-- -*- M2-comint -*- {* hash: -1055676355 *}
i1 : R = ZZ/32003[a,b,c,d]
o1 = R
o1 : PolynomialRing
i2 : X = ideal(a^3+b^3+c^3+d^3)
3 3 3 3
o2 = ideal(a + b + c + d )
o2 : Ideal of R
i3 : KK = coefficientRing R
o3 = KK
o3 : QuotientRing
i4 : S = KK [s,t,p_0..p_3,q_0..q_3]
o4 = S
o4 : PolynomialRing
i5 : F = map(S,R,
s*matrix{{p_0..p_3}} +
t*matrix{{q_0..q_3}}
)
o5 = map(S,R,{s*p + t*q , s*p + t*q , s*p + t*q , s*p + t*q })
0 0 1 1 2 2 3 3
o5 : RingMap S <--- R
i6 : FX = F X
3 3 3 3 3 3 3 3 2 2 2 2 3 3
o6 = ideal(s p + s p + s p + s p + 3s t*p q + 3s*t p q + t q +
0 1 2 3 0 0 0 0 0
------------------------------------------------------------------------
2 2 2 2 3 3 2 2 2 2 3 3 2 2
3s t*p q + 3s*t p q + t q + 3s t*p q + 3s*t p q + t q + 3s t*p q
1 1 1 1 1 2 2 2 2 2 3 3
------------------------------------------------------------------------
2 2 3 3
+ 3s*t p q + t q )
3 3 3
o6 : Ideal of S
i7 : cFX = last coefficients(gens FX, Variables => {s,t})
o7 = {3} | p_0^3+p_1^3+p_2^3+p_3^3 |
{3} | 3p_0^2q_0+3p_1^2q_1+3p_2^2q_2+3p_3^2q_3 |
{3} | 3p_0q_0^2+3p_1q_1^2+3p_2q_2^2+3p_3q_3^2 |
{3} | q_0^3+q_1^3+q_2^3+q_3^3 |
4 1
o7 : Matrix S <--- S
i8 : S1 = KK[p_0..p_3,q_0..q_3]
o8 = S1
o8 : PolynomialRing
i9 : cFX = substitute(cFX, S1)
o9 = {3} | p_0^3+p_1^3+p_2^3+p_3^3 |
{3} | 3p_0^2q_0+3p_1^2q_1+3p_2^2q_2+3p_3^2q_3 |
{3} | 3p_0q_0^2+3p_1q_1^2+3p_2q_2^2+3p_3q_3^2 |
{3} | q_0^3+q_1^3+q_2^3+q_3^3 |
4 1
o9 : Matrix S1 <--- S1
i10 : S1bar = S1/ideal cFX
o10 = S1bar
o10 : QuotientRing
i11 : GR = coefficientRing R[x_0..x_5]
o11 = GR
o11 : PolynomialRing
i12 : M = substitute(
exteriorPower(2, matrix{{p_0..p_3},{q_0..q_3}}),
S1bar)
o12 = | -p_1q_0+p_0q_1 -p_2q_0+p_0q_2 -p_2q_1+p_1q_2 -p_3q_0+p_0q_3
-----------------------------------------------------------------------
-p_3q_1+p_1q_3 -p_3q_2+p_2q_3 |
1 6
o12 : Matrix S1bar <--- S1bar
i13 : gr = map (S1bar, GR, M)
o13 = map(S1bar,GR,{- p q + p q , - p q + p q , - p q + p q , - p q + p q , - p q + p q , - p q + p q })
1 0 0 1 2 0 0 2 2 1 1 2 3 0 0 3 3 1 1 3 3 2 2 3
o13 : RingMap S1bar <--- GR
i14 : fano = trim ker gr
2 2 2
o14 = ideal (x x - x x + x x , x x x , x x x , x x + x x , x x x , x x +
2 3 1 4 0 5 3 4 5 1 2 5 0 4 1 5 0 2 4 0 4
-----------------------------------------------------------------------
2 3 3 3 2 2 2 2 2 2 2
x x , x + x + x , x x + x x , x x - x x , x x + x x , x x x , x x
1 5 3 4 5 1 3 2 4 0 3 2 5 1 3 2 4 0 1 3 0 3
-----------------------------------------------------------------------
2 2 2 2 2 2 2 2 2 3 3
+ x x , x x + x x , x x + x x , x x + x x , x x - x x , x + x -
2 5 1 2 3 4 0 2 3 5 1 2 3 4 0 2 3 5 1 2
-----------------------------------------------------------------------
3 2 2 2 2 3 3 3
x , x x - x x , x x - x x , x - x - x )
5 0 1 4 5 0 1 4 5 0 2 4
o14 : Ideal of GR
i15 : codim fano
o15 = 5
i16 : degree fano
o16 = 27
i17 : betti fano
0 1
o17 = total: 1 20
0: 1 .
1: . 1
2: . 19
o17 : BettiTally
i18 : needsPackage "Text"
o18 = Text
o18 : Package
i19 : document {
Key => Fano2,
TT "Fano2(k,X,GR) or Fano2(k,X)", " -- computes
the ideal of a Fano scheme in the Grassmannian.",
PARA{},
"Given an ideal X representing a projective variety
in P^r, a positive integer k<r, and optionally a
ring GR with (exactly) r+1 choose k+1 variables,
representing the ambient space of the Grassmannian of
k-planes in P^r, this routine returns the ideal in
GR of the Fano scheme that parametrizes the k-planes
lying on X. If the optional third argument is not
present, the routine fabricates its own local ring,
and returns an ideal over it."
}
i20 : document {
Key => symbol Grassmannian2,
TT "Grassmannian2(k,r,R) or
Grassmannian2(k,r)",
"-- Given natural numbers k <= r,
and optionally a ring R with at least binomial(r+1,k+1)
variables, the routine defines the ideal of the
Grassmannian of projective k-planes in P^r, using
the first binomial(r+1,k+1) variables of R.
If R is not given, the routine makes and uses
ZZ/31991[vars(0..binomial(r+1,k+1)-1]."
}
i21 : Fano2 = method()
o21 = Fano2
o21 : MethodFunction
i22 : Fano2(ZZ,Ideal,Ring) := (k,X,GR) -> (
-- Get info about the base ring of X:
-- The coefficient ring (to make new rings of
-- the same characteristic, for example)
-- and the number of variables
KK:=coefficientRing ring X;
r := (numgens ring X) - 1;
-- Next make private variables for our
-- intermediate rings, to avoid interfering
-- with something outside:
t:=symbol t;
p:=symbol p;
-- And rings
S1 := KK[t_0..t_k];
S2 := KK[p_0..p_(k*r+k+r)];
S := tensor(S1,S2);
-- Over S we have a generic point of a generic
-- line, represented by a row vector, which
-- we use to define a map from the base ring
-- of X
F := map(S,ring X,
genericMatrix(S,S_0,1,k+1)*
genericMatrix(S,S_(k+1),k+1,r+1)
);
-- We now apply F to the ideal of X
FX := F X;
-- and the condition we want becomes the condition
-- that FX vanishes identically in the t_i.
-- The following line produces the matrix of
-- coefficients of the monomials in the
-- variables labelled 0..k:
cFX := last coefficients (gens FX, Variables => toList apply(0..k, i -> S_i));
-- We can get rid of the variables t_i
-- to ease the computation:
cFX = substitute(cFX, S2);
-- The ring we want is the quotient
S2bar := S2/ideal cFX;
-- Now we want to move to the Grassmannian,
-- represented by the ring GR
-- We define a map sending the variables of GR
-- to the minors of the generic matrix in the
-- p_i regarded as elements of S1bar
gr := map(S2bar,GR,
exteriorPower(k+1,
genericMatrix(S2bar,S2bar_0,k+1,r+1)
)
);
-- and the defining ideal of the Fano variety is
ker gr
)
o22 = {*Function[stdio:52:33-100:6]*}
o22 : FunctionClosure
i23 : Fano2(ZZ, Ideal) := (k,X) -> (
KK:=coefficientRing ring X;
r := (numgens ring X) - 1;
-- We can specify a private ring with binomial(r+1,k+1)
-- variables as follows
GR := KK[Variables => binomial(r+1,k+1)];
-- the work is done by
Fano2(k,X,GR)
)
o23 = {*Function[stdio:102:26-109:12]*}
o23 : FunctionClosure
i24 : Grassmannian2 = method()
o24 = Grassmannian2
o24 : MethodFunction
i25 : Grassmannian2(ZZ,ZZ,Ring) := (k,r,R) ->(
KK := coefficientRing R;
RPr := KK[Variables => r+1];
Pr := ideal(0_RPr);
Fano2(k,Pr)
)
o25 = {*Function[stdio:112:37-116:16]*}
o25 : FunctionClosure
i26 : Grassmannian2(ZZ,ZZ) := (r,k) -> (
R := ZZ/31991[
vars(0..(binomial(r+1,k+1)-1))
];
Grassmannian2(k,r,R)
)
o26 = {*Function[stdio:118:30-122:26]*}
o26 : FunctionClosure
i27 : KK = ZZ/31991
o27 = KK
o27 : QuotientRing
i28 : R = KK[a,b,c,d]
o28 = R
o28 : PolynomialRing
i29 : X = ideal(a*b-c*d)
o29 = ideal(a*b - c*d)
o29 : Ideal of R
i30 : I = Fano2(1,X)
2
o30 = ideal (p p , p p , p p + 15995p p - 15995p , p p + p p , p p -
3 4 2 4 1 4 0 5 5 0 4 4 5 2 3
-----------------------------------------------------------------------
2
15995p p - 15995p , p p , p p - p p , p p , p p - p p , p p + p p ,
0 5 5 1 3 0 3 3 5 1 2 0 2 2 5 0 1 1 5
-----------------------------------------------------------------------
2 2
p - p )
0 5
o30 : Ideal of KK[p , p , p , p , p , p ]
0 1 2 3 4 5
i31 : dim I
o31 = 2
i32 : degree I
o32 = 4
i33 : KK = ZZ/31991
o33 = KK
o33 : QuotientRing
i34 : P5 = KK[a..f]
o34 = P5
o34 : PolynomialRing
i35 : MVero = genericSymmetricMatrix(P5,a,3)
o35 = | a b c |
| b d e |
| c e f |
3 3
o35 : Matrix P5 <--- P5
i36 : Vero = minors(2,MVero)
2 2
o36 = ideal (- b + a*d, - b*c + a*e, - c*d + b*e, - b*c + a*e, - c + a*f, -
-----------------------------------------------------------------------
2
c*e + b*f, - c*d + b*e, - c*e + b*f, - e + d*f)
o36 : Ideal of P5
i37 : catalecticant = (R,v,m,n) ->
map(R^m,n,(i,j)-> R_(i+j+v))
o37 = catalecticant
o37 : FunctionClosure
i38 : catalecticant(P5,1,2,4)
o38 = | b c d e |
| c d e f |
2 4
o38 : Matrix P5 <--- P5
i39 : M13 = catalecticant(P5,0,2,1) |
catalecticant(P5,2,2,3)
o39 = | a c d e |
| b d e f |
2 4
o39 : Matrix P5 <--- P5
i40 : S13 = minors(2,M13)
2
o40 = ideal (- b*c + a*d, - b*d + a*e, - d + c*e, - b*e + a*f, - d*e + c*f,
-----------------------------------------------------------------------
2
- e + d*f)
o40 : Ideal of P5
i41 : M22 = catalecticant(P5,0,2,2) | catalecticant(P5,3,2,2)
o41 = | a b d e |
| b c e f |
2 4
o41 : Matrix P5 <--- P5
i42 : S22 = minors(2, M22)
2
o42 = ideal (- b + a*c, - b*d + a*e, - c*d + b*e, - b*e + a*f, - c*e + b*f,
-----------------------------------------------------------------------
2
- e + d*f)
o42 : Ideal of P5
i43 : Verores = res coker gens Vero
1 6 8 3
o43 = P5 <-- P5 <-- P5 <-- P5 <-- 0
0 1 2 3 4
o43 : ChainComplex
i44 : S22res = res coker gens S22
1 6 8 3
o44 = P5 <-- P5 <-- P5 <-- P5 <-- 0
0 1 2 3 4
o44 : ChainComplex
i45 : S13res = res coker gens S13
1 6 8 3
o45 = P5 <-- P5 <-- P5 <-- P5 <-- 0
0 1 2 3 4
o45 : ChainComplex
i46 : betti Verores
0 1 2 3
o46 = total: 1 6 8 3
0: 1 . . .
1: . 6 8 3
o46 : BettiTally
i47 : betti S22res
0 1 2 3
o47 = total: 1 6 8 3
0: 1 . . .
1: . 6 8 3
o47 : BettiTally
i48 : betti S13res
0 1 2 3
o48 = total: 1 6 8 3
0: 1 . . .
1: . 6 8 3
o48 : BettiTally
i49 : FVero = Fano2(1, Vero)
2
o49 = ideal (p , p p , p p , p p , p p , p p , p p , p p , p p ,
14 13 14 12 14 11 14 10 14 9 14 8 14 7 14 6 14
-----------------------------------------------------------------------
2
p p , p p , p p , p p , p p , p p , p , p p , p p , p p ,
5 14 4 14 3 14 2 14 1 14 0 14 13 12 13 11 13 10 13
-----------------------------------------------------------------------
p p , p p , p p , p p , p p , p p , p p , p p , p p , p p ,
9 13 8 13 7 13 6 13 5 13 4 13 3 13 2 13 1 13 0 13
-----------------------------------------------------------------------
2
p , p p , p p , p p , p p , p p , p p , p p , p p , p p ,
12 11 12 10 12 9 12 8 12 7 12 6 12 5 12 4 12 3 12
-----------------------------------------------------------------------
2
p p , p p , p p , p , p p , p p , p p , p p , p p , p p ,
2 12 1 12 0 12 11 10 11 9 11 8 11 7 11 6 11 5 11
-----------------------------------------------------------------------
2
p p , p p , p p , p p , p p , p , p p , p p , p p , p p ,
4 11 3 11 2 11 1 11 0 11 10 9 10 8 10 7 10 6 10
-----------------------------------------------------------------------
2
p p , p p , p p , p p , p p , p p , p , p p , p p , p p , p p ,
5 10 4 10 3 10 2 10 1 10 0 10 9 8 9 7 9 6 9 5 9
-----------------------------------------------------------------------
2
p p , p p , p p , p p , p p , p , p p , p p , p p , p p , p p , p p ,
4 9 3 9 2 9 1 9 0 9 8 7 8 6 8 5 8 4 8 3 8 2 8
-----------------------------------------------------------------------
2 2
p p , p p , p , p p , p p , p p , p p , p p , p p , p p , p , p p ,
1 8 0 8 7 6 7 5 7 4 7 3 7 2 7 1 7 0 7 6 5 6
-----------------------------------------------------------------------
2 2
p p , p p , p p , p p , p p , p , p p , p p , p p , p p , p p , p ,
4 6 3 6 2 6 1 6 0 6 5 4 5 3 5 2 5 1 5 0 5 4
-----------------------------------------------------------------------
2 2 2
p p , p p , p p , p p , p , p p , p p , p p , p , p p , p p , p , p p ,
3 4 2 4 1 4 0 4 3 2 3 1 3 0 3 2 1 2 0 2 1 0 1
-----------------------------------------------------------------------
2
p )
0
o49 : Ideal of KK[p , p , p , p , p , p , p , p , p , p , p , p , p , p , p ]
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
i50 : betti gens FVero
0 1
o50 = total: 1 120
0: 1 .
1: . 120
o50 : BettiTally
i51 : FS13 = Fano2(1, S13)
2
o51 = ideal (p , p p , p p , p p , p p , p p , p p , p p , p p ,
14 13 14 12 14 11 14 10 14 9 14 8 14 7 14 6 14
-----------------------------------------------------------------------
2
p p , p p , p p , p p , p p , p p , p , p p , p p , p p ,
5 14 4 14 3 14 2 14 1 14 0 14 13 12 13 11 13 10 13
-----------------------------------------------------------------------
p p , p p , p p , p p , p p , p p , p p , p p , p p , p p ,
9 13 8 13 7 13 6 13 5 13 4 13 3 13 2 13 1 13 0 13
-----------------------------------------------------------------------
2
p , p p , p p , p p , p p , p p , p p , p p , p p , p p ,
12 11 12 10 12 9 12 8 12 7 12 6 12 5 12 4 12 3 12
-----------------------------------------------------------------------
p p , p p , p p , p p , p p , p p - p p , p p , p p -
2 12 1 12 0 12 9 11 8 11 7 11 10 11 5 11 4 11
-----------------------------------------------------------------------
2
p p , p p - p p , p p , p - p p , p p , p p , p p - p p ,
6 11 2 11 3 11 0 11 10 6 11 9 10 8 10 7 10 6 11
-----------------------------------------------------------------------
p p - p p , p p , p p - p p , p p - p p , p p - p p ,
6 10 3 11 5 10 4 10 3 11 3 10 1 11 2 10 1 11
-----------------------------------------------------------------------
2 2
p p , p , p p , p p , p p , p p , p p , p p , p p , p p , p p , p ,
0 10 9 8 9 7 9 6 9 5 9 4 9 3 9 2 9 1 9 0 9 8
-----------------------------------------------------------------------
2
p p , p p , p p , p p , p p , p p , p p , p p , p - p p , p p -
7 8 6 8 5 8 4 8 3 8 2 8 1 8 0 8 7 6 11 6 7
-----------------------------------------------------------------------
p p , p p , p p - p p , p p - p p , p p - p p , p p - p p ,
3 11 5 7 4 7 3 11 3 7 1 11 2 7 1 11 1 7 1 10
-----------------------------------------------------------------------
2
p p , p - p p , p p , p p - p p , p p - p p , p p - p p , p p ,
0 7 6 1 11 5 6 4 6 1 11 3 6 1 10 2 6 1 10 0 6
-----------------------------------------------------------------------
2 2
p , p p , p p , p p , p p , p p , p - p p , p p - p p , p p -
5 4 5 3 5 2 5 1 5 0 5 4 1 11 3 4 1 10 2 4
-----------------------------------------------------------------------
2 2
p p , p p - p p , p p , p - p p , p p - p p , p p , p - p p , p p
1 10 1 4 1 6 0 4 3 1 6 2 3 1 6 0 3 2 1 6 1 2
-----------------------------------------------------------------------
- p p , p p , p p )
1 3 0 2 0 1
o51 : Ideal of KK[p , p , p , p , p , p , p , p , p , p , p , p , p , p , p ]
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
i52 : hilbertPolynomial coker gens FS13
o52 = - 2*P + 4*P
0 1
o52 : ProjectiveHilbertPolynomial
i53 : FS22 = Fano2(1, S22)
2
o53 = ideal (p , p p , p p , p p , p p , p p , p p , p p , p p ,
14 13 14 12 14 11 14 10 14 9 14 8 14 7 14 6 14
-----------------------------------------------------------------------
2
p p , p p , p p , p p , p p , p p , p , p p , p p , p p ,
5 14 4 14 3 14 2 14 1 14 0 14 13 12 13 11 13 10 13
-----------------------------------------------------------------------
p p , p p , p p , p p , p p , p p , p p , p p , p p , p p ,
9 13 8 13 7 13 6 13 5 13 4 13 3 13 2 13 1 13 0 13
-----------------------------------------------------------------------
p p , p p - p p , p p - p p , p p - p p , p p - p p ,
9 12 8 12 11 12 7 12 10 12 5 12 10 12 4 12 6 12
-----------------------------------------------------------------------
2
p p , p p , p p , p - p p , p p - p p , p p , p p -
2 12 1 12 0 12 11 10 12 10 11 6 12 9 11 8 11
-----------------------------------------------------------------------
p p , p p - p p , p p - p p , p p - p p , p p - p p ,
10 12 7 11 6 12 6 11 3 12 5 11 6 12 4 11 3 12
-----------------------------------------------------------------------
2
p p , p p , p p , p - p p , p p , p p - p p , p p - p p ,
2 11 1 11 0 11 10 3 12 9 10 8 10 6 12 7 10 3 12
-----------------------------------------------------------------------
2
p p - p p , p p - p p , p p - p p , p p , p p , p p , p ,
6 10 3 11 5 10 3 12 4 10 3 11 2 10 1 10 0 10 9
-----------------------------------------------------------------------
2
p p , p p , p p , p p , p p , p p , p p , p p , p p , p - p p , p p
8 9 7 9 6 9 5 9 4 9 3 9 2 9 1 9 0 9 8 10 12 7 8
-----------------------------------------------------------------------
- p p , p p - p p , p p - p p , p p - p p , p p - p p , p p ,
6 12 6 8 3 12 5 8 6 12 4 8 3 12 3 8 3 11 2 8
-----------------------------------------------------------------------
2
p p , p p , p - p p , p p - p p , p p - p p , p p - p p , p p
1 8 0 8 7 3 12 6 7 3 11 5 7 3 12 4 7 3 11 3 7
-----------------------------------------------------------------------
2
- p p , p p , p p , p p , p - p p , p p - p p , p p - p p ,
3 10 2 7 1 7 0 7 6 3 10 5 6 3 11 4 6 3 10
-----------------------------------------------------------------------
2
p p , p p , p p , p - p p , p p - p p , p p - p p , p p , p p ,
2 6 1 6 0 6 5 3 12 4 5 3 11 3 5 3 10 2 5 1 5
-----------------------------------------------------------------------
2 2
p p , p - p p , p p - p p , p p , p p , p p , p p , p p , p p , p ,
0 5 4 3 10 3 4 3 6 2 4 1 4 0 4 2 3 1 3 0 3 2
-----------------------------------------------------------------------
2 2
p p , p p , p , p p , p )
1 2 0 2 1 0 1 0
o53 : Ideal of KK[p , p , p , p , p , p , p , p , p , p , p , p , p , p , p ]
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
i54 : hilbertPolynomial coker gens FS22
o54 = - 3*P + 4*P
0 1
o54 : ProjectiveHilbertPolynomial
i55 :
