Text
The above example shows that the Billera-Schenck-Stillman spline complex is the cokernel of the natural map between the complex
of ideals (given by idealsComplex) and the cellular chain complex of $\Delta$ relative to its boundary.
- Example
- V = {{1, 0, 0}, {-1, 0, 0}, {0, 1, 0}, {0, -1, 0}, {0, 0, 1}, {0, 0, -1}, {-2, -2, -2}, {-2, 2, -2}, {2, 2, -2}, {2, -2, -2}, {-2, -2, 2}, {-2, 2, 2}, {2, 2, 2}, {2, -2, 2}};
- F = {{0, 1, 2, 3, 4, 5}, {0, 8, 9, 12, 13}, {1, 6, 7, 10, 11}, {2, 7, 8, 11, 12}, {3, 6, 9, 10, 13}, {4, 10, 11, 12, 13}, {5, 6, 7, 8, 9}, {0, 2, 8, 12}, {0, 3, 9, 13}, {0, 4, 12, 13}, {0, 5, 8, 9}, {1, 2, 7, 11}, {1, 3, 6, 10}, {1, 4, 10, 11}, {1, 5, 6, 7}, {2, 4, 11, 12}, {3, 4, 10, 13}, {3, 5, 6, 9}, {2, 5, 7, 8}, {0, 2, 4, 12}, {0, 2, 5, 8}, {0, 3, 4, 13}, {0, 3, 5, 9}, {1, 2, 4, 11}, {1, 2, 5, 7}, {1, 3, 4, 10}, {1, 3, 5, 6}};
- C = splineComplex(V,F,1);
- associatedPrimes annihilator HH_2 C
+ CannedExample
+ i25 : V = {{1, 0, 0}, {-1, 0, 0}, {0, 1, 0}, {0, -1, 0}, {0, 0, 1}, {0, 0, -1}, {-2, -2, -2}, {-2, 2, -2}, {2, 2, -2}, {2, -2, -2}, {-2, -2, 2}, {-2, 2, 2}, {2, 2, 2}, {2, -2, 2}};
+
+ i26 : F = {{0, 1, 2, 3, 4, 5}, {0, 8, 9, 12, 13}, {1, 6, 7, 10, 11}, {2, 7, 8, 11, 12}, {3, 6, 9, 10, 13}, {4, 10, 11, 12, 13}, {5, 6, 7, 8, 9}, {0, 2, 8, 12}, {0, 3, 9, 13}, {0, 4, 12, 13}, {0, 5, 8, 9}, {1, 2, 7, 11}, {1, 3, 6, 10}, {1, 4, 10, 11}, {1, 5, 6, 7}, {2, 4, 11, 12}, {3, 4, 10, 13}, {3, 5, 6, 9}, {2, 5, 7, 8}, {0, 2, 4, 12}, {0, 2, 5, 8}, {0, 3, 4, 13}, {0, 3, 5, 9}, {1, 2, 4, 11}, {1, 2, 5, 7}, {1, 3, 4, 10}, {1, 3, 5, 6}};
+
+ i27 : C = splineComplex(V,F,1);
+
+ i28 : associatedPrimes annihilator HH_2 C
+
+ o28 = {ideal (t , t , t ), ideal (t , t , t ), ideal (t , t , t ), ideal (t ,
+ 3 2 0 3 2 1 3 1 0 2
+ -----------------------------------------------------------------------
+ t , t - t ), ideal (t , t , t + t ), ideal (t , t - t , t ), ideal
+ 1 0 3 2 1 0 3 2 1 3 0
+ -----------------------------------------------------------------------
+ (t , t + t , t ), ideal (t - t , t , t ), ideal (t + t , t , t ),
+ 2 1 3 0 2 3 1 0 2 3 1 0
+ -----------------------------------------------------------------------
+ ideal (t - 2t , t - 2t , t - 2t ), ideal (t - 2t , t - 2t , t +
+ 2 3 1 3 0 3 2 3 1 3 0
+ -----------------------------------------------------------------------
+ 2t ), ideal (t - 2t , t + 2t , t - 2t ), ideal (t - 2t , t + 2t ,
+ 3 2 3 1 3 0 3 2 3 1 3
+ -----------------------------------------------------------------------
+ t + 2t ), ideal (t + 2t , t + 2t , t - 2t ), ideal (t + 2t , t +
+ 0 3 2 3 1 3 0 3 2 3 1
+ -----------------------------------------------------------------------
+ 2t , t + 2t ), ideal (t + 2t , t - 2t , t - 2t ), ideal (t + 2t ,
+ 3 0 3 2 3 1 3 0 3 2 3
+ -----------------------------------------------------------------------
+ t - 2t , t + 2t ), ideal (t , t , t , t )}
+ 1 3 0 3 3 2 1 0
+
+ o28 : List
Text
The above example showcases a fairly complex three dimensional polyhedral complex.
It is a three-dimensional analog of the Morgan-Scot partition in the sense that
Text
Given a monomial ideal I, this function computes the j-multiplicity of I
following the method of Jeffries-Montaño.
- Example
- R = QQ[x,y]
- I = (ideal"xy5,x2y3,x3y2")^4
- elapsedTime monjMult I
- elapsedTime jMult I
+ CannedExample
+ i1 : R = QQ[x,y]
+
+ o1 = R
+
+ o1 : PolynomialRing
+
+ i2 : I = (ideal"xy5,x2y3,x3y2")^4
+
+ 4 20 5 18 6 17 6 16 7 15 8 14 7 14 8 13 9 12 10 11 8 12 9 11 10 10 11 9 12 8
+ o2 = ideal (x y , x y , x y , x y , x y , x y , x y , x y , x y , x y , x y , x y , x y , x y , x y )
+
+ o2 : Ideal of R
+
+ i3 : elapsedTime monjMult I
+ -- 0.0945551 seconds elapsed
+
+ o3 = 192
+
+ i4 : elapsedTime jMult I
+ -- 1.15819 seconds elapsed
SeeAlso
multiplicitySequence
jMult
Next, we provide several examples to show the interplay between computing a set of Noetherian operators and then getting the back the original ideal.
The first example shows an ideal that can be described with two different sets of Noetherian operators (this example appeared in Example 7.8 of @ HREF("https://arxiv.org/abs/2001.04700", "Primary ideals and their differential equations")@).
- Example
- R = QQ[x_1,x_2,x_3,x_4]
- MM = matrix {{x_3,x_1,x_2},{x_1,x_2,x_4}}
- P = minors(2,MM)
- M = ideal{x_1^2,x_2^2,x_3^2,x_4^2}
- Q = joinIdeals(P,M);
- L1 = noetherianOperators(Q) -- A set of Noetherian operators
- Q1 = getIdealFromNoetherianOperators(L1, P);
- Q == Q1
- L2 = noetherianOperators(M) -- Another set of Noetherian operators
- Q2 = getIdealFromNoetherianOperators(L2, P);
- Q == Q2
+ CannedExample
+ i1 : R = QQ[x_1,x_2,x_3,x_4]
+
+ o1 = R
+
+ o1 : PolynomialRing
+
+ i2 : MM = matrix {{x_3,x_1,x_2},{x_1,x_2,x_4}}
+
+ o2 = | x_3 x_1 x_2 |
+ | x_1 x_2 x_4 |
+
+ 2 3
+ o2 : Matrix R <--- R
+
+ i3 : P = minors(2,MM)
+
+ 2 2
+ o3 = ideal (- x + x x , - x x + x x , - x + x x )
+ 1 2 3 1 2 3 4 2 1 4
+
+ o3 : Ideal of R
+
+ i4 : M = ideal{x_1^2,x_2^2,x_3^2,x_4^2}
+
+ 2 2 2 2
+ o4 = ideal (x , x , x , x )
+ 1 2 3 4
+
+ o4 : Ideal of R
+
+ i5 : Q = joinIdeals(P,M);
+
+ o5 : Ideal of R
+
+ i6 : L1 = noetherianOperators(Q) -- A set of Noetherian operators
+
+ 2 2 3 2 2 3 3 2 2
+ o6 = {1, dx_2, dx_1, dx_2 , dx_1*dx_2, dx_1 , dx_2 , dx_1*dx_2 , dx_1 dx_2, dx_1 , 72x x dx_1 dx_2 + 180x x dx_1 dx_2 +
+ 1 3 2 3
+ ----------------------------------------------------------------------------------------------------------------------------
+ 3
+ 72x x dx_1*dx_2 }
+ 3 4
+
+ o6 : List
+
+ i7 : Q1 = getIdealFromNoetherianOperators(L1, P);
+
+ o7 : Ideal of R
+
+ i8 : Q == Q1
+
+ o8 = true
+
+ i9 : L2 = noetherianOperators(M) -- Another set of Noetherian operators
+
+ o9 = {1, dx_4, dx_3, dx_2, dx_1, dx_3*dx_4, dx_2*dx_4, dx_1*dx_4, dx_2*dx_3, dx_1*dx_3, dx_1*dx_2, dx_2*dx_3*dx_4,
+ ----------------------------------------------------------------------------------------------------------------------------
+ dx_1*dx_3*dx_4, dx_1*dx_2*dx_4, dx_1*dx_2*dx_3, dx_1*dx_2*dx_3*dx_4}
+
+ o9 : List
+
+ i10 : Q2 = getIdealFromNoetherianOperators(L2, P);
+
+ o10 : Ideal of R
+
+ i11 : Q == Q2
+
+ o11 = true
Text
The following example was given as the running example in the Introduction of @ HREF("https://arxiv.org/abs/2001.04700", "Primary ideals and their differential equations")@.
- Example
- Q = ideal(3*x_1^2*x_2^2-x_2^3*x_3-x_1^3*x_4-3*x_1*x_2*x_3*x_4+2*x_3^2*x_4^2,3*x_1^3*x_2*x_4-3*x_1*x_2^2*x_3*x_4-3*x_1^2*x_3*x_4^2+3*x_2*x_3^2*x_4^2+2*x_2^3-2*x_3*x_4^2,3*x_2^4*x_3-6*x_1*x_2^2*x_3*x_4+3*x_1^2*x_3*x_4^2+x_2^3-x_3*x_4^2,4*x_1*x_2^3*x_3+x_1^4*x_4-6*x_1^2*x_2*x_3*x_4-3*x_2^2*x_3^2*x_4+4*x_1*x_3^2*x_4^2,x_2^5-x_1*x_2^3*x_4-x_2^2*x_3*x_4^2+x_1*x_3*x_4^3,x_1*x_2^4-x_2^3*x_3*x_4-x_1*x_2*x_3*x_4^2+x_3^2*x_4^3,x_1^4*x_2-x_2^3*x_3^2-2*x_1^3*x_3*x_4+2*x_1*x_2*x_3^2*x_4,x_1^5-4*x_1^3*x_2*x_3+3*x_1*x_2^2*x_3^2+2*x_1^2*x_3^2*x_4-2*x_2*x_3^3*x_4,3*x_1^4*x_3*x_4-6*x_1^2*x_2*x_3^2*x_4+3*x_2^2*x_3^3*x_4+2*x_1^3*x_2+6*x_1*x_2^2*x_3-6*x_1^2*x_3*x_4-2*x_2*x_3^2*x_4,4*x_2^3*x_3^3+4*x_1^3*x_3^2*x_4-12*x_1*x_2*x_3^3*x_4+4*x_3^4*x_4^2-x_1^4+6*x_1^2*x_2*x_3+3*x_2^2*x_3^2-8*x_1*x_3^2*x_4)
- L = noetherianOperators(Q)
- Q' = getIdealFromNoetherianOperators(L, P);
- Q == Q'
+ CannedExample
+ i12 : Q = ideal(3*x_1^2*x_2^2-x_2^3*x_3-x_1^3*x_4-3*x_1*x_2*x_3*x_4+2*x_3^2*x_4^2,3*x_1^3*x_2*x_4-3*x_1*x_2^2*x_3*x_4-3*x_1^2*x_3*x_4^2+3*x_2*x_3^2*x_4^2+2*x_2^3-2*x_3*x_4^2,3*x_2^4*x_3-6*x_1*x_2^2*x_3*x_4+3*x_1^2*x_3*x_4^2+x_2^3-x_3*x_4^2,4*x_1*x_2^3*x_3+x_1^4*x_4-6*x_1^2*x_2*x_3*x_4-3*x_2^2*x_3^2*x_4+4*x_1*x_3^2*x_4^2,x_2^5-x_1*x_2^3*x_4-x_2^2*x_3*x_4^2+x_1*x_3*x_4^3,x_1*x_2^4-x_2^3*x_3*x_4-x_1*x_2*x_3*x_4^2+x_3^2*x_4^3,x_1^4*x_2-x_2^3*x_3^2-2*x_1^3*x_3*x_4+2*x_1*x_2*x_3^2*x_4,x_1^5-4*x_1^3*x_2*x_3+3*x_1*x_2^2*x_3^2+2*x_1^2*x_3^2*x_4-2*x_2*x_3^3*x_4,3*x_1^4*x_3*x_4-6*x_1^2*x_2*x_3^2*x_4+3*x_2^2*x_3^3*x_4+2*x_1^3*x_2+6*x_1*x_2^2*x_3-6*x_1^2*x_3*x_4-2*x_2*x_3^2*x_4,4*x_2^3*x_3^3+4*x_1^3*x_3^2*x_4-12*x_1*x_2*x_3^3*x_4+4*x_3^4*x_4^2-x_1^4+6*x_1^2*x_2*x_3+3*x_2^2*x_3^2-8*x_1*x_3^2*x_4)
+
+ 2 2 3 3 2 2 3 2 2 2 2 2 3 2 4 2
+ o12 = ideal (3x x - x x - x x - 3x x x x + 2x x , 3x x x - 3x x x x - 3x x x + 3x x x + 2x - 2x x , 3x x - 6x x x x +
+ 1 2 2 3 1 4 1 2 3 4 3 4 1 2 4 1 2 3 4 1 3 4 2 3 4 2 3 4 2 3 1 2 3 4
+ ---------------------------------------------------------------------------------------------------------------------------
+ 2 2 3 2 3 4 2 2 2 2 2 5 3 2 2 3 4 3
+ 3x x x + x - x x , 4x x x + x x - 6x x x x - 3x x x + 4x x x , x - x x x - x x x + x x x , x x - x x x -
+ 1 3 4 2 3 4 1 2 3 1 4 1 2 3 4 2 3 4 1 3 4 2 1 2 4 2 3 4 1 3 4 1 2 2 3 4
+ ---------------------------------------------------------------------------------------------------------------------------
+ 2 2 3 4 3 2 3 2 5 3 2 2 2 2 3 4 2 2
+ x x x x + x x , x x - x x - 2x x x + 2x x x x , x - 4x x x + 3x x x + 2x x x - 2x x x , 3x x x - 6x x x x +
+ 1 2 3 4 3 4 1 2 2 3 1 3 4 1 2 3 4 1 1 2 3 1 2 3 1 3 4 2 3 4 1 3 4 1 2 3 4
+ ---------------------------------------------------------------------------------------------------------------------------
+ 2 3 3 2 2 2 3 3 3 2 3 4 2 4 2 2 2 2
+ 3x x x + 2x x + 6x x x - 6x x x - 2x x x , 4x x + 4x x x - 12x x x x + 4x x - x + 6x x x + 3x x - 8x x x )
+ 2 3 4 1 2 1 2 3 1 3 4 2 3 4 2 3 1 3 4 1 2 3 4 3 4 1 1 2 3 2 3 1 3 4
+
+ o12 : Ideal of R
+
+ i13 : L = noetherianOperators(Q)
+
+ 2
+ o13 = {1, dx_1, dx_1 - 2x dx_2}
+ 2
+
+ o13 : List
+
+ i14 : Q' = getIdealFromNoetherianOperators(L, P);
+
+ o14 : Ideal of R
+
+ i15 : Q == Q'
+
+ o15 = true
Text
The next example was given by Palamodov to show that there exists primary ideals that cannot be described by using differential operators with constant coefficients.
Example
Outputs => {FunctionClosure => {"which takes the ideal of a (general) point ", TEX///$p\in\mathbb{P}^5$///, " and returns the unique rational curve of degree ", TEX///$e$///, ", ", TEX///$(3e-1)$///, "-secant to ", TEX///$S$///, ", and passing through ", TEX///$p$///, " (an error is thrown if such a curve does not exist or is not unique)"}},
EXAMPLE {"-- A general cubic fourfold of discriminant 26"|newline|"X = specialCubicFourfold(\"Farkas-Verra C26\",ZZ/33331);", "describe X", "time f = detectCongruence X;", "p = point ring X -- random point on P^5", "time C = f p -- 5-secant conic to the surface", "assert(codim C == 4 and degree C == 2 and codim(C+(first ideals X)) == 5 and degree(C+(first ideals X)) == 5 and isSubset(C, p))"},
PARA{"The same method can be also applied to ", ofClass SpecialGushelMukaiFourfold, ". In this case it will detect and return a congruence of (2e-1)-secant curves of degree e inside the unique del Pezzo fivefold containing the GM fourfold."},
-EXAMPLE{"-- A general GM fourfold of discriminant 20"|newline|"X = specialGushelMukaiFourfold(\"surface of degree 9 and genus 2\",ZZ/33331);", "describe X", "time f = detectCongruence X;", "Y = source map X; -- del Pezzo fivefold containing X", "p = point Y -- random point on Y", "time C = f p -- 3-secant conic to the surface", "S = sub(first ideals X,Y);", "assert(dim C -1 == 1 and degree C == 2 and dim(C+S)-1 == 0 and degree(C+S) == 3 and isSubset(C, p))"},
+EXAMPLE{PRE(///i7 : -- A general GM fourfold of discriminant 20
+ X = specialGushelMukaiFourfold("surface of degree 9 and genus 2",ZZ/33331);
+/// | ///
+o7 : SpecialGushelMukaiFourfold (Gushel-Mukai fourfold containing a surface of degree 9 and sectional genus 2)
+/// | ///
+i8 : describe X
+/// | ///
+o8 = Special Gushel-Mukai fourfold of discriminant 20
+ containing a surface in PP^8 of degree 9 and sectional genus 2
+ cut out by 19 hypersurfaces of degree 2
+ and with class in G(1,4) given by 6*s_(3,1)+3*s_(2,2)
+ Type: ordinary
+ (case 17 of Table 1 in arXiv:2002.07026)
+/// | ///
+i9 : time f = detectCongruence X;
+S: surface of degree 9 and sectional genus 2 in PP^8 cut out by 19 hypersurfaces of degree 2
+phi: quadratic rational map from 5-dimensional subvariety of PP^8 to PP^13
+Z=phi(del Pezzo fivefold)
+number lines containing in Z and passing through the point phi(p): 7
+number 1-secant lines to S passing through p: 6
+number 3-secant conics to S passing through p: 1
+ -- used 19.3376 seconds
+/// | ///
+i10 : Y = source map X; -- del Pezzo fivefold containing X
+/// | ///
+i11 : p = point Y -- random point on Y
+/// | ///
+o11 = ideal (t + 14118t , t + 3234t , t - 16296t , t - 5674t , t -
+ 7 8 6 8 5 8 4 8 3
+ -----------------------------------------------------------------------
+ 12127t , t - 1329t , t + 3304t , t + 779t )
+ 8 2 8 1 8 0 8
+/// | ///
+o11 : Ideal of Y
+/// | ///
+i12 : time C = f p -- 3-secant conic to the surface
+ -- used 0.347885 seconds
+/// | ///
+o12 = ideal (t - 1000t + 8254t - 12393t , t + 10116t + 7449t - 15895t ,
+ 5 6 7 8 4 6 7 8
+ -----------------------------------------------------------------------
+ t + 10858t + 13401t + 13664t , t - 11215t + 13587t - 5150t , t -
+ 3 6 7 8 2 6 7 8 1
+ -----------------------------------------------------------------------
+ 1898t + 4900t + 14451t , t - 7830t + 1802t - 14129t )
+ 6 7 8 0 6 7 8
+/// | ///
+o12 : Ideal of Y
+/// | ///
+i13 : S = sub(first ideals X,Y);
+/// | ///
+o13 : Ideal of Y
+/// | ///
+i14 : assert(dim C -1 == 1 and degree C == 2 and dim(C+S)-1 == 0 and degree(C+S) == 3 and isSubset(C, p))
+///)},
SeeAlso => {coneOfLines}}
document {Key => {SpecialCubicFourfold},
huge rank and presentation matrices. In particular, the method directImageComplex becomes slow down.
The following is an example of direct images of the structure sheaf on a twisted cubic.
- Example
- kk=ZZ/101; d=6;
- needsPackage "Resultants";
- vd=veronese(1,d,kk);
- R=target vd; S=source vd;
-
- M=R^{1:-1}; I=ann M; J=ker vd;
-
- RM=time directImageComplex(I,M,matrix vd);
-
- for i from min RM to max RM list (rank RM_i)
+ CannedExample
+ i1 : kk=ZZ/101; d=6;
+
+ i3 : needsPackage "Resultants";
+
+ i4 : vd=veronese(1,d,kk);
+
+ o4 : RingMap kk[t ..t ] <--- kk[x ..x ]
+ 0 1 0 6
+
+ i5 : R=target vd; S=source vd;
+
+ i7 : M=R^{1:-1}; I=ann M; J=ker vd;
+
+ o8 : Ideal of R
+
+ o9 : Ideal of S
+
+ i10 : RM=time directImageComplex(I,M,matrix vd);
+ -- used 46.0364 seconds
+
+ i11 : for i from min RM to max RM list (rank RM_i)
+
+ o11 = {36, 180, 360, 360, 180, 36}
+
+ o11 : List
Text
RM looks complicated since it is consisted of universal bundles on P^6, which are of high rank.
- Example
- retTable=time actionOnDirectImage(I,M,matrix vd);
- keys retTable
+ CannedExample
+ i12 : retTable=time actionOnDirectImage(I,M,matrix vd);
+ We take a finite projection from P^n to P^d as {x , x } -- used 47.1632 seconds
+ 0,0 0,6
+
+ i13 : keys retTable
+
+ o13 = {0}
+
+ o13 : List
Text
We see that 0 is the only key, in other words, there is no other R^i vd_{*} except i=0.
To see whether it gives an action on S/J, we can use the test function isAction.
Note that list retTable#i is consisted of maps of chain complexes R^i(\pi \cdot phi)_{*}(y_j) : C\to C(1) where
C represents the direct image R^i(\pi \cdot phi)_{*}F. In general, it does not give a right action on C itself.
The induced maps on cohomology groups provide a right action.
- Example
- isAction(J, apply(dim S, i->prune HH^0 retTable#0#i))
+ CannedExample
+ i14 : isAction(J, apply(dim S, i->prune HH^0 retTable#0#i))
+
+ o14 = true
Text
The following is a little more complicated example with nontrivial higher direct images.
Let X be the product of two quartic curves C, and f : X \to C be the second projection.
Let P, Q be two distinct points of C, and let L = O(P\times C + Q\times C + D) be a line bundle
on X where D is the diagonal. We want to compute the higher direct images R^i f_{*}L.
We choose C as the Fermat quartic, and choose P, Q as points on the intersection of C and the line V(x_2).
- Example
- kk=ZZ/1009;
- (S,E)=productOfProjectiveSpaces({2,2},CoefficientField=>kk);
- f1=S_0^4+S_1^4+S_2^4; f2=S_3^4+S_4^4+S_5^4;
- IX=ideal (f1,f2);
- -- 2 points on C. Note that the line passing through P and Q is V(x_2).
- P=matrix{{1,192,0}}; Q=matrix{{192,1,0}};
- varX=matrix{{S_0,S_1,S_2}}; varY=matrix{{S_3,S_4,S_5}};
- PX=P||varX; QX=Q||varX;
-
- I1=ideal(f2, minors(2,PX)); -- ideal for P*C
- I2=ideal(f2, minors(2,QX)); -- ideal for Q*C
- I3=ideal(f1,f2, minors(2, varX||varY)); -- ideal for the diagonal D
- I=intersect(I1,I2,I3); -- ideal for P*C+Q*C+D
- SX=S/IX; Ldual=I/IX;
-
- betti (LX=Hom(Ldual, SX^1))
+ CannedExample
+ i15 : kk=ZZ/1009;
+
+ i16 : (S,E)=productOfProjectiveSpaces({2,2},CoefficientField=>kk);
+
+ i17 : f1=S_0^4+S_1^4+S_2^4; f2=S_3^4+S_4^4+S_5^4;
+
+ i19 : IX=ideal (f1,f2);
+
+ o19 : Ideal of S
+
+ i20 : P=matrix{{1,192,0}}; Q=matrix{{192,1,0}};
+
+ 1 3
+ o20 : Matrix ZZ <--- ZZ
+
+ 1 3
+ o21 : Matrix ZZ <--- ZZ
+
+ i22 : varX=matrix{{S_0,S_1,S_2}}; varY=matrix{{S_3,S_4,S_5}};
+
+ 1 3
+ o22 : Matrix S <--- S
+
+ 1 3
+ o23 : Matrix S <--- S
+
+ i24 : PX=P||varX; QX=Q||varX;
+
+ 2 3
+ o24 : Matrix S <--- S
+
+ 2 3
+ o25 : Matrix S <--- S
+
+ i26 : I1=ideal(f2, minors(2,PX)); -- ideal for P*C
+
+ o26 : Ideal of S
+
+ i27 : I2=ideal(f2, minors(2,QX)); -- ideal for Q*C
+
+ o27 : Ideal of S
+
+ i28 : I3=ideal(f1,f2, minors(2, varX||varY)); -- ideal for the diagonal D
+
+ o28 : Ideal of S
+
+ i29 : I=intersect(I1,I2,I3); -- ideal for P*C+Q*C+D
+
+ o29 : Ideal of S
+
+ i30 : SX=S/IX; Ldual=I/IX;
+
+ i32 : betti (LX=Hom(Ldual, SX^1))
+
+ 0 1
+ o32 = total: 10 26
+ 0: 1 .
+ 1: 1 1
+ 2: . 1
+ 3: 8 24
+
+ o32 : BettiTally
Text
To compute its Tate resolution on the ambient space P^2 \times P^2, we first consider it as a sheaf on P^2 \times P^2,
and then take a linear presentation matrix via a truncation.
- Example
- phi = map(SX,S,vars SX);
- betti (L=prune pushForward(phi,LX))
- Ltr = (truncate ({2,2},L))**S^{{2,2}};
- betti res Ltr
+ CannedExample
+ i33 : phi = map(SX,S,vars SX);
+
+ o33 : RingMap SX <--- S
+
+ i34 : betti (L=prune pushForward(phi,LX))
+
+ 0 1
+ o34 = total: 10 27
+ 0: 1 .
+ 1: 1 1
+ 2: . 1
+ 3: 8 25
+
+ o34 : BettiTally
+
+ i35 : Ltr = (truncate ({2,2},L))**S^{{2,2}};
+
+ i36 : betti res Ltr
+
+ 0 1 2 3 4
+ o36 = total: 60 176 184 80 12
+ 0: 60 176 184 80 12
+
+ o36 : BettiTally
Text
We read off (a finite subquotient of) the Tate resolution of Rf_{*}L as follows.
- Example
- Q=symExt(presentation Ltr, E);
- T=(res (coker Q,LengthLimit=>12))**E^{{2,2}}[4];
- cohomologyMatrix (T, -{5,5},{3,3})
- sT=strand(T,{0,0},{0});
-
- sTFull=new ChainComplex;
- sTFull.ring = ring sT;
- ma=6;
- sTFull=(dual res (coker transpose (sT[ma]).dd_0, LengthLimit=>2*ma))[-ma];
- (S',E')=productOfProjectiveSpaces({2},CoefficientField=>kk);
- projOnE=map(E', E, toList(3:0)|(gens E'));
- mi=min sTFull; ma=max sTFull;
- W=new ChainComplex; W.ring = E';
- apply(toList(mi..ma),i-> W_i = E'^(-apply(degrees sTFull_i,d->d_{1})));
- apply(toList(mi+1..ma),i->W.dd_i = map(W_(i-1),W_i,projOnE sTFull.dd_i));
- betti W
+ CannedExample
+ i37 : Q=symExt(presentation Ltr, E);
+
+ 184 60
+ o37 : Matrix E <--- E
+
+ i38 : T=(res (coker Q,LengthLimit=>12))**E^{{2,2}}[4];
+
+ i39 : cohomologyMatrix (T, -{5,5},{3,3})
+
+ o39 = | 0 0 0 0 0 0 0 96k 0 |
+ | 0 0 0 0 0 0 0 60k 88k |
+ | 60h 48h 36h 24h 12h 3h+3 12 0 0 |
+ | 36h2+20h 28h2+16h 20h2+12h 12h2+8h 4h2+4h 5h+1 12h+4 0 0 |
+ | 92h2 72h2 52h2 32h2 12h2 8h 28h 0 0 |
+ | 168h2 132h2 96h2 60h2 24h2 12h 48h 0 0 |
+ | 244h2 192h2 140h2 88h2 36h2 16h 68h 0 0 |
+ | 320h2 252h2 184h2 116h2 48h2 20h 88h 0 0 |
+ | 396h2 312h2 228h2 144h2 60h2 24h 108h 0 0 |
+
+ 9 9
+ o39 : Matrix (ZZ[h, k]) <--- (ZZ[h, k])
+
+ i40 : sT=strand(T,{0,0},{0});
+
+ i41 : sTFull=new ChainComplex;
+
+ i42 : sTFull.ring = ring sT;
+
+ i43 : ma=6;
+
+ i44 : sTFull=(dual res (coker transpose (sT[ma]).dd_0, LengthLimit=>2*ma))[-ma];
+
+ i45 : (S',E')=productOfProjectiveSpaces({2},CoefficientField=>kk);
+
+ i46 : projOnE=map(E', E, toList(3:0)|(gens E'));
+
+ o46 : RingMap E' <--- E
+
+ i47 : mi=min sTFull; ma=max sTFull;
+
+ i49 : W=new ChainComplex; W.ring = E';
+
+ i51 : apply(toList(mi..ma),i-> W_i = E'^(-apply(degrees sTFull_i,d->d_{1})));
+
+ i52 : apply(toList(mi+1..ma),i->W.dd_i = map(W_(i-1),W_i,projOnE sTFull.dd_i));
+
+ i53 : betti W
+
+ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
+ o53 = total: 24 20 16 12 9 8 9 12 16 20 24 28 32
+ 0: 22 18 14 10 6 3 1 . . . . . .
+ 1: 2 2 2 2 3 5 8 12 16 20 24 28 32
+
+ o53 : BettiTally
Text
One can check that W has two strands (corresponding to R^0f_{*}L and R^1f_{*}L, respectively).
By taking the Beilinson functor, one can check that R^0f_{*}L is the structure sheaf on C,
and R^1f_{*}L is a torsion sheaf supported on two points lying on the intersection of C and
the line V(x_2) other than P, Q.
- Example
- R0fL = prune HH^0 beilinson W
- R1fL = prune HH^1 beilinson W
-
- degree R0fL
- primaryDecomposition ann R0fL
-
- degree R1fL
- primaryDecomposition ann R1fL
+ CannedExample
+ i54 : R0fL = prune HH^0 beilinson W
+
+ o54 = cokernel | x_(0,0)^4+x_(0,1)^4+x_(0,2)^4 |
+
+ 1
+ o54 : S'-module, quotient of S'
+
+ i55 : R1fL = prune HH^1 beilinson W
+
+ o55 = cokernel | -131x_(0,2) 0 0 0 x_(0,0) x_(0,1) 0 0 -131x_(0,2) 0 -131x_(0,2) 0 |
+ | x_(0,2) x_(0,1) x_(0,0) 0 0 0 0 0 0 0 0 0 |
+ | 0 0 0 x_(0,2) -x_(0,1) x_(0,0)+439x_(0,1) 0 0 0 0 0 0 |
+ | 0 0 0 0 0 0 x_(0,2) x_(0,1) x_(0,0) 0 0 0 |
+ | 0 0 0 0 0 0 0 0 0 x_(0,2) x_(0,1) x_(0,0) |
+
+ 5
+ o55 : S'-module, quotient of S'
+
+ i56 : degree R0fL
+
+ o56 = 4
+
+ i57 : primaryDecomposition ann R0fL
+
+ 4 4 4
+ o57 = {ideal(x + x + x )}
+ 0,0 0,1 0,2
+
+ o57 : List
+
+ i58 : degree R1fL
+
+ o58 = 2
+
+ i59 : primaryDecomposition ann R1fL
+
+ 2
+ o59 = {ideal (x , x + 247x ), ideal (x , x + 192x ), ideal (x , x , x )}
+ 0,2 0,0 0,1 0,2 0,0 0,1 0,1 0,0 0,2
+
+ o59 : List
Text
These module also can be seen as in the following way via a finite linear projection.
We take a further projection \pi:C\to P^1, and check whether these modules induce an action
on the direct image under \pi, in other words, provide {O_C}-module structures. As results,
these actions make (the sheafification of) M0 and M1 into {O_C}-modules which are identical to
R^0f_{*}L and R^1f_{*}L.
- Example
- J=ideal (S'_0^4+S'_1^4+S'_2^4);
- retTable=actionOnDirectImage(J,W);
- keys retTable
+ CannedExample
+ i60 : J=ideal (S'_0^4+S'_1^4+S'_2^4);
+
+ o60 : Ideal of S'
+
+ i61 : retTable=actionOnDirectImage(J,W);
+ We take a finite projection from P^n to P^d as {x , x }
+ 0,1 0,2
+ i62 : keys retTable
+
+ o62 = {0, 1}
+
+ o62 : List
Text
We see that 0, 1 appear as keys, in other words, both R^0f_{*}L and R^1f_{*}L survives.
- Example
- prunedActionList = i->apply(dim S',j->prune HH^i retTable#i#j);
- apply(keys retTable, i->isAction(J,prunedActionList(i)))
-
- M0=source (prunedActionList(0))_0
- (rank M0, degree M0, betti res M0)
- isIsomorphic(truncate(regularity M0, M0), truncate(regularity M0, dual dual M0))
- dual dual M0
-
- M1=source (prunedActionList(1))_0
- (rank M1, degree M1, betti res M1)
+ CannedExample
+ i63 : prunedActionList = i->apply(dim S',j->prune HH^i retTable#i#j);
+
+ i64 : apply(keys retTable, i->isAction(J,prunedActionList(i)))
+
+ o64 = {true, true}
+
+ o64 : List
+
+ i65 : M0=source (prunedActionList(0))_0
+
+ o65 = cokernel {2} | 0 -x_(0,1) 0 |
+ {2} | x_(0,0) 0 0 |
+ {2} | -x_(0,1) x_(0,0) 0 |
+ {2} | 0 0 0 |
+ {2} | 0 0 x_(0,0) |
+ {2} | 0 0 -x_(0,1) |
+ {3} | 0 0 0 |
+ 7
+ o65 : kk[x ..x ]-module, quotient of (kk[x ..x ])
+ 0,0 0,1 0,0 0,1
+
+ i66 : (rank M0, degree M0, betti res M0)
+
+ 0 1
+ o66 = (4, 4, total: 7 3)
+ 2: 6 3
+ 3: 1 .
+
+ o66 : Sequence
+
+ i67 : isIsomorphic(truncate(regularity M0, M0), truncate(regularity M0, dual dual M0))
+
+ o67 = true
+
+ i68 : dual dual M0
+
+ 4
+ o68 = (kk[x ..x ])
+ 0,0 0,1
+
+ o68 : kk[x ..x ]-module, free, degrees {3, 2, 1, 0}
+ 0,0 0,1
+
+ i69 : M1=source (prunedActionList(1))_0
+
+ o69 = cokernel {2} | x_(0,1) 0 |
+ {2} | 0 x_(0,1) |
+
+ 2
+ o69 : kk[x ..x ]-module, quotient of (kk[x ..x ])
+ 0,0 0,1 0,0 0,1
+
+ i70 : (rank M1, degree M1, betti res M1)
+
+ 0 1
+ o70 = (0, 2, total: 2 2)
+ 2: 2 2
+
+ o70 : Sequence
Text
Note that the sheafification of M0 (=R^0(\pi \cdot f)_{*}L) is a rank 4 vector bundle O \oplus O(-1) \oplus O(-2) \oplus O(-3) on P^1,
and the sheafification of M1 (= R^1(\pi \cdot f)_{*} L) is a torsion sheaf on P^1 supported on the double point at [1:0].
Then $uT$ is a Frobenius map on $E$, and the primes $P$ computed by this function are precisely those for which $uT$ restricts to a nonzero Frobenius map of the annihilator of $P$ on $E$.
The following is a simple example, which is split with the coordinate axes in $\mathbb{A}^2$.
- Example
- R = ZZ/3[s,t];
- u = s^2*t^2;
- compatibleIdeals u
+ CannedExample
+ i1 : R = ZZ/3[s,t];
+
+ i2 : u = s^2*t^2;
+
+ i3 : compatibleIdeals u
+
+ o3 = {ideal s, ideal (t, s), ideal t}
+
+ o3 : List
Text
Here is a more substantial example.
- Example
- R = ZZ/2[a,b,c,d,e,f];
- u = d*(b*e - d*c)*(d - a*e - b*f + a*c*f);
- print \ compatibleIdeals u;
+ CannedExample
+ i4 : R = ZZ/2[a,b,c,d,e,f];
+
+ i5 : u = d*(b*e - d*c)*(d - a*e - b*f + a*c*f);
+
+ i6 : print \ compatibleIdeals u;
+ ideal d
+ ideal (e, d)
+ ideal (f, e, d)
+ ideal (f, e, d, b)
+ ideal (f, e, d, b, a)
+ ideal (f, e, d, c, b, a)
+ ideal (f, e, d, c, b)
+ ideal (f, e, d, a*c + b)
+ ideal (d, b, a)
+ ideal (e, d, b, a)
+ ideal (e, d, c, b, a)
+ ideal (d, b, a, c*f + e)
+ ideal (e, d, b)
+ ideal (e, d, c, b)
+ ideal (e, d, a*c + b)
+ ideal (d, b)
+ ideal (d, b, c*f + e)
+ ideal (d, a*c*f + a*e + b*f)
+ ideal (a*e + d, c*d + b*e, a*c + b)
+ ideal (c*f + e, b*f + d, a*e + d, c*d + b*e, a*c + b)
+ ideal(c*d + b*e)
+ ideal (c*f + e, b*f + d, c*d + b*e)
+ ideal(a*c*f + a*e + b*f + d)
Text
The option {\tt FrobeniusRootStrategy} is passed to internal @TO frobeniusRoot@ calls.
///
The interesting part of the output may be the actual lineages of the basis polynomials,
in addition to the Groebner basis itself.
Here is a verbose example when the Groebner basis is trivial.
- Example
- QQ[a..d]
- I=ideal( -c^3+a^2+b*d, a*b*c-1,a*b*c)
- T = tgb(I,2,Verbose=>true)
+ CannedExample
+ i10 : QQ[a..d]
+
+ o10 = QQ[a, b, c, d]
+
+ o10 : PolynomialRing
+
+ i11 : I=ideal( -c^3+a^2+b*d, a*b*c-1,a*b*c)
+
+ 3 2
+ o11 = ideal (- c + a + b*d, a*b*c - 1, a*b*c)
+
+ o11 : Ideal of QQ[a, b, c, d]
+
+ i12 : T = tgb(I,2,Verbose=>true)
+ You turned on Verbose! You will be notified of each new S-polynomial task created and each new GB element added to the HashTable as we go.
+ Scheduling a task for lineage (0-1)
+ Scheduling a task for lineage (0-2)
+ Scheduling a task for lineage (1-2)
+ Scheduling task for lineage ((0-1)-0)
+ Scheduling task for lineage ((0-1)-1)
+ Scheduling task for lineage ((0-1)-2)
+ Adding the following remainder to GB: -1 from lineage (1-2)
+ Adding the following remainder to GB: -a^3*b-a*b^2*d+c^2 from lineage (0-1)
+ Scheduling task for lineage ((0-2)-0)
+ Scheduling task for lineage ((0-2)-1)
+ Scheduling task for lineage ((0-2)-2)
+ Scheduling task for lineage ((0-2)-(1-2))
+ Scheduling task for lineage ((0-2)-(0-1))
+ Adding the following remainder to GB: -a^3*b-a*b^2*d from lineage (0-2)
+ Found 1 or -1 in the Groebner basis; reducing now.
+
+ o12 = HashTable{(0-1) => null}
+ (0-2) => null
+ (1-2) => 1
+ 0 => null
+ 1 => null
+ 2 => null
+
+ o12 : HashTable
Text
In particular, the lineages of null values tell us what S-polynomials didn't reduce to zero until $1$ was found as
a remainder.
triangulation. Notice that {\tt tri} is a triangulation of the
polytope which is the convex hull of the columns of $A$, which are
the only points allowed in the triangulation.
- Example
- A = transpose matrix {{0,3},{0,1},{-1,-1},{1,-1},{-4,-2},{4,-2}}
- tri = {{0,1,2}, {1,3,5}, {2,3,4}, {0,1,5},
- {0,2,4}, {3,4,5}, {1,2,3}}
- isRegularTriangulation(A,tri)
+ CannedExample
+ i1 : A = transpose matrix {{0,3},{0,1},{-1,-1},{1,-1},{-4,-2},{4,-2}}
+
+ o1 = | 0 0 -1 1 -4 4 |
+ | 3 1 -1 -1 -2 -2 |
+
+ 2 6
+ o1 : Matrix ZZ <--- ZZ
+
+ i2 : tri = {{0,1,2}, {1,3,5}, {2,3,4}, {0,1,5},
+ {0,2,4}, {3,4,5}, {1,2,3}}
+
+ o2 = {{0, 1, 2}, {1, 3, 5}, {2, 3, 4}, {0, 1, 5}, {0, 2, 4}, {3, 4, 5}, {1,
+ ------------------------------------------------------------------------
+ 2, 3}}
+
+ o2 : List
+
+ i3 : isRegularTriangulation(A,tri)
+
+ o3 = false
Text
Setting debugLevel to either 1,2, or 5 will give more detail about
what files are written to Topcom, and what the executable is.
projects / macaulay2.git / commitdiff