 4 0 4 1 2 5 0 3 5 3 2 3 4 1 4 2 5 1 3 5 2 4 1 3 4 1 2 5 0 3 5 1 6 0 2 6 2 3 4 1 4 2 5 0 4 5 1 2 6 0 3 6 3 4 2 4 2 3 5 1 4 5 2 6 1 3 6 2 4 2 3 5 1 4 5 0 5 1 3 6 0 4 6 3 4 3 5 2 4 5 1 5 2 3 6 1 4 6 4 3 4 5 2 5 3 6 2 4 6
ZZ ZZ
o2 : RingMap ------[t , t , t , t , t , t , t ] <--- ------[x , x , x , x , x , x , x , x , x , x ]
300007 0 1 2 3 4 5 6 300007 0 1 2 3 4 5 6 7 8 9
|
i3 : time J = kernel(phi,2)
-- used 0.186348 seconds
o3 = ideal (x x - x x + x x , x x - x x + x x , x x - x x + x x , x x
6 7 5 8 4 9 3 7 2 8 1 9 3 5 2 6 0 9 3 4
------------------------------------------------------------------------
- x x + x x , x x - x x + x x )
1 6 0 8 2 4 1 5 0 7
ZZ
o3 : Ideal of ------[x , x , x , x , x , x , x , x , x , x ]
300007 0 1 2 3 4 5 6 7 8 9
|
i4 : time degreeOfRationalMap phi
-- used 0.0271019 seconds
o4 = 1
|
i5 : time projectiveDegrees phi
-- used 0.545305 seconds
o5 = {1, 3, 9, 17, 21, 15, 5}
o5 : List
|
i6 : time projectiveDegrees(phi,NumDegrees=>0)
-- used 0.138814 seconds
o6 = {5}
o6 : List
|
i7 : time phi = toMap(phi,Dominant=>J)
-- used 0.00384842 seconds
ZZ
------[x , x , x , x , x , x , x , x , x , x ]
ZZ 300007 0 1 2 3 4 5 6 7 8 9 3 2 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 2 2
o7 = map(------[t , t , t , t , t , t , t ],----------------------------------------------------------------------------------------------------,{- t + 2t t t - t t - t t + t t t , - t t + t t + t t t - t t t - t t + t t t , - t t + t t + t t t - t t - t t t + t t t , - t + 2t t t - t t - t t + t t t , - t t + t t t + t t t - t t t - t t + t t t , - t t t + t t + t t - t t t - t t t + t t t , - t t + t t + t t t - t t t - t t + t t t , - t t + t t t + t t t - t t - t t t + t t t , - t t + t t + t t t - t t - t t t + t t t , - t + 2t t t - t t - t t + t t t })
300007 0 1 2 3 4 5 6 (x x - x x + x x , x x - x x + x x , x x - x x + x x , x x - x x + x x , x x - x x + x x ) 2 1 2 3 0 3 1 4 0 2 4 2 3 1 3 1 2 4 0 3 4 1 5 0 2 5 2 3 2 4 1 3 4 0 4 1 2 5 0 3 5 3 2 3 4 1 4 2 5 1 3 5 2 4 1 3 4 1 2 5 0 3 5 1 6 0 2 6 2 3 4 1 4 2 5 0 4 5 1 2 6 0 3 6 3 4 2 4 2 3 5 1 4 5 2 6 1 3 6 2 4 2 3 5 1 4 5 0 5 1 3 6 0 4 6 3 4 3 5 2 4 5 1 5 2 3 6 1 4 6 4 3 4 5 2 5 3 6 2 4 6
6 7 5 8 4 9 3 7 2 8 1 9 3 5 2 6 0 9 3 4 1 6 0 8 2 4 1 5 0 7
ZZ
------[x , x , x , x , x , x , x , x , x , x ]
ZZ 300007 0 1 2 3 4 5 6 7 8 9
o7 : RingMap ------[t , t , t , t , t , t , t ] <--- ----------------------------------------------------------------------------------------------------
300007 0 1 2 3 4 5 6 (x x - x x + x x , x x - x x + x x , x x - x x + x x , x x - x x + x x , x x - x x + x x )
6 7 5 8 4 9 3 7 2 8 1 9 3 5 2 6 0 9 3 4 1 6 0 8 2 4 1 5 0 7
|
i8 : time psi = inverseMap phi
-- used 0.83545 seconds
ZZ
------[x , x , x , x , x , x , x , x , x , x ]
300007 0 1 2 3 4 5 6 7 8 9 ZZ 3 2 2 2 2 2 2 2 2 2 2 2 3 2 2 2 2 2 2 3 2 2 2 2 2
o8 = map(----------------------------------------------------------------------------------------------------,------[t , t , t , t , t , t , t ],{x - 2x x x + x x - x x x + x x + x x + x x x - x x x + x x - 2x x x - x x x - 2x x , x x - x x - x x x + x x x + x x x + x x - 2x x x - x x x + x x x , x x - x x x + x x - x x x + x x - x x x - x x x , x - x x x + x x x + x x x - 2x x x - x x x , x x - x x x + x x + x x - x x x - x x x - x x x , x x - x x - x x x + x x + x x x + x x x - 2x x x - x x x + x x x , x - 2x x x - x x x + x x + x x + x x + x x + x x x - 2x x x - x x x - x x x - 2x x })
(x x - x x + x x , x x - x x + x x , x x - x x + x x , x x - x x + x x , x x - x x + x x ) 300007 0 1 2 3 4 5 6 2 1 2 3 0 3 1 2 5 0 5 1 6 0 2 6 0 4 6 1 7 0 2 7 0 4 7 0 9 2 3 1 3 1 2 6 0 3 6 0 5 6 1 8 0 2 8 0 4 8 0 1 9 2 3 1 3 6 0 6 0 3 8 1 9 0 2 9 0 4 9 3 1 3 8 0 6 8 1 2 9 0 3 9 0 5 9 3 6 2 3 8 0 8 2 9 1 3 9 0 6 9 0 7 9 3 6 3 8 2 6 8 1 8 2 3 9 2 5 9 1 6 9 1 7 9 0 8 9 6 3 6 8 5 6 8 2 8 4 8 3 9 5 9 2 6 9 4 6 9 2 7 9 4 7 9 0 9
6 7 5 8 4 9 3 7 2 8 1 9 3 5 2 6 0 9 3 4 1 6 0 8 2 4 1 5 0 7
ZZ
------[x , x , x , x , x , x , x , x , x , x ]
300007 0 1 2 3 4 5 6 7 8 9 ZZ
o8 : RingMap ---------------------------------------------------------------------------------------------------- <--- ------[t , t , t , t , t , t , t ]
(x x - x x + x x , x x - x x + x x , x x - x x + x x , x x - x x + x x , x x - x x + x x ) 300007 0 1 2 3 4 5 6
6 7 5 8 4 9 3 7 2 8 1 9 3 5 2 6 0 9 3 4 1 6 0 8 2 4 1 5 0 7
|
i9 : time isInverseMap(phi,psi)
-- used 0.00965082 seconds
o9 = true
|
i10 : time degreeOfRationalMap psi
-- used 0.0630312 seconds
o10 = 1
|
i11 : time projectiveDegrees psi
-- used 1.68195 seconds
o11 = {5, 15, 21, 17, 9, 3, 1}
o11 : List
|
We repeat the example using the type RationalMap and using deterministic methods.
i12 : time phi = rationalMap minors(3,matrix{{t_0..t_4},{t_1..t_5},{t_2..t_6}})
-- used 0.00451608 seconds
o12 = -- rational map --
ZZ
source: Proj(------[t , t , t , t , t , t , t ])
300007 0 1 2 3 4 5 6
ZZ
target: Proj(------[x , x , x , x , x , x , x , x , x , x ])
300007 0 1 2 3 4 5 6 7 8 9
defining forms: {
3 2 2
- t + 2t t t - t t - t t + t t t ,
2 1 2 3 0 3 1 4 0 2 4
2 2 2
- t t + t t + t t t - t t t - t t + t t t ,
2 3 1 3 1 2 4 0 3 4 1 5 0 2 5
2 2 2
- t t + t t + t t t - t t - t t t + t t t ,
2 3 2 4 1 3 4 0 4 1 2 5 0 3 5
3 2 2
- t + 2t t t - t t - t t + t t t ,
3 2 3 4 1 4 2 5 1 3 5
2 2
- t t + t t t + t t t - t t t - t t + t t t ,
2 4 1 3 4 1 2 5 0 3 5 1 6 0 2 6
2 2
- t t t + t t + t t - t t t - t t t + t t t ,
2 3 4 1 4 2 5 0 4 5 1 2 6 0 3 6
2 2 2
- t t + t t + t t t - t t t - t t + t t t ,
3 4 2 4 2 3 5 1 4 5 2 6 1 3 6
2 2
- t t + t t t + t t t - t t - t t t + t t t ,
2 4 2 3 5 1 4 5 0 5 1 3 6 0 4 6
2 2 2
- t t + t t + t t t - t t - t t t + t t t ,
3 4 3 5 2 4 5 1 5 2 3 6 1 4 6
3 2 2
- t + 2t t t - t t - t t + t t t
4 3 4 5 2 5 3 6 2 4 6
}
o12 : RationalMap (cubic rational map from PP^6 to PP^9)
|
i13 : time phi = rationalMap(phi,Dominant=>2)
-- used 0.28005 seconds
o13 = -- rational map --
ZZ
source: Proj(------[t , t , t , t , t , t , t ])
300007 0 1 2 3 4 5 6
ZZ
target: subvariety of Proj(------[x , x , x , x , x , x , x , x , x , x ]) defined by
300007 0 1 2 3 4 5 6 7 8 9
{
x x - x x + x x ,
6 7 5 8 4 9
x x - x x + x x ,
3 7 2 8 1 9
x x - x x + x x ,
3 5 2 6 0 9
x x - x x + x x ,
3 4 1 6 0 8
x x - x x + x x
2 4 1 5 0 7
}
defining forms: {
3 2 2
- t + 2t t t - t t - t t + t t t ,
2 1 2 3 0 3 1 4 0 2 4
2 2 2
- t t + t t + t t t - t t t - t t + t t t ,
2 3 1 3 1 2 4 0 3 4 1 5 0 2 5
2 2 2
- t t + t t + t t t - t t - t t t + t t t ,
2 3 2 4 1 3 4 0 4 1 2 5 0 3 5
3 2 2
- t + 2t t t - t t - t t + t t t ,
3 2 3 4 1 4 2 5 1 3 5
2 2
- t t + t t t + t t t - t t t - t t + t t t ,
2 4 1 3 4 1 2 5 0 3 5 1 6 0 2 6
2 2
- t t t + t t + t t - t t t - t t t + t t t ,
2 3 4 1 4 2 5 0 4 5 1 2 6 0 3 6
2 2 2
- t t + t t + t t t - t t t - t t + t t t ,
3 4 2 4 2 3 5 1 4 5 2 6 1 3 6
2 2
- t t + t t t + t t t - t t - t t t + t t t ,
2 4 2 3 5 1 4 5 0 5 1 3 6 0 4 6
2 2 2
- t t + t t + t t t - t t - t t t + t t t ,
3 4 3 5 2 4 5 1 5 2 3 6 1 4 6
3 2 2
- t + 2t t t - t t - t t + t t t
4 3 4 5 2 5 3 6 2 4 6
}
o13 : RationalMap (cubic rational map from PP^6 to 6-dimensional subvariety of PP^9)
|
i14 : time phi^(-1)
-- used 0.798784 seconds
o14 = -- rational map --
ZZ
source: subvariety of Proj(------[x , x , x , x , x , x , x , x , x , x ]) defined by
300007 0 1 2 3 4 5 6 7 8 9
{
x x - x x + x x ,
6 7 5 8 4 9
x x - x x + x x ,
3 7 2 8 1 9
x x - x x + x x ,
3 5 2 6 0 9
x x - x x + x x ,
3 4 1 6 0 8
x x - x x + x x
2 4 1 5 0 7
}
ZZ
target: Proj(------[t , t , t , t , t , t , t ])
300007 0 1 2 3 4 5 6
defining forms: {
3 2 2 2 2 2
x - 2x x x + x x - x x x + x x + x x + x x x - x x x + x x - 2x x x - x x x - 2x x ,
2 1 2 3 0 3 1 2 5 0 5 1 6 0 2 6 0 4 6 1 7 0 2 7 0 4 7 0 9
2 2 2
x x - x x - x x x + x x x + x x x + x x - 2x x x - x x x + x x x ,
2 3 1 3 1 2 6 0 3 6 0 5 6 1 8 0 2 8 0 4 8 0 1 9
2 2 2
x x - x x x + x x - x x x + x x - x x x - x x x ,
2 3 1 3 6 0 6 0 3 8 1 9 0 2 9 0 4 9
3
x - x x x + x x x + x x x - 2x x x - x x x ,
3 1 3 8 0 6 8 1 2 9 0 3 9 0 5 9
2 2 2
x x - x x x + x x + x x - x x x - x x x - x x x ,
3 6 2 3 8 0 8 2 9 1 3 9 0 6 9 0 7 9
2 2 2
x x - x x - x x x + x x + x x x + x x x - 2x x x - x x x + x x x ,
3 6 3 8 2 6 8 1 8 2 3 9 2 5 9 1 6 9 1 7 9 0 8 9
3 2 2 2 2 2
x - 2x x x - x x x + x x + x x + x x + x x + x x x - 2x x x - x x x - x x x - 2x x
6 3 6 8 5 6 8 2 8 4 8 3 9 5 9 2 6 9 4 6 9 2 7 9 4 7 9 0 9
}
o14 : RationalMap (cubic birational map from 6-dimensional subvariety of PP^9 to PP^6)
|
i15 : time degrees phi^(-1)
-- used 0.339945 seconds
o15 = {5, 15, 21, 17, 9, 3, 1}
o15 : List
|
i16 : time degrees phi
-- used 0.000146842 seconds
o16 = {1, 3, 9, 17, 21, 15, 5}
o16 : List
|
i17 : time describe phi
-- used 0.00301943 seconds
o17 = rational map defined by forms of degree 3
source variety: PP^6
target variety: 6-dimensional variety of degree 5 in PP^9 cut out by 5 hypersurfaces of degree 2
dominance: true
birationality: true (the inverse map is known)
projective degrees: {1, 3, 9, 17, 21, 15, 5}
coefficient ring: ZZ/300007
|
i18 : time describe phi^(-1)
-- used 0.0127567 seconds
o18 = rational map defined by forms of degree 3
source variety: 6-dimensional variety of degree 5 in PP^9 cut out by 5 hypersurfaces of degree 2
target variety: PP^6
dominance: true
birationality: true (the inverse map is known)
projective degrees: {5, 15, 21, 17, 9, 3, 1}
number of minimal representatives: 1
dimension base locus: 4
degree base locus: 24
coefficient ring: ZZ/300007
|
i19 : time (f,g) = graph phi^-1; f;
-- used 0.0177277 seconds
o20 : MultihomogeneousRationalMap (birational map from 6-dimensional subvariety of PP^9 x PP^6 to 6-dimensional subvariety of PP^9)
|
i21 : time degrees f
-- used 2.04709 seconds
o21 = {904, 508, 268, 130, 56, 20, 5}
o21 : List
|
i22 : time degree f
-- used 0.000023432 seconds
o22 = 1
|
i23 : time describe f
-- used 0.00171776 seconds
o23 = rational map defined by multiforms of degree {1, 0}
source variety: 6-dimensional subvariety of PP^9 x PP^6 cut out by 20 hypersurfaces of degrees ({1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{2, 0},{2, 0},{2, 0},{2, 0},{2, 0})
target variety: 6-dimensional variety of degree 5 in PP^9 cut out by 5 hypersurfaces of degree 2
dominance: true
birationality: true
projective degrees: {904, 508, 268, 130, 56, 20, 5}
coefficient ring: ZZ/300007
|
A rudimentary version of Cremona has been already used in an essential way in the paper doi:10.1016/j.jsc.2015.11.004 (it was originally named bir.m2).
Author
- Giovanni Staglianò <giovannistagliano@gmail.com>
Certification 
Version 4.2.2 of this package was accepted for publication in volume 8 of the journal The Journal of Software for Algebra and Geometry on 11 June 2018, in the article A Macaulay2 package for computations with rational maps. That version can be obtained from the journal or from the Macaulay2 source code repository, http://github.com/Macaulay2/M2/blob/master/M2/Macaulay2/packages/Cremona.m2, commit number 2e87a29e4b5b68af1bd8917a9c76d4008ff9fc5b.
Version
This documentation describes version 4.3 of Cremona.
Source code
The source code from which this documentation is derived is in the file Cremona.m2. The auxiliary files accompanying it are in the directory Cremona/.
Exports
-
Types
- RationalMap -- the class of all rational maps between absolutely irreducible projective varieties over a field
-
Functions and commands
- abstractRationalMap -- make an abstract rational map
- approximateInverseMap -- random map related to the inverse of a birational map
- ChernSchwartzMacPherson -- Chern-Schwartz-MacPherson class of a projective scheme
- degreeOfRationalMap -- degree of a rational map between projective varieties
- EulerCharacteristic -- topological Euler characteristic of a (smooth) projective variety
- exceptionalLocus -- exceptional locus of a birational map
- forceImage -- declare which is the image of a rational map
- forceInverseMap -- declare that two rational maps are one the inverse of the other
- graph -- closure of the graph of a rational map
- inverseMap -- inverse of a birational map
- isBirational -- whether a rational map is birational
- isDominant -- whether a rational map is dominant
- isInverseMap -- checks whether a rational map is the inverse of another
- isMorphism -- whether a rational map is a morphism
- parametrize -- parametrization of linear varieties and hyperquadrics
- point -- pick a random rational point on a projective variety
- projectiveDegrees -- projective degrees of a rational map between projective varieties
- quadroQuadricCremonaTransformation -- quadro-quadric Cremona transformations
- rationalMap -- makes a rational map
- segre -- Segre embedding
- SegreClass -- Segre class of a closed subscheme of a projective variety
- specialCremonaTransformation -- special Cremona transformations whose base locus has dimension at most three
- specialCubicTransformation -- special cubic transformations whose base locus has dimension at most three
- specialQuadraticTransformation -- special quadratic transformations whose base locus has dimension three
- toMap -- rational map defined by a linear system
-
Methods
- abstractRationalMap(RationalMap), see abstractRationalMap -- make an abstract rational map
- approximateInverseMap(RationalMap), see approximateInverseMap -- random map related to the inverse of a birational map
- approximateInverseMap(RationalMap,ZZ), see approximateInverseMap -- random map related to the inverse of a birational map
- coefficientRing(RationalMap) -- coefficient ring of a rational map
- coefficients(RationalMap) -- coefficient matrix of a rational map
- degree(RationalMap) -- degree of a rational map
- degreeOfRationalMap(RationalMap) -- degree of a rational map
- degrees(RationalMap) -- projective degrees of a rational map
- multidegree(RationalMap), see degrees(RationalMap) -- projective degrees of a rational map
- entries(RationalMap) -- the entries of the matrix associated to a rational map
- exceptionalLocus(RationalMap), see exceptionalLocus -- exceptional locus of a birational map
- flatten(RationalMap) -- write source and target as nondegenerate varieties
- forceImage(RationalMap,Ideal), see forceImage -- declare which is the image of a rational map
- forceInverseMap(RationalMap,RationalMap), see forceInverseMap -- declare that two rational maps are one the inverse of the other
- graph(RationalMap), see graph -- closure of the graph of a rational map
- ideal(RationalMap) -- base locus of a rational map
- image(RationalMap), see image(RationalMap,ZZ) -- closure of the image of a rational map
- image(RationalMap,ZZ) -- closure of the image of a rational map
- inverse(RationalMap) -- inverse of a birational map
- inverseMap(RationalMap), see inverseMap -- inverse of a birational map
- isBirational(RationalMap), see isBirational -- whether a rational map is birational
- isDominant(RationalMap), see isDominant -- whether a rational map is dominant
- isInverseMap(RationalMap,RationalMap) -- checks whether two rational maps are one the inverse of the other
- isIsomorphism(RationalMap) -- whether a birational map is an isomorphism
- isMorphism(RationalMap), see isMorphism -- whether a rational map is a morphism
- map(RationalMap) -- get the ring map defining a rational map
- matrix(RationalMap) -- the matrix associated to a rational map
- projectiveDegrees(RationalMap) -- projective degrees of a rational map
- rationalMap(RationalMap), see rationalMap -- makes a rational map
- RationalMap ! -- calculates every possible thing
- compose(RationalMap,RationalMap), see RationalMap * RationalMap -- composition of rational maps
- RationalMap * RationalMap -- composition of rational maps
- RationalMap == RationalMap -- equality of rational maps
- RationalMap == ZZ, see RationalMap == RationalMap -- equality of rational maps
- ZZ == RationalMap, see RationalMap == RationalMap -- equality of rational maps
- RationalMap ^ ZZ -- power
- RationalMap ^*, see RationalMap ^** Ideal -- inverse image via a rational map
- RationalMap ^** Ideal -- inverse image via a rational map
- RationalMap _* -- direct image via a rational map
- RationalMap Ideal, see RationalMap _* -- direct image via a rational map
- RationalMap | Ideal -- restriction of a rational map
- RationalMap | Ring, see RationalMap | Ideal -- restriction of a rational map
- RationalMap | RingElement, see RationalMap | Ideal -- restriction of a rational map
- RationalMap || Ideal -- restriction of a rational map
- RationalMap || Ring, see RationalMap || Ideal -- restriction of a rational map
- RationalMap || RingElement, see RationalMap || Ideal -- restriction of a rational map
- segre(RationalMap), see segre -- Segre embedding
- SegreClass(RationalMap), see SegreClass -- Segre class of a closed subscheme of a projective variety
- source(RationalMap) -- coordinate ring of the source for a rational map
- target(RationalMap) -- coordinate ring of the target for a rational map
- toExternalString(RationalMap) -- convert to a readable string
-
Symbols
- BlowUpStrategy
- CodimBsInv
- Dominant
- MathMode -- whether to ensure correctness of output
- NumDegrees