Fundamental and Advanced Algorithms in Singular

Fundamental and Advanced Algorithms in Singular Janko Boehm, Yue Ren University of Kaiserslautern

3C in G Workshop on Computational Algebra Cambridge, April 18, 2017

Janko Boehm (TU-KL)

Algorithms in Singular

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Singular Computer algebra system for polynomial computations, over 30 development teams worldwide, over 140 libraries for advanced topics.

https://www.singular.uni-kl.de/

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Singular Computer algebra system for polynomial computations, over 30 development teams worldwide, over 140 libraries for advanced topics.

https://www.singular.uni-kl.de/ Special emphasis on algebraic geometry, commutative and non-commutative algebra, singularity theory, packages for convex and tropical geometry. Janko Boehm (TU-KL)


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Outline Warm-up, Gröbner bases Normalization, Adjoint Curves, Classification of Singularities Parallel Computations Resolution of Singularities Modular Methods Massively Parallel Computations

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Outline Warm-up, Gröbner bases Normalization, Adjoint Curves, Classification of Singularities Parallel Computations Resolution of Singularities Modular Methods Massively Parallel Computations Primary Decomposition Standard Bases and Associated Graded Ring Convex Geometry Computation of Tropical Varieties

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Outline Warm-up, Gröbner bases Normalization, Adjoint Curves, Classification of Singularities Parallel Computations Resolution of Singularities Modular Methods Massively Parallel Computations Primary Decomposition Standard Bases and Associated Graded Ring Convex Geometry Computation of Tropical Varieties Computing the GIT-Fan Feynman Integrals and Tropical Mirror Symmetry Janko Boehm (TU-KL)


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Parametrizing Rational Curves Example

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Parametrizing Rational Curves Example We consider the degree-5 curve with equation x 5 + 10x 4 y + 20x 3 y 2 + 130x 2 y 3 − 20xy 4 + 20y 5 − 2x 4 z

− 40x 3 yz − 150x 2 y 2 z − 90xy 3 z − 40y 4 z + x 3 z 2 + 30x 2 yz 2 + 110xy 2 z 2 + 20y 3 z 2 = 0.

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Genus Formula. Janko Boehm (TU-KL)

pg (C ) = pa (C ) − δ(C ) = pa (C ) − ∑P ∈Sing(Γ) δP (C ) Algorithms in Singular

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Parametrizing Rational Curves

Example > ring R = 0, (x,y,z), dp; > poly f = x5+10x4y+20x3y2+130x2y3-20xy4+20y5-2x4z-40x3yz-150x2y2z -90xy3z-40y4z+x3z2+30x2yz2+110xy2z2+20y3z2; > LIB "paraplanecurves.lib"; > genus(f); 0 > paraPlaneCurve(f);

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The Parametrization Algorithm Example > ideal AI = adjointIdeal(f); // requires normalization, integral bases > AI; [1]=y3-y2z [2]=xy2-xyz [3]=x2y-xyz [4]=x3-x2z > def Rn = mapToRatNormCurve(f,AI); > setring(Rn);

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The Parametrization Algorithm Example > ideal AI = adjointIdeal(f); // requires normalization, integral bases > AI; [1]=y3-y2z [2]=xy2-xyz [3]=x2y-xyz [4]=x3-x2z > def Rn = mapToRatNormCurve(f,AI); > setring(Rn); > RNC; RNC[1]=y(2)*y(3)-y(1)*y(4) RNC[2]=20*y(1)*y(2)-20*y(2)^2+130*y(1)*y(4) +20*y(2)*y(4)+10*y(3)*y(4)+y(4)^2 RNC[3]=20*y(1)^2-20*y(1)*y(2)+130*y(1)*y(3) +10*y(3)^2+20*y(1)*y(4)+y(3)*y(4) Janko Boehm (TU-KL)


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The Parametrization Algorithm

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The Parametrization Algorithm Example > LIB "sing.lib"; > radical(slocus(RNC)); [1]=y(4) [2]=y(3) [2]=y(2) [1]=y(1) > rncAntiCanonicalMap(RNC); [1]=2*y(2)+13*y(4) [2]=y(4)

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The Parametrization Algorithm Example > LIB "sing.lib"; > radical(slocus(RNC)); [1]=y(4) [2]=y(3) [2]=y(2) [1]=y(1) > rncAntiCanonicalMap(RNC); [1]=2*y(2)+13*y(4) [2]=y(4)

Remark May require quadratic field extension in even-degree case.

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The Main Computational Tool: Gröbner Bases Division with remainder in one variable successively eliminates the highest power.

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The Main Computational Tool: Gröbner Bases Division with remainder in one variable successively eliminates the highest power. In more than one variable we have to fix a monomial ordering (a total ordering compatible with multiplication).

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The Main Computational Tool: Gröbner Bases Division with remainder in one variable successively eliminates the highest power. In more than one variable we have to fix a monomial ordering (a total ordering compatible with multiplication). Divide x 2 − y 2 durch x 2 + y und xy + x with respect to lexicographic ordering. x 2 − y 2 = 1 · x 2 + y + −y 2 − y x2 + y −y 2 − y

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The Main Computational Tool: Gröbner Bases Division with remainder in one variable successively eliminates the highest power. In more than one variable we have to fix a monomial ordering (a total ordering compatible with multiplication). Divide x 2 − y 2 durch x 2 + y und xy + x with respect to lexicographic ordering. x 2 − y 2 = 1 · x 2 + y + −y 2 − y x2 + y −y 2 − y so remainder 6= 0, but

x 2 − y 2 = −y x 2 + y + x (xy + x ) ∈ I := x 2 + y , xy + x

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x 2 − y 2 = −y x 2 + y + x (xy + x ) ∈ I := x 2 + y , xy + x Problem: Lead terms cancel, division algorithm can’t do that.

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x 2 − y 2 = −y x 2 + y + x (xy + x ) ∈ I := x 2 + y , xy + x Problem: Lead terms cancel, division algorithm can’t do that. Solution: Add y 2 + y to the divisor set. The result is a Gr¨ obner basis G of I . Then

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x 2 − y 2 = −y x 2 + y + x (xy + x ) ∈ I := x 2 + y , xy + x Problem: Lead terms cancel, division algorithm can’t do that. Solution: Add y 2 + y to the divisor set. The result is a Gr¨ obner basis G of I . Then f ∈ I ⇐⇒ NF (f , G ) = 0 Janko Boehm (TU-KL)


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The Main Computational Tool: Gröbner Bases Example Gr¨ obner Bases can be used to: eliminate variables (→ birational geometry), ideal intersections, compute ideal quotients

(I : J ) = {a ∈ R | aJ ⊂ I } for ideals I , J ⊂ R, saturations, syzygies (→ homological algebra). Greuel, G.-M., Pfister, G.: A Singular Introduction to Commutative Algebra. Springer. Janko Boehm (TU-KL)


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Normalization Setup: A = K [X ]/I domain.

Definition The normalization A of A is the integral closure of A in its quotient field Q (A).

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Definition The normalization A of A is the integral closure of A in its quotient field Q (A). We call A normal if A = A.

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Theorem (Noether) A is a finitely generated A-module.

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Example

Curve I = x 3 + x 2 − y 2 ⊂ K [x, y ] A = K [x, y ]/I x y

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∼ = 7→ 7→

K [t 2 − 1, t 3 − t ] t2 − 1 t3 − t


⊂

K [t ] ∼ =A

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Example

Curve I = x 3 + x 2 − y 2 ⊂ K [x, y ] A = K [x, y ]/I x Dy

∼ = 7→ 7→ E

K [t 2 − 1, t 3 − t ] t2 − 1 t3 − t

⊂

K [t ] ∼ =A

As an A-module A = 1, yx . Janko Boehm (TU-KL)


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Normalization Lemma If J ⊂ A is an ideal and 0 6= g ∈ J, then

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Normalization Lemma If J ⊂ A is an ideal and 0 6= g ∈ J, then A a

,→ 7→

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HomA (J, J ) a· ϕ

∼ =

1 g (gJ :A

7→

ϕ (g ) g


J)

⊂

A

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,→ 7→

HomA (J, J ) a· ϕ

∼ =

1 g (gJ :A

7→

ϕ (g ) g

J)

⊂

A

Algorithm Starting from A0 = A and J0 = J,

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,→ 7→

HomA (J, J ) a· ϕ

∼ =

1 g (gJ :A

7→

ϕ (g ) g

J)

⊂

A

Algorithm Starting from A0 = A and J0 = J, setting Ai +1 = g1 (gJi :Ai Ji )

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Ji =

√

JAi

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,→ 7→

HomA (J, J ) a· ϕ

∼ =

1 g (gJ :A

7→

ϕ (g ) g

J)

⊂

A

Algorithm Starting from A0 = A and J0 = J, setting Ai +1 = g1 (gJi :Ai Ji )

Ji =

√

JAi

we get a chain of extensions of reduced Noetherian rings A = A0 ⊂ · · · ⊂ Ai ⊂ · · · ⊂ Am = Am+1 . Terminates since A is Noetherian. Janko Boehm (TU-KL)


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Grauert-Remmert criterion Non-normal locus N (A) is contained in singular locus Sing(A).

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Theorem (Grauert-Remmert) Let 0 6= J ⊂ A be an ideal with J =

Janko Boehm (TU-KL)

√

J


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√

J and

N (A) ⊂ V (J ).

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√

J and

N (A) ⊂ V (J ). Then A is normal iff the inclusion A ,→ a 7→

HomA (J, J ) a·

is an isomorphism.

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√

J and


HomA (J, J ) a·

is an isomorphism. p =⇒ For J = Jac(I ) algorithm terminates with Am = Am+1 = A,

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√

J and


HomA (J, J ) a·

is an isomorphism. p =⇒ For J = Jac(I ) algorithm terminates with Am = Am+1 = A, since:

Lemma √ N (Ai ) ⊂ V ( JAi ) Janko Boehm (TU-KL)


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Local Techniques for Normalization Theorem (BDLSS, 2011) Suppose Sing(A) = {P1 , . . . , Pr }

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Local Techniques for Normalization Theorem (BDLSS, 2011) Suppose Sing(A) = {P1 , . . . , Pr } and

A ⊂ Bi ⊂ A is the ring given by the normalization algorithm applied to Pi instead of J

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A ⊂ Bi ⊂ A is the ring given by the normalization algorithm applied to Pi instead of J. Then

(Bi )Pi = APi (Bi )Q = AQ for all Pi 6= Q ∈ Spec A,

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A ⊂ Bi ⊂ A is the ring given by the normalization algorithm applied to Pi instead of J. Then

(Bi )Pi = APi (Bi )Q = AQ for all Pi 6= Q ∈ Spec A, and A = B1 + . . . + Br . We call Bi the minimal local contribution to A at Pi . Janko Boehm (TU-KL)


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Adjoint ideals Setup: Γ ⊂ Pr integral, non-degenerate projective curve, π : Γ → Γ normalization map, I (Γ) $ I ⊂ k [x0 , ..., xr ] saturated homogeneous ideal.

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Adjoint ideals Setup: Γ ⊂ Pr integral, non-degenerate projective curve, π : Γ → Γ normalization map, I (Γ) $ I ⊂ k [x0 , ..., xr ] saturated homogeneous ideal. Let H be pullback of hyperplane, ∆(I ) pullback of Proj(S /I ).

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Adjoint ideals Setup: Γ ⊂ Pr integral, non-degenerate projective curve, π : Γ → Γ normalization map, I (Γ) $ I ⊂ k [x0 , ..., xr ] saturated homogeneous ideal. Let H be pullback of hyperplane, ∆(I ) pullback of Proj(S /I ). Then 0 → IeOΓ → π∗ (IeO ) → F → 0 Γ

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Adjoint ideals Setup: Γ ⊂ Pr integral, non-degenerate projective curve, π : Γ → Γ normalization map, I (Γ) $ I ⊂ k [x0 , ..., xr ] saturated homogeneous ideal. Let H be pullback of hyperplane, ∆(I ) pullback of Proj(S /I ). Then 0 → IeOΓ → π∗ (IeOΓ ) → F → 0 gives for m 0 linear maps $m 0 → Im /I (Γ)m → H 0 Γ, OΓ (mH − ∆(I )) → H 0 (Γ, F ) → 0

Definition I is an adjoint ideal of Γ if $m surjective for m 0.

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Adjoint ideals Setup: Γ ⊂ Pr integral, non-degenerate projective curve, π : Γ → Γ normalization map, I (Γ) $ I ⊂ k [x0 , ..., xr ] saturated homogeneous ideal. Let H be pullback of hyperplane, ∆(I ) pullback of Proj(S /I ). Then 0 → IeOΓ → π∗ (IeOΓ ) → F → 0 gives for m 0 linear maps $m 0 → Im /I (Γ)m → H 0 Γ, OΓ (mH − ∆(I )) → H 0 (Γ, F ) → 0

Definition I is an adjoint ideal of Γ if $m surjective for m 0. h0 (Γ, F ) =

∑P ∈Sing(Γ) `(IP OΓ,P /IP )

=⇒

Theorem I adjoint ⇐⇒ IP OΓ,P = IP for all P ∈ Sing(Γ). Conductor is largest ideal with this property. Janko Boehm (TU-KL)


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Adjoint ideals Definition Gorenstein adjoint ideal is the unique largest homogeneous ideal G ⊂ K [x0 , . . . , xr ] with GP = COΓ,P for all P ∈ Sing(Γ).

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Adjoint ideals Definition Gorenstein adjoint ideal is the unique largest homogeneous ideal G ⊂ K [x0 , . . . , xr ] with GP = COΓ,P for all P ∈ Sing(Γ). Applications:

Example If Γ is plane curve of degree n, then Gn−3 cuts out canonical linear series.

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Example If Γ is plane rational of degree n then Gn−2 maps Γ to rational normal curve of degree n − 2 in Pn−2 .

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Example If Γ is plane rational of degree n then Gn−2 maps Γ to rational normal curve of degree n − 2 in Pn−2 .

Example Brill-Noether-Algorithm for computing Riemann-Roch spaces. Janko Boehm (TU-KL)


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Example Minimal generators of G for rational curve of degree 5:

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Example

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Local-to-global algorithm Definition The local adjoint ideal of Γ at P ∈ Sing Γ is the largest homogeneous ideal G(P ) ⊂ k [x0 , . . . , xr ] with G(P )P = COΓ,P

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Lemma (BDLP, 2015) G=

\ P ∈Sing Γ

G(P )

The G(P ) can be computed in parallel via normalization.

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Lemma (BDLP, 2015) G=

\ P ∈Sing Γ

G(P )

The G(P ) can be computed in parallel via normalization.

Algorithm (BDLP, 2015) If

1 dU

is the minimal local contribution at P then G(P ) = (d : U )h

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Special types of singularities If Γ ⊂ P2 has a singularity of type An at P = (0 : 0 : 1), then given by f = T 2 + W n +1

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with T , W ∈ C[[x, y ]].


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with T , W ∈ C[[x, y ]].

Compute Tj = T + O (j + 1) inductively.

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with T , W ∈ C[[x, y ]].


Lemma 1 If P = (0, 0) is of type An and s = n+ , then 2 G(P ) = hx s , Ts −1 , y s ih ⊂ C[x, y , z ]

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with T , W ∈ C[[x, y ]].


Lemma 1 If P = (0, 0) is of type An and s = n+ , then 2 G(P ) = hx s , Ts −1 , y s ih ⊂ C[x, y , z ] Similar results for Dn , En and other singularities in Arnold’s list.

Example

f = x 4 − y 2 + x 5 with A3 singularity. Then G(P ) = x 2 , y .

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Parallel Computations in Singular Example > LIB("parallel.lib","random.lib"); > ring R = 0,x(1..4),dp; > ideal I = randomid(maxideal(3),3,100);

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Parallel Computations in Singular Example > > > >

LIB("parallel.lib","random.lib"); ring R = 0,x(1..4),dp; ideal I = randomid(maxideal(3),3,100); proc sizeStd(ideal I, string monord){ def R = basering; list RL = ringlist(R); RL[3][1][1] = monord; def S = ring(RL); setring(S); return(size(std(imap(R,I))));}

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> list commands = "sizeStd","sizeStd"; > list args = list(I,"lp"),list(I,"dp");

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> list commands = "sizeStd","sizeStd"; > list args = list(I,"lp"),list(I,"dp"); > parallelWaitFirst(commands, args); [1] empty list [2] 11

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> list commands = "sizeStd","sizeStd"; > list args = list(I,"lp"),list(I,"dp"); > parallelWaitFirst(commands, args); [1] empty list [2] 11 > parallelWaitAll(commands, args); [1] 55 [2] 11 Janko Boehm (TU-KL)


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Parallelization

There are algorithms whose basic strategy is inherently parallel, whereas others are sequential in nature.

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Parallelization

There are algorithms whose basic strategy is inherently parallel, whereas others are sequential in nature.

Example Normalization is inherently sequential. Local-to-global algorithms for normalization and adjoint ideal are parallel, if the singular locus decomposes. Villamayor’s constructive version of Hironaka’s desingularization theorem is inherently parallel by the iterative use of blow-ups in charts. Modular methods can be used to turn sequential algorithms over Q into parallel ones.

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Example: Resolution of Singularities Theorem (Hironaka, 1964) For every algebraic variety over a field K with char K = 0 a desingularization can be obtained by a finite sequence of blow-ups along smooth centers.

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Example: Resolution of Singularities Theorem (Hironaka, 1964) For every algebraic variety over a field K with char K = 0 a desingularization can be obtained by a finite sequence of blow-ups along smooth centers.

Example Blow-up of the node resolves the singularity

←−

by replacing it by a line of points corresponding to its tangent directions, hence separating the two branches of the curve. Janko Boehm (TU-KL)


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Hironaka Resolution of Singularities Example: x 2 − y 2z 2 = 0

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Hironaka Resolution of Singularities Example: x 2 − y 2z 2 = 0

Resolution Step Search for Center of Blowup

Gluing Step

Blowup in Charts

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Traversal of Tree of Charts required to draw information from resolution data


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Hironaka Resolution of Singularities Example > > > > > > >

LIB "resolve.lib"; ring R= 0,(x,y,z),dp; ideal I = x2-y2z2; list L = resolve(I); def S1 = L[1][1]; setring S1; showBO(BO); ==== Ambient Space: [1]=0 ==== Ideal of Variety: [1]=y(1)^2-1 ==== Exceptional Divisors: ...

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Modular computations Many exact computations in computer algebra are carried out over Q and extensions thereof.

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Modular computations Many exact computations in computer algebra are carried out over Q and extensions thereof. Modular techniques are an important tool to improve performance of algorithms over Q.

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Modular computations Many exact computations in computer algebra are carried out over Q and extensions thereof. Modular techniques are an important tool to improve performance of algorithms over Q. Fundamental approach: 1

Compute modulo primes.

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Modular computations Many exact computations in computer algebra are carried out over Q and extensions thereof. Modular techniques are an important tool to improve performance of algorithms over Q. Fundamental approach: 1 2

Compute modulo primes. Reconstruct result over Q.

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Benefits: Avoid intermediate coefficient growth.

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Benefits: Avoid intermediate coefficient growth. Obtain parallel version of the algorithm.

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Benefits: Avoid intermediate coefficient growth. Obtain parallel version of the algorithm.

Goal: General reconstruction scheme for algorithms in commutative algebra, algebraic geometry, number theory. Janko Boehm (TU-KL)


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Modular computations Example Compute 13 3 1 + = 4 3 12 using modular techniques:

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3 4

7→

Z/5

×

Z/7

×

Z/11

×

Z/101

(2

,

6

,

9

,

26 )

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∼ =

Z/38885

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3 4

7→

Z/5

×

Z/7

×

Z/11

×

Z/101

(2

,

6

,

9

,

26 )

4

,

34 )

∼ =

Z/38885

+ 1 3

7→

(2

Janko Boehm (TU-KL)

,

5

,


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3 4

7→

Z/5

×

Z/7

×

Z/11

×

Z/101

(2

,

6

,

9

,

26 )

4

,

34 )

2

,

60 )

∼ =

Z/38885

+ 1 3

7→

(2

,

5

,

q (4

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,

4

,


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3 4

7→

Z/5

×

Z/7

×

Z/11

×

Z/101

(2

,

6

,

9

,

26 )

4

,

34 )

2

,

60 )

∼ =

Z/38885

7→

22684

+ 1 3

7→

(2

,

5

,

q (4

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,

4

,


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3 4

7→

Z/5

×

Z/7

×

Z/11

×

Z/101

(2

,

6

,

9

,

26 )

4

,

34 )

2

,

60 )

∼ =

Z/38885

7→

22684

+ 1 3

7→

(2

,

5

,

q (4

,

4

,

How to obtain a rational number from 22684? Janko Boehm (TU-KL)


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Rational reconstruction Theorem (Kornerup, Gregory, 1983) The Farey map gcd(a, b ) = 1 a ∈ Q b gcd(b, N ) = 1

|a | , |b | ≤

p

(N − 1)/2 a b

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−→

Z/N

7−→

a·b

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|a | , |b | ≤

p

(N − 1)/2 a b

−→

Z/N

7−→

a·b

−1

is injective.

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|a | , |b | ≤

p

(N − 1)/2 a b

−→

Z/N

7−→

a·b

−1

is injective. Efficient algorithm for preimage.

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|a | , |b | ≤

p

(N − 1)/2 a b

−→

Z/N

7−→

a·b

−1


Example Indeed, in the above example gcd(a, b ) = 1 a b ∈ Q gcd(b, 38885) = 1

Janko Boehm (TU-KL)

|a| , |b | ≤ 139


−→

Z/38885

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|a | , |b | ≤

p

(N − 1)/2 a b

−→

Z/N

7−→

a·b

−1


Example Indeed, in the above example gcd(a, b ) = 1 a b ∈ Q gcd(b, 38885) = 1

|a| , |b | ≤ 139 13 12

Janko Boehm (TU-KL)


−→

Z/38885

7−→

22684 April 18, 2017

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Basic concept for modular computations 1

Compute result over Z/pi for distinct primes p1 , . . . , pr .

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2

For N = p1 · . . . · pr compute lift w.r.t Chinese remainder isomorphism Z/N ∼ = Z/p1 × . . . × Z/pr

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2


3

If exists, compute preimage w.r.t injective Farey map.

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2


3


4

Verify correctness of lift.

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2


3


4


This will yield correct result, provided N is large enough s.t. the Q-result is in source of Farey map, and

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2


3


4


This will yield correct result, provided N is large enough s.t. the Q-result is in source of Farey map, and none of the pi is bad.

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2


3


4


This will yield correct result, provided N is large enough s.t. the Q-result is in source of Farey map, and none of the pi is bad.

Definition A prime p is called bad if the result over Q does not reduce modulo p to the result over Z/p. Janko Boehm (TU-KL)


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Bad primes in Gröbner basis computations For G ⊂ K [X ] = K [x1 , . . . , xn ] and a monomial ordering >, let LM(G ) be the set of lead monomials of G .

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Bad primes in Gröbner basis computations For G ⊂ K [X ] = K [x1 , . . . , xn ] and a monomial ordering >, let LM(G ) be the set of lead monomials of G . For G ⊂ Z[X ] define Gp := G ⊂ Z/p [X ].

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Theorem (Arnold, 2003) Suppose F = {f1 , ..., fr } ⊂ Z[X ] with fi primitve,

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Theorem (Arnold, 2003) Suppose F = {f1 , ..., fr } ⊂ Z[X ] with fi primitve, and G is the reduced Gröbner basis of hF i ⊂ Q[X ],

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Theorem (Arnold, 2003) Suppose F = {f1 , ..., fr } ⊂ Z[X ] with fi primitve, and G is the reduced Gröbner basis of hF i ⊂ Q[X ], G (p ) is the reduced Gr¨ obner basis of hFp i, and

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Theorem (Arnold, 2003) Suppose F = {f1 , ..., fr } ⊂ Z[X ] with fi primitve, and G is the reduced Gröbner basis of hF i ⊂ Q[X ], G (p ) is the reduced Gr¨ obner basis of hFp i, and GZ a minimal strong Gr¨ obnerbasis of hF i ⊂ Z[X ].

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Theorem (Arnold, 2003) Suppose F = {f1 , ..., fr } ⊂ Z[X ] with fi primitve, and G is the reduced Gröbner basis of hF i ⊂ Q[X ], G (p ) is the reduced Gr¨ obner basis of hFp i, and GZ a minimal strong Gr¨ obnerbasis of hF i ⊂ Z[X ]. Then p does not divide any lead coefficient in GZ ⇐⇒ LM G = LM G (p )

⇐⇒ Gp = G (p )

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Theorem (Arnold, 2003) Suppose F = {f1 , ..., fr } ⊂ Z[X ] with fi primitve, and G is the reduced Gröbner basis of hF i ⊂ Q[X ], G (p ) is the reduced Gr¨ obner basis of hFp i, and GZ a minimal strong Gr¨ obnerbasis of hF i ⊂ Z[X ]. Then p does not divide any lead coefficient in GZ ⇐⇒ LM G = LM G (p )

⇐⇒ Gp = G (p ) that is, p is not bad. Janko Boehm (TU-KL)


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Bad primes in Gröbner basis computations Example > > > > >

option("redSB"); ring R = integer,(x, y, z),lp; poly f = x7y5 + x2yz9 + xz11 + y3z9; ideal I = groebner(ideal(diff(f, x), diff(f, y), diff(f,z))); apply(list(I[1..size(I)]),leadcoef);

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Bad primes in Gröbner basis computations Example > option("redSB"); > ring R = integer,(x, y, z),lp; > poly f = x7y5 + x2yz9 + xz11 + y3z9; > ideal I = groebner(ideal(diff(f, x), diff(f, y), diff(f,z))); > apply(list(I[1..size(I)]),leadcoef); 13781115527868730344777310464613260 83521912290113517241074608876444 60 12 4 12 12 45349632 12 1473863040 12 22674816 12 3888 12 12 12 13608 12 108 54 6 2 27 3 1 4 2 2 1 216 1 2 3 1 540 12 108 27 3 1 9 3 1 1 1 1 1 7 1 5 1 1

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and the bad primes are the prime factors p = 2, 3, 5, 7, 11, 13, 257, 247072949, 328838088993550682027

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and the bad primes are the prime factors p = 2, 3, 5, 7, 11, 13, 257, 247072949, 328838088993550682027 Note: The lead coefficients of the Gr¨ obner basis over Q involve only the prime factors 2, 3, 5, 7, 13. Janko Boehm (TU-KL)


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Bad primes Classification of bad primes: Type 1: Input modulo p not valid (no problem)

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Bad primes Classification of bad primes: Type 1: Input modulo p not valid (no problem) Type 2: Failure in the course of the algorithm (e.g. matrix not invertible modulo p, wastes computation time if happens)

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Bad primes Classification of bad primes: Type 1: Input modulo p not valid (no problem) Type 2: Failure in the course of the algorithm (e.g. matrix not invertible modulo p, wastes computation time if happens) Type 3: Computable invariant with known expected value (e.g. dimension) is wrong (have to do expensive test for each prime, although set of bad primes usually is finite)

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Bad primes Classification of bad primes: Type 1: Input modulo p not valid (no problem) Type 2: Failure in the course of the algorithm (e.g. matrix not invertible modulo p, wastes computation time if happens) Type 3: Computable invariant with known expected value (e.g. dimension) is wrong (have to do expensive test for each prime, although set of bad primes usually is finite) Type 4: Computable invariant with unknown expected value (e.g. lead ideal in Gröbner basis computations) is wrong (to detect by a majority vote, have to compute invariant for each modular result and store modular results)

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Bad primes Classification of bad primes: Type 1: Input modulo p not valid (no problem) Type 2: Failure in the course of the algorithm (e.g. matrix not invertible modulo p, wastes computation time if happens) Type 3: Computable invariant with known expected value (e.g. dimension) is wrong (have to do expensive test for each prime, although set of bad primes usually is finite) Type 4: Computable invariant with unknown expected value (e.g. lead ideal in Gröbner basis computations) is wrong (to detect by a majority vote, have to compute invariant for each modular result and store modular results) Type 5: otherwise.

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Example of type 5 bad prime For ideal I ⊂ Q[X ] and prime p define Ip = (I ∩ Z[X ])p .

Example Consider the algorithm I 7→ I =

p

I + Jac(I ) for

x 6 + y 6 + 7x 5 z + x 3 y 2 z − 31x 4 z 2 − 224x 3 z 3 + 244x 2 z 4 + 1632xz 5 + 576z 6

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p

I + Jac(I ) for


Then w.r.t dp

Janko Boehm (TU-KL)

LM(I ) = x 6 = LM(I5 )


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p

I + Jac(I ) for


LM(I ) = x 6 = LM(I5 )

Then w.r.t dp U (0) = U (5) =

q q

I + Jac(I ) = hy , x − 4z i ∩ hy , x + 6z i

I5 + Jac(I5 ) = y , x 2 − z 2 = hy , x − z i ∩ hy , x + z i

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p

I + Jac(I ) for


LM(I ) = x 6 = LM(I5 )

Then w.r.t dp U (0) = U (5) =

q q



U (0)5 = y , (x + z )2

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p

I + Jac(I ) for


LM(I ) = x 6 = LM(I5 )

Then w.r.t dp U (0) = U (5) =

q q



U (0)5 = y , (x + z )2

Hence

Janko Boehm (TU-KL)

U (0)5 6 = U (5)

LM(U (0)) = y , x 2 = LM(U (5)) Algorithms in Singular

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Error tolerant reconstruction Goal: Reconstruct

Janko Boehm (TU-KL)

a b

from r ∈ Z/N in the presence of bad primes.


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Error tolerant reconstruction Goal: Reconstruct ba from r ∈ Z/N in the presence of bad primes. Algorithm: Find (x, y ) with yx = ba in the lattice Λ = h(N, 0), (r , 1)i ⊂ Z2

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Lemma (BDFP, 2015) All (x, y ) ∈ Λ with x 2 + y 2 < N are collinear.

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Proof. Let λ = (x, y ), µ = (c, d ) ∈ Λ with x 2 + y 2 , c 2 + d 2 < N. Then y µ − d λ = (yc − xd, 0) ∈ Λ, so N |(yc − xd ). By Cauchy–Schwarz |yc − xd | < N, hence yc = xd.

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Proof. Let λ = (x, y ), µ = (c, d ) ∈ Λ with x 2 + y 2 , c 2 + d 2 < N. Then y µ − d λ = (yc − xd, 0) ∈ Λ, so N |(yc − xd ). By Cauchy–Schwarz |yc − xd | < N, hence yc = xd. Now suppose N = N0 · M with gcd(N 0 , M ) = 1. Janko Boehm (TU-KL)


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Error tolerant reconstruction Think of N 0 as the product of the good primes with correct result s, and of M as the product of the bad primes with wrong result t.

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Theorem (BDFP, 2015) If

r 7→ (s, t )

and

with respect to Z/N ∼ = Z/N 0 × Z/M a ≡ s mod N 0 b

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r 7→ (s, t )

and

with respect to Z/N ∼ = Z/N 0 × Z/M a ≡ s mod N 0 b

then (aM, bM ) ∈ Λ.

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r 7→ (s, t )

with respect to Z/N ∼ = Z/N 0 × Z/M

and

a ≡ s mod N 0 b

then (aM, bM ) ∈ Λ. So if

(a2 + b 2 )M < N 0 , then (by the lemma) x a = for all (x, y ) ∈ Λ with (x 2 + y 2 ) < N y b and such vectors exist. Janko Boehm (TU-KL)


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r 7→ (s, t )

with respect to Z/N ∼ = Z/N 0 × Z/M

and

a ≡ s mod N 0 b

then (aM, bM ) ∈ Λ. So if

(a2 + b 2 )M < N 0 , then (by the lemma) x a = for all (x, y ) ∈ Λ with (x 2 + y 2 ) < N y b and such vectors exist. Moreover, if gcd(a, b ) = 1 and (x, y ) is a shortest vector 6= 0 in Λ, we also have gcd(x, y )|M. Janko Boehm (TU-KL)


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Error tolerant reconstruction via Gauss-Lagrange Hence, if N 0 M, the Gauss-Lagrange-Algorithm for finding a shortest vector (x, y ) ∈ Λ gives ba independently of t, provided x 2 + y 2 < N.

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Algorithm (Error tolerant reconstruction) function ErrorTolerantReconstruction(r::Integer, N::Integer) a1 = [N, 0] a2 = [r, 1] while dot(a1, a1) > dot(a2, a2) q = dot(a1, a2)//dot(a2, a2) a1, a2 = a2, a1 - Integer(round(q))*a2 end if dot(a1, a1) < N return a1[1]//a1[2] else return false end end

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Algorithm (Error tolerant reconstruction) function ErrorTolerantReconstruction(r::Integer, N::Integer) a1 = [N, 0] a2 = [r, 1] while dot(a1, a1) > dot(a2, a2) q = dot(a1, a2)//dot(a2, a2) a1, a2 = a2, a1 - Integer(round(q))*a2 end if dot(a1, a1) < N return a1[1]//a1[2] else return false end end

Singular-kernel

Julia

Singular-interpreter

0.001

0.005

0.055

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(in seconds, bitlength 500) April 18, 2017

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Reconstruction via Gauss-Lagrange Example We reconstruct

13 12

from 22684 ∈ Z/38885

by determining a shortest vector in the lattice

h(38885, 0), (22684, 1)i ⊂ Z2 via Gauss-Lagrange

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Reconstruction via Gauss-Lagrange Example We reconstruct

13 12

from 22684 ∈ Z/38885

by determining a shortest vector in the lattice

h(38885, 0), (22684, 1)i ⊂ Z2 via Gauss-Lagrange

(38885, 0) = 2 · (22684, 1) + (−6483, −2), (22684, 1) = −3 · (−6483, −2) + (3235, −5), (−6483, −2) = 2 · (3235, −5) + (−13, −12), (3235, −5) = −134 · (−13, −12) + (1493, −1613). Janko Boehm (TU-KL)


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Reconstruction via Gauss-Lagrange Example Now introduce an error in the modular results: Z/5

×

Z/7

×

Z/11

×

Z/101

∼ =

Z/38885

(4

,

4

,

2

,

60 )

7→

22684

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×

Z/7

×

Z/11

×

Z/101

∼ =

Z/38885

(4

,

4

,

2

,

60 )

7→

22684

(4

,

2

,

2

60 )

7→

464

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×

Z/7

×

Z/11

×

Z/101

∼ =

Z/38885

(4

,

4

,

2

,

60 )

7→

22684

(4

,

2

,

2

60 )

7→

464

Error tolerant reconstruction computes

(38885, 0) = 84 · (464, 1) + (−91, −84), (464, 1) = −3 · (−91, −84) + (191, −251)

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×

Z/7

×

Z/11

×

Z/101

∼ =

Z/38885

(4

,

4

,

2

,

60 )

7→

22684

(4

,

2

,

2

60 )

7→

464


(38885, 0) = 84 · (464, 1) + (−91, −84), (464, 1) = −3 · (−91, −84) + (191, −251) hence yields

Janko Boehm (TU-KL)

91 7 · 13 13 = = . 84 7 · 12 12


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×

Z/7

×

Z/11

×

Z/101

∼ =

Z/38885

(4

,

4

,

2

,

60 )

7→

22684

(4

,

2

,

2

60 )

7→

464


(38885, 0) = 84 · (464, 1) + (−91, −84), (464, 1) = −3 · (−91, −84) + (191, −251) hence yields Note that

91 7 · 13 13 = = . 84 7 · 12 12

(132 + 122 ) · 7 = 2191 < 5555 = 5 · 11 · 101.

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General reconstruction scheme Setup: For ideal I ⊂ Q[X ] compute ideal (or module) U (0) associated to I by deterministic algorithm.

Algorithm For Ip compute result U (p ) over Z/p for p in finite set of primes P .

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Algorithm For Ip compute result U (p ) over Z/p for p in finite set of primes P . Reduce P according to majority vote on LM(U (p )).

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Algorithm For Ip compute result U (p ) over Z/p for p in finite set of primes P . Reduce P according to majority vote on LM(U (p )). For N = ∏p ∈P p compute termwise CRT–lift U (N ) to Z/N.

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Algorithm For Ip compute result U (p ) over Z/p for p in finite set of primes P . Reduce P according to majority vote on LM(U (p )). For N = ∏p ∈P p compute termwise CRT–lift U (N ) to Z/N. Lift U (N ) by error tolerant rational reconstruction to U.

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Algorithm For Ip compute result U (p ) over Z/p for p in finite set of primes P . Reduce P according to majority vote on LM(U (p )). For N = ∏p ∈P p compute termwise CRT–lift U (N ) to Z/N. Lift U (N ) by error tolerant rational reconstruction to U. Test Up = U (p ) for random prime p.

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Algorithm For Ip compute result U (p ) over Z/p for p in finite set of primes P . Reduce P according to majority vote on LM(U (p )). For N = ∏p ∈P p compute termwise CRT–lift U (N ) to Z/N. Lift U (N ) by error tolerant rational reconstruction to U. Test Up = U (p ) for random prime p. Verify U = U (0).

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Algorithm For Ip compute result U (p ) over Z/p for p in finite set of primes P . Reduce P according to majority vote on LM(U (p )). For N = ∏p ∈P p compute termwise CRT–lift U (N ) to Z/N. Lift U (N ) by error tolerant rational reconstruction to U. Test Up = U (p ) for random prime p. Verify U = U (0). If lift, test or verification fails, then enlarge P .

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Algorithm For Ip compute result U (p ) over Z/p for p in finite set of primes P . Reduce P according to majority vote on LM(U (p )). For N = ∏p ∈P p compute termwise CRT–lift U (N ) to Z/N. Lift U (N ) by error tolerant rational reconstruction to U. Test Up = U (p ) for random prime p. Verify U = U (0). If lift, test or verification fails, then enlarge P .

Theorem (BDFP, 2015) If the set of bad primes for computing U (0) from I is finite, then this algorithm terminates with the correct result. Janko Boehm (TU-KL)


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Timings in Singular for Adjoint Ideal 1 Plane curve fn of degree n with (n− 2 ) singularities of type A1 .

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Timings in Singular for Adjoint Ideal

locNormal Maple-IB LA IQ locIQ ADE modLocIQ

probablisitic

parallel

1 Plane curve fn of degree n with (n− 2 ) singularities of type A1 .

f5

f6

f7

2.1 5.1 98 1.3 1.3 .18 6.4 6.2 .36 .21

56 47 4400 54 54 1.2 19 18 1.6 0.48

318 3800 3800 49 150 104 51 5.2

(1) (1) [33] [33] (74) (74)

(1) (1) [53] [53] (153) (153)

(1) (1) [75] [75] (230) (230)

[primes] (cores) Janko Boehm (TU-KL)


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References J. Boehm, W. Decker, C. Fieker, G. Pfister. The use of bad primes in rational reconstruction, Math. Comp. 84 (2015). J. Boehm, W. Decker, S. Laplagne, G. Pfister, A. Steenpaß, S. Steidel. Parallel algorithms for normalization, J. Symb. Comp. 51 (2013). J. Boehm, W. Decker, G. Pfister, S. Laplagne. Local to global algorithms for the Gorenstein adjoint ideal of a curve, arXiv:1505.05040. P. Kornerup, R. T. Gregory, Mapping integers and Hensel codes onto Farey fractions, BIT 23 (1983). E. Arnold, Modular algorithms for computing Gr¨ obner bases, J. Symb. Comp. 35 (2003). G.-M. Greuel, S. Laplagne, S. Seelisch, Normalization of rings, J. Symb. Comp. (2010).

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Singular and GPI-Space Features of GPI-Space developed by Fraunhofer Institute for Industrial Mathematics ITWM, Kaiserslautern: Distributed run-time system suitable for massively parallel computations.

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Singular and GPI-Space Features of GPI-Space developed by Fraunhofer Institute for Industrial Mathematics ITWM, Kaiserslautern: Distributed run-time system suitable for massively parallel computations. Virtual memory layer.

Janko Boehm (TU-KL)


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Singular and GPI-Space Features of GPI-Space developed by Fraunhofer Institute for Industrial Mathematics ITWM, Kaiserslautern: Distributed run-time system suitable for massively parallel computations. Virtual memory layer. Modeling with Petri nets. Auto-parallelization engine. Proof of concept integration of Singular in GPI-Space.

Janko Boehm (TU-KL)


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GPI-Space: A Petri net

Clock at time t = 4:

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GPI-Space: A Petri net

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Singular and GPI-Space Features of GPI-Space developed by Fraunhofer Institute for Industrial Mathematics ITWM, Kaiserslautern: Distributed run-time system suitable for massively parallel computations. Virtual memory layer. Modeling with Petri nets.

Janko Boehm (TU-KL)


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Singular and GPI-Space Features of GPI-Space developed by Fraunhofer Institute for Industrial Mathematics ITWM, Kaiserslautern: Distributed run-time system suitable for massively parallel computations. Virtual memory layer. Modeling with Petri nets. Auto-parallelization engine. Proof of concept integration of Singular in GPI-Space.

Janko Boehm (TU-KL)


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GPI-Space: Scheduler

Janko Boehm (TU-KL)


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Singular and GPI-Space Features of GPI-Space developed by Fraunhofer Institute for Industrial Mathematics ITWM, Kaiserslautern: Distributed run-time system suitable for massively parallel computations. Virtual memory layer. Modeling with Petri nets. Auto-parallelization engine.

Janko Boehm (TU-KL)


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Singular and GPI-Space Features of GPI-Space developed by Fraunhofer Institute for Industrial Mathematics ITWM, Kaiserslautern: Distributed run-time system suitable for massively parallel computations. Virtual memory layer. Modeling with Petri nets. Auto-parallelization engine. Integration of Singular in GPI-Space. Cluster at ITWM with ≈ 104 nodes.

Janko Boehm (TU-KL)


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Singular and GPI-Space Example Algorithm for determining smoothness by local descent in codimension relative to a smooth complete intersection (as in Hironaka’s resolution of singularities). Descent to any desired size of minors in Jacobian criterion.

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Singular and GPI-Space Example Algorithm for determining smoothness by local descent in codimension relative to a smooth complete intersection (as in Hironaka’s resolution of singularities). Descent to any desired size of minors in Jacobian criterion.

Boehm, J., Fr¨ uhbis-Kr¨ uger: A smoothness test for higher codimensions. arXiv:1603.09241 JSC (to appear). Janko Boehm (TU-KL)


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Good Quotients The goal of Geometric Invariant Theory (GIT) is to assign to a given algebraic variety X with action of a reductive group G a reasonable quotient space X //G .

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Good Quotients The goal of Geometric Invariant Theory (GIT) is to assign to a given algebraic variety X with action of a reductive group G a reasonable quotient space X //G . Two main problems:

Janko Boehm (TU-KL)


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Good Quotients The goal of Geometric Invariant Theory (GIT) is to assign to a given algebraic variety X with action of a reductive group G a reasonable quotient space X //G . Two main problems: 1

The orbit space X /G is not a good candidate for X //G : C∗ × C → C,

Janko Boehm (TU-KL)


t · x = tx

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t · x = tx

Instead, for X affine define X //G = Spec K [X ]G as the spectrum of the (finitely generated) invariant ring of the functions on X . For general X , glue the quotients of an affine covering.

Janko Boehm (TU-KL)


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t · x = tx

Instead, for X affine define X //G = Spec K [X ]G

2

as the spectrum of the (finitely generated) invariant ring of the functions on X . For general X , glue the quotients of an affine covering. The quotient X //G may not carry much information.

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t · x = tx

Instead, for X affine define X //G = Spec K [X ]G

2

as the spectrum of the (finitely generated) invariant ring of the functions on X . For general X , glue the quotients of an affine covering. The quotient X //G may not carry much information. Hence pass to open subset U ⊂ X . Janko Boehm (TU-KL)


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Good Quotients Example C∗ × C2 → C2 ,

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t · (x, y ) = (tx, ty )


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t · (x, y ) = (tx, ty )

U = C2

U//C∗ = {pt }

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t · (x, y ) = (tx, ty )

U = C2

U = C2 \{0}

U//C∗ = {pt }

U//C∗ = P1

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GIT-Fan In general, there are many choices for these open subsets U ⊂ X leading to different quotients. To describe this behaviour, Dolgachev and Hu introduced the GIT-fan, a polyhedral fan describing the variation of GIT-quotients.

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GIT-Fan In general, there are many choices for these open subsets U ⊂ X leading to different quotients. To describe this behaviour, Dolgachev and Hu introduced the GIT-fan, a polyhedral fan describing the variation of GIT-quotients. We focus on the action of an algebraic torus G = (C∗ )k on an affine variety X ⊂ Cr .

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GIT-Fan In general, there are many choices for these open subsets U ⊂ X leading to different quotients. To describe this behaviour, Dolgachev and Hu introduced the GIT-fan, a polyhedral fan describing the variation of GIT-quotients. We focus on the action of an algebraic torus G = (C∗ )k on an affine variety X ⊂ Cr . Setup: ideal a ⊂ C[T1 , . . . , Tr ] defining X , matrix Q = (q1 , . . . , qr ) ∈ Zk ×r such that a is homogeneous w.r.t. grading deg(Ti ) = qi ∈ Zk .

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GIT-Fan In general, there are many choices for these open subsets U ⊂ X leading to different quotients. To describe this behaviour, Dolgachev and Hu introduced the GIT-fan, a polyhedral fan describing the variation of GIT-quotients. We focus on the action of an algebraic torus G = (C∗ )k on an affine variety X ⊂ Cr . Setup: ideal a ⊂ C[T1 , . . . , Tr ] defining X , matrix Q = (q1 , . . . , qr ) ∈ Zk ×r such that a is homogeneous w.r.t. grading deg(Ti ) = qi ∈ Zk .

Example For C∗ × C2 → C2 , U1 = C2 U2 = C2 \{0} Janko Boehm (TU-KL)

t · (x, y ) = (tx, ty ) Λ(h0i , (1, 1)) = Algorithms in Singular

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Computing GIT-Fans

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Computing GIT-Fans In case of a torus acting on an affine variety, Berchthold/Hausen and Keicher have developed a method for computing the GIT-fan.

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Computing GIT-Fans In case of a torus acting on an affine variety, Berchthold/Hausen and Keicher have developed a method for computing the GIT-fan. Decomposition into torus orbits corresponding to faces γ ≺ Qr≥0 : Cr =

[ γ

O (γ)

O (γ) = (C∗ )r · ∑e ∈γ ei = {(z1 , . . . , zr ) ∈ Cr | zi 6= 0 ⇔ ei ∈ γ} i

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[ γ

O (γ)


Proposition Face γ ≺ Qr≥0 is called an a-face if the following equivalent conditions are satisfied:

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[ γ

O (γ)


Proposition Face γ ≺ Qr≥0 is called an a-face if the following equivalent conditions are satisfied: 1

X ∩ O (γ) 6= ∅

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[ γ

O (γ)


Proposition Face γ ≺ Qr≥0 is called an a-face if the following equivalent conditions are satisfied: 1 2 3

X ∩ O (γ) 6= ∅ There is x ∈ X with xi 6= 0 ⇔ ei ∈ γ a|Ti =0 for ei ∈/ γ : hT1 · · · Tr i∞ 6= h1i

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[ γ

O (γ)


Proposition Face γ ≺ Qr≥0 is called an a-face if the following equivalent conditions are satisfied: 1 2 3

X ∩ O (γ) 6= ∅ There is x ∈ X with xi 6= 0 ⇔ ei ∈ γ a|Ti =0 for ei ∈/ γ : hT1 · · · Tr i∞ 6= h1i

The orbit cones are the Q (γ) = cone(qi | ei ∈ γ) with γ an a-face. Janko Boehm (TU-KL)


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Computing GIT-Fans 1

Determine a-faces.

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Determine a-faces.

2

Compute set of orbit cones Ω = {Q (γ) | γ an a-face} where Q (γ) = cone(qi | ei ∈ γ) ⊂ Γ = Q (Qr≥0 ) = cone(q1 , . . . , qr ) ⊂ Qk is projection of γ with respect to Q.

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Determine a-faces.

2

Compute set of orbit cones Ω = {Q (γ) | γ an a-face} where Q (γ) = cone(qi | ei ∈ γ) ⊂ Γ = Q (Qr≥0 ) = cone(q1 , . . . , qr ) ⊂ Qk is projection of γ with respect to Q.

3

Determine GIT-fan: Λ(a, Q ) = {λΩ (w ) | w ∈ Γ}

where

λ Ω (w ) =

\

η

w ∈η ∈Ω

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GIT-Algorithm Algorithm Input: Ideal a ⊂ C[T1 , . . . , Tr ] and matrix Q ∈ Zk ×r of full rank such that a is homogeneous w.r.t. multigrading by Q. Output: The set of maximal cones of Λ(a, Q ).

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GIT-Algorithm Algorithm Input: Ideal a ⊂ C[T1 , . . . , Tr ] and matrix Q ∈ Zk ×r of full rank such that a is homogeneous w.r.t. multigrading by Q. Output: The set of maximal cones of Λ(a, Q ). 1:

A := {γ ≺ Qr≥0 | γ is an a-face}

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GIT-Algorithm Algorithm Input: Ideal a ⊂ C[T1 , . . . , Tr ] and matrix Q ∈ Zk ×r of full rank such that a is homogeneous w.r.t. multigrading by Q. Output: The set of maximal cones of Λ(a, Q ). 1: 2:

A := {γ ≺ Qr≥0 | γ is an a-face} Ω := {Q (γ) | γ ∈ A}

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GIT-Algorithm Algorithm Input: Ideal a ⊂ C[T1 , . . . , Tr ] and matrix Q ∈ Zk ×r of full rank such that a is homogeneous w.r.t. multigrading by Q. Output: The set of maximal cones of Λ(a, Q ). 1: 2: 3:

A := {γ ≺ Qr≥0 | γ is an a-face} Ω := {Q (γ) | γ ∈ A} Choose a vector w0 ∈ Q (Qr≥0 ) such that dim(λΩ (w0 )) = k.

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GIT-Algorithm Algorithm Input: Ideal a ⊂ C[T1 , . . . , Tr ] and matrix Q ∈ Zk ×r of full rank such that a is homogeneous w.r.t. multigrading by Q. Output: The set of maximal cones of Λ(a, Q ). 1: 2: 3: 4:

A := {γ ≺ Qr≥0 | γ is an a-face} Ω := {Q (γ) | γ ∈ A} Choose a vector w0 ∈ Q (Qr≥0 ) such that dim(λΩ (w0 )) = k. C := {λΩ (w0 )}

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GIT-Algorithm Algorithm Input: Ideal a ⊂ C[T1 , . . . , Tr ] and matrix Q ∈ Zk ×r of full rank such that a is homogeneous w.r.t. multigrading by Q. Output: The set of maximal cones of Λ(a, Q ). 1: 2: 3: 4: 5:

A := {γ ≺ Qr≥0 | γ is an a-face} Ω := {Q (γ) | γ ∈ A} Choose a vector w0 ∈ Q (Qr≥0 ) such that dim(λΩ (w0 )) = k. C := {λΩ (w0 )} F := {(η, λΩ (w0 )) | η ≺ λΩ (w0 ) interior facet}.

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GIT-Algorithm Algorithm Input: Ideal a ⊂ C[T1 , . . . , Tr ] and matrix Q ∈ Zk ×r of full rank such that a is homogeneous w.r.t. multigrading by Q. Output: The set of maximal cones of Λ(a, Q ). 1: 2: 3: 4: 5: 6:

A := {γ ≺ Qr≥0 | γ is an a-face} Ω := {Q (γ) | γ ∈ A} Choose a vector w0 ∈ Q (Qr≥0 ) such that dim(λΩ (w0 )) = k. C := {λΩ (w0 )} F := {(η, λΩ (w0 )) | η ≺ λΩ (w0 ) interior facet}. while there is (η, λ) ∈ F do

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GIT-Algorithm Algorithm Input: Ideal a ⊂ C[T1 , . . . , Tr ] and matrix Q ∈ Zk ×r of full rank such that a is homogeneous w.r.t. multigrading by Q. Output: The set of maximal cones of Λ(a, Q ). 1: 2: 3: 4: 5: 6: 7:

A := {γ ≺ Qr≥0 | γ is an a-face} Ω := {Q (γ) | γ ∈ A} Choose a vector w0 ∈ Q (Qr≥0 ) such that dim(λΩ (w0 )) = k. C := {λΩ (w0 )} F := {(η, λΩ (w0 )) | η ≺ λΩ (w0 ) interior facet}. while there is (η, λ) ∈ F do Find w ∈ Q (γ) such that w 6∈ λ and λΩ (w ) ∩ λ = η.

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GIT-Algorithm Algorithm Input: Ideal a ⊂ C[T1 , . . . , Tr ] and matrix Q ∈ Zk ×r of full rank such that a is homogeneous w.r.t. multigrading by Q. Output: The set of maximal cones of Λ(a, Q ). 1: 2: 3: 4: 5: 6: 7: 8:

A := {γ ≺ Qr≥0 | γ is an a-face} Ω := {Q (γ) | γ ∈ A} Choose a vector w0 ∈ Q (Qr≥0 ) such that dim(λΩ (w0 )) = k. C := {λΩ (w0 )} F := {(η, λΩ (w0 )) | η ≺ λΩ (w0 ) interior facet}. while there is (η, λ) ∈ F do Find w ∈ Q (γ) such that w 6∈ λ and λΩ (w ) ∩ λ = η. C := C ∪ {λΩ (w )}

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GIT-Algorithm Algorithm Input: Ideal a ⊂ C[T1 , . . . , Tr ] and matrix Q ∈ Zk ×r of full rank such that a is homogeneous w.r.t. multigrading by Q. Output: The set of maximal cones of Λ(a, Q ). 1: 2: 3: 4: 5: 6: 7: 8: 9:

A := {γ ≺ Qr≥0 | γ is an a-face} Ω := {Q (γ) | γ ∈ A} Choose a vector w0 ∈ Q (Qr≥0 ) such that dim(λΩ (w0 )) = k. C := {λΩ (w0 )} F := {(η, λΩ (w0 )) | η ≺ λΩ (w0 ) interior facet}. while there is (η, λ) ∈ F do Find w ∈ Q (γ) such that w 6∈ λ and λΩ (w ) ∩ λ = η. C := C ∪ {λΩ (w )} F := F {(τ, λΩ (w )) | τ ⊂ λΩ (w ) interior facet}

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GIT-Algorithm Algorithm Input: Ideal a ⊂ C[T1 , . . . , Tr ] and matrix Q ∈ Zk ×r of full rank such that a is homogeneous w.r.t. multigrading by Q. Output: The set of maximal cones of Λ(a, Q ). 1: 2: 3: 4: 5: 6: 7: 8: 9: 10:

A := {γ ≺ Qr≥0 | γ is an a-face} Ω := {Q (γ) | γ ∈ A} Choose a vector w0 ∈ Q (Qr≥0 ) such that dim(λΩ (w0 )) = k. C := {λΩ (w0 )} F := {(η, λΩ (w0 )) | η ≺ λΩ (w0 ) interior facet}. while there is (η, λ) ∈ F do Find w ∈ Q (γ) such that w 6∈ λ and λΩ (w ) ∩ λ = η. C := C ∪ {λΩ (w )} F := F {(τ, λΩ (w )) | τ ⊂ λΩ (w ) interior facet} return C Janko Boehm (TU-KL)


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Fast Monomial Containment Test Generalization of (Sturmfels, 1996), where degree reverse lex (dp) is used:

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Proposition Let > be a monomial ordering on R = K [Y1 , . . . , Yn ] and G a Gröbner basis of I . Suppose that for all f ∈ G Yn | f

Janko Boehm (TU-KL)

⇐⇒

Yn | LM> (f ).


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Proposition Let > be a monomial ordering on R = K [Y1 , . . . , Yn ] and G a Gröbner basis of I . Suppose that for all f ∈ G Yn | f Then

⇐⇒

Yn | LM> (f ).

f i f ∈ G and i ≥ 0 maximal such that Yn | f Yni is a Gröbner basis for I : Yn∞ .

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⇐⇒

Yn | LM> (f ).


Algorithm To compute I : (Y1 · . . . · Yn )∞ ,

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⇐⇒

Yn | LM> (f ).


Algorithm To compute I : (Y1 · . . . · Yn )∞ , replace any remainder r 6= 0 in Buchberger’s algorithm by r a where aj is maximal s.t. Yj j | r . Y1a1 · . . . · Ynan Janko Boehm (TU-KL)


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Timings

Saturation in product of variables for ideal a with 225 generators in 40 variables with variables not in J equal to 0:

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Timings

Saturation in product of variables for ideal a with 225 generators in 40 variables with variables not in J equal to 0:

{1, . . . , 40}\J {3, 4, 5, 7, . . . , 15} {9, 11, 12, 13, 15} {11, 12, 13, 15} {9, 11, 14, 15} {9, 11, 15} {9, 11, 13}

40 − |J | 28 35 36 36 37 37

a-face no no no yes yes no

divgbsat 1 1 1 64 1170 1

gbsat 761 57200 44100 121000 114000 31400

sat 517 ∗ ∗ ∗ ∗ ∗

rabinowitsch 342 ∗ ∗ ∗ ∗ ∗

(in seconds, * did not finish in > 2 days)

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Symmetry Groups of Torus Actions Definition A symmetry group of the action of (C∗ )k on X is a subgroup G ⊂ Sr of the symmetric group such that there are group actions

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Symmetry Groups of Torus Actions Definition A symmetry group of the action of (C∗ )k on X is a subgroup G ⊂ Sr of the symmetric group such that there are group actions G × K[T1 , . . . , Tr ] → K[T1 , . . . , Tr ], (σ, Tj ) 7→ σ(Tj ) = cσ,j · Tσ(j ) G × Qr → Qr , (σ, ej ) 7→ σ(ej ) = eσ(j ) k G × Q → Qk , (σ, v ) 7→ Aσ · v

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Symmetry Groups of Torus Actions Definition A symmetry group of the action of (C∗ )k on X is a subgroup G ⊂ Sr of the symmetric group such that there are group actions G × K[T1 , . . . , Tr ] → K[T1 , . . . , Tr ], (σ, Tj ) 7→ σ(Tj ) = cσ,j · Tσ(j ) G × Qr → Qr , (σ, ej ) 7→ σ(ej ) = eσ(j ) k G × Q → Qk , (σ, v ) 7→ Aσ · v with Aσ ∈ GL(k, Q) and cσ ∈ Tr

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Symmetry Groups of Torus Actions Definition A symmetry group of the action of (C∗ )k on X is a subgroup G ⊂ Sr of the symmetric group such that there are group actions G × K[T1 , . . . , Tr ] → K[T1 , . . . , Tr ], (σ, Tj ) 7→ σ(Tj ) = cσ,j · Tσ(j ) G × Qr → Qr , (σ, ej ) 7→ σ(ej ) = eσ(j ) k G × Q → Qk , (σ, v ) 7→ Aσ · v with Aσ ∈ GL(k, Q) and cσ ∈ Tr such that G · a = a and that for each σ ∈ G the following diagram is commutative:

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Symmetry Groups of Torus Actions Definition A symmetry group of the action of (C∗ )k on X is a subgroup G ⊂ Sr of the symmetric group such that there are group actions G × K[T1 , . . . , Tr ] → K[T1 , . . . , Tr ], (σ, Tj ) 7→ σ(Tj ) = cσ,j · Tσ(j ) G × Qr → Qr , (σ, ej ) 7→ σ(ej ) = eσ(j ) k G × Q → Qk , (σ, v ) 7→ Aσ · v with Aσ ∈ GL(k, Q) and cσ ∈ Tr such that G · a = a and that for each σ ∈ G the following diagram is commutative: Qr

7−→ −→

Q↓ Qk

−→

ej

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Aσ

eσ (j ) Qr ↓Q Qk


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Representation of GIT-Cones

Perfect hash function for cones with compatible group action

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Perfect hash function for cones with compatible group action  hΩ : Λ(a, Q ) → {0, 1}Ω ,

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λ 7→ 


Ω →( {0, 1} 1 λ⊂ϑ ϑ 7→ 0 λ 6⊂ ϑ

 

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G × {0, 1}Ω → {0, 1}Ω ,

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λ 7→ 

Ω →( {0, 1} 1 λ⊂ϑ ϑ 7→ 0 λ 6⊂ ϑ

(g , b ) 7 →


 

Ω → {0, 1} ϑ 7 → b (g −1 · ϑ )

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G × {0, 1}Ω → {0, 1}Ω ,

λ 7→ 

Ω →( {0, 1} 1 λ⊂ϑ ϑ 7→ 0 λ 6⊂ ϑ

(g , b ) 7 →

 

Ω → {0, 1} ϑ 7 → b (g −1 · ϑ )

such that g · hΩ ( λ ) = hΩ (g · λ ).

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Symmetric GIT-Algorithm Algorithm (System of representatives of the G -orbits on Λ(a, Q )(k )) 1:

S := system of representatives of G -orbits of faces(Qr≥0 )

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Symmetric GIT-Algorithm Algorithm (System of representatives of the G -orbits on Λ(a, Q )(k )) 1: 2:

S := system of representatives of G -orbits of faces(Qr≥0 ) A := {γ ∈ S | γ is a-face}

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Symmetric GIT-Algorithm Algorithm (System of representatives of the G -orbits on Λ(a, Q )(k )) 1: 2: 3:

S := system of representatives of G -orbits of faces(Qr≥0 ) A := {γ ∈ S | γ is a-face} S Ω := γ∈A G · Q (γ)

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Symmetric GIT-Algorithm Algorithm (System of representatives of the G -orbits on Λ(a, Q )(k )) 1: 2: 3: 4:

S := system of representatives of G -orbits of faces(Qr≥0 ) A := {γ ∈ S | γ is a-face} S Ω := γ∈A G · Q (γ) Ω := set of minimal elements of Ω(k )

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Symmetric GIT-Algorithm Algorithm (System of representatives of the G -orbits on Λ(a, Q )(k )) 1: 2: 3: 4: 5:

S := system of representatives of G -orbits of faces(Qr≥0 ) A := {γ ∈ S | γ is a-face} S Ω := γ∈A G · Q (γ) Ω := set of minimal elements of Ω(k ) Choose w0 ∈ Q (Γ) such that dim(λΩ (w0 )) = k.

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Symmetric GIT-Algorithm Algorithm (System of representatives of the G -orbits on Λ(a, Q )(k )) 1: 2: 3: 4: 5: 6:

S := system of representatives of G -orbits of faces(Qr≥0 ) A := {γ ∈ S | γ is a-face} S Ω := γ∈A G · Q (γ) Ω := set of minimal elements of Ω(k ) Choose w0 ∈ Q (Γ) such that dim(λΩ (w0 )) = k. C := {λΩ (w0 )}, H := {hΩ (λΩ (w0 ))}

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Symmetric GIT-Algorithm Algorithm (System of representatives of the G -orbits on Λ(a, Q )(k )) 1: 2: 3: 4: 5: 6: 7:

S := system of representatives of G -orbits of faces(Qr≥0 ) A := {γ ∈ S | γ is a-face} S Ω := γ∈A G · Q (γ) Ω := set of minimal elements of Ω(k ) Choose w0 ∈ Q (Γ) such that dim(λΩ (w0 )) = k. C := {λΩ (w0 )}, H := {hΩ (λΩ (w0 ))} F := {(η, v ) | η ≺ λΩ (w0 ) interior facet with inner normal v }

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Symmetric GIT-Algorithm Algorithm (System of representatives of the G -orbits on Λ(a, Q )(k )) 1: 2: 3: 4: 5: 6: 7: 8: 9:

S := system of representatives of G -orbits of faces(Qr≥0 ) A := {γ ∈ S | γ is a-face} S Ω := γ∈A G · Q (γ) Ω := set of minimal elements of Ω(k ) Choose w0 ∈ Q (Γ) such that dim(λΩ (w0 )) = k. C := {λΩ (w0 )}, H := {hΩ (λΩ (w0 ))} F := {(η, v ) | η ≺ λΩ (w0 ) interior facet with inner normal v } while there is (η, v ) ∈ F do Find w ∈ Q (Γ) such that η ≺ λΩ (w ) is a facet and −v ∈ λΩ (w )∨ .

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Symmetric GIT-Algorithm Algorithm (System of representatives of the G -orbits on Λ(a, Q )(k )) 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11:

S := system of representatives of G -orbits of faces(Qr≥0 ) A := {γ ∈ S | γ is a-face} S Ω := γ∈A G · Q (γ) Ω := set of minimal elements of Ω(k ) Choose w0 ∈ Q (Γ) such that dim(λΩ (w0 )) = k. C := {λΩ (w0 )}, H := {hΩ (λΩ (w0 ))} F := {(η, v ) | η ≺ λΩ (w0 ) interior facet with inner normal v } while there is (η, v ) ∈ F do Find w ∈ Q (Γ) such that η ≺ λΩ (w ) is a facet and −v ∈ λΩ (w )∨ . if G · hΩ (λΩ (w )) ∩ H = ∅ then C := C ∪ {λΩ (w )}, H := H ∪ {hΩ (λΩ (w ))}

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Symmetric GIT-Algorithm Algorithm (System of representatives of the G -orbits on Λ(a, Q )(k )) 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12:

S := system of representatives of G -orbits of faces(Qr≥0 ) A := {γ ∈ S | γ is a-face} S Ω := γ∈A G · Q (γ) Ω := set of minimal elements of Ω(k ) Choose w0 ∈ Q (Γ) such that dim(λΩ (w0 )) = k. C := {λΩ (w0 )}, H := {hΩ (λΩ (w0 ))} F := {(η, v ) | η ≺ λΩ (w0 ) interior facet with inner normal v } while there is (η, v ) ∈ F do Find w ∈ Q (Γ) such that η ≺ λΩ (w ) is a facet and −v ∈ λΩ (w )∨ . if G · hΩ (λΩ (w )) ∩ H = ∅ then C := C ∪ {λΩ (w )}, H := H ∪ {hΩ (λΩ (w ))} ˜ v˜ ) | η˜ ≺ λΩ (w ) interior facet w. inner normal v˜ } F := F {(η,

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Symmetric GIT-Algorithm Algorithm (System of representatives of the G -orbits on Λ(a, Q )(k )) 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14:

S := system of representatives of G -orbits of faces(Qr≥0 ) A := {γ ∈ S | γ is a-face} S Ω := γ∈A G · Q (γ) Ω := set of minimal elements of Ω(k ) Choose w0 ∈ Q (Γ) such that dim(λΩ (w0 )) = k. C := {λΩ (w0 )}, H := {hΩ (λΩ (w0 ))} F := {(η, v ) | η ≺ λΩ (w0 ) interior facet with inner normal v } while there is (η, v ) ∈ F do Find w ∈ Q (Γ) such that η ≺ λΩ (w ) is a facet and −v ∈ λΩ (w )∨ . if G · hΩ (λΩ (w )) ∩ H = ∅ then C := C ∪ {λΩ (w )}, H := H ∪ {hΩ (λΩ (w ))} ˜ v˜ ) | η˜ ≺ λΩ (w ) interior facet w. inner normal v˜ } F := F {(η, else F := F \ {(η, v )} Janko Boehm (TU-KL)


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Symmetric GIT-Algorithm Algorithm (System of representatives of the G -orbits on Λ(a, Q )(k )) 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15:

S := system of representatives of G -orbits of faces(Qr≥0 ) A := {γ ∈ S | γ is a-face} S Ω := γ∈A G · Q (γ) Ω := set of minimal elements of Ω(k ) Choose w0 ∈ Q (Γ) such that dim(λΩ (w0 )) = k. C := {λΩ (w0 )}, H := {hΩ (λΩ (w0 ))} F := {(η, v ) | η ≺ λΩ (w0 ) interior facet with inner normal v } while there is (η, v ) ∈ F do Find w ∈ Q (Γ) such that η ≺ λΩ (w ) is a facet and −v ∈ λΩ (w )∨ . if G · hΩ (λΩ (w )) ∩ H = ∅ then C := C ∪ {λΩ (w )}, H := H ∪ {hΩ (λΩ (w ))} ˜ v˜ ) | η˜ ≺ λΩ (w ) interior facet w. inner normal v˜ } F := F {(η, else F := F \ {(η, v )} return C Janko Boehm (TU-KL)


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An Example with D4 -Symmetry Example a = hT1 T3 − T2 T4 i ⊂ K[T1 , . . . , T4 ] deg(Tj ) = qj 1 −1 −1 1 Q = (q1 , . . . , q4 ) = 1 1 −1 −1

Janko Boehm (TU-KL)


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An Example with D4 -Symmetry Example a = hT1 T3 − T2 T4 i ⊂ K[T1 , . . . , T4 ] deg(Tj ) = qj 1 −1 −1 1 Q = (q1 , . . . , q4 ) = 1 1 −1 −1 T2

G = D4 = h(1, 2)(3, 4), (1, 2, 3, 4)i ⊂ S4

1

-1

T1

1

-1

T3

Janko Boehm (TU-KL)


T4

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An Example with D4 -Symmetry Example a = hT1 T3 − T2 T4 i ⊂ K[T1 , . . . , T4 ] deg(Tj ) = qj 1 −1 −1 1 Q = (q1 , . . . , q4 ) = 1 1 −1 −1 T2

G = D4 = h(1, 2)(3, 4), (1, 2, 3, 4)i ⊂ S4

T1

1

-1

1

-1

T3

A(1,2)(3,4) = Janko Boehm (TU-KL)

−1 0 0 1

T4

A(1,2,3,4) =


0 −1 1 0

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Example with D4 -symmetry Example γ γ0 γ1 γ2 γ20 γ3 γ4

= cone(0) = cone(e1 ) = cone(e1 , e2 ) = cone(e1 , e3 ) = cone(e1 , e2 , e3 ) = cone(e1 , e2 , e3 , e4 )

Janko Boehm (TU-KL)

|G · γ | 1 4 4 2 4 1

a|Ti =0 for ei ∈/ γ 0 0 0 hT1 T3 i hT1 T3 i hT1 T3 − T2 T4 i


a-face true true true false false true

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|G · γ | 1 4 4 2 4 1


Q (γ0 ) = cone(0),

Q (γ1 ) = cone

Janko Boehm (TU-KL)

1 1


,

Q (γ2 ) = cone


1 1


−1 , , 1

Q (γ4 ) = Q2

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|G · γ | 1 4 4 2 4 1


Q (γ0 ) = cone(0),

Q (γ1 ) = cone

1 1


,

Q (γ2 ) = cone

1 1

−1 , , 1

q2

Q (γ4 ) = Q2

q1 λ(w0 )

0 w0 = 1

(0, 0) q3

Janko Boehm (TU-KL)



q4 April 18, 2017

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Mori Dream Spaces A projective variety X over C is called a Mori dream space if its Cox ring R (X ) = ∑[D ]∈Cl(X ) H 0 (X , OX (D )) is finitely generated.

Example Fano varieties. Projective toric varieties (⇔ R (X ) polynomial ring). Like toric varieties, admit construction as GIT-quotient (Hu, Keel, 2000): X = Xˆ //G where

Xˆ

Janko Boehm (TU-KL)

⊂

X := Spec R (X ) open invariant G := Spec C[Cl(X )]


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Mori Dream Spaces A projective variety X over C is called a Mori dream space if its Cox ring R (X ) = ∑[D ]∈Cl(X ) H 0 (X , OX (D )) is finitely generated.

Example Fano varieties. Projective toric varieties (⇔ R (X ) polynomial ring). Like toric varieties, admit construction as GIT-quotient (Hu, Keel, 2000): X = Xˆ //G where

Xˆ

⊂

X := Spec R (X ) open invariant G := Spec C[Cl(X )]

Remark The GIT-fan yields the Mori chamber decomposition, which describes all birational modifications (analogous to the GKZ-fan of a toric varietiy). Janko Boehm (TU-KL)


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Moduli Spaces of Stable Maps For the Deligne-Mumford compactification moduli space of stable curves of genus 0 with n marked points M 0,n (only double points, on each component ≥ 3 marked or double points) we have:

Janko Boehm (TU-KL)


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Moduli Spaces of Stable Maps For the Deligne-Mumford compactification moduli space of stable curves of genus 0 with n marked points M 0,n (only double points, on each component ≥ 3 marked or double points) we have: M 0,n for n ≤ 6 is a Mori dream space: Castravet, 2009, for n = 6.

Janko Boehm (TU-KL)


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Moduli Spaces of Stable Maps For the Deligne-Mumford compactification moduli space of stable curves of genus 0 with n marked points M 0,n (only double points, on each component ≥ 3 marked or double points) we have: M 0,n for n ≤ 6 is a Mori dream space: Castravet, 2009, for n = 6.

M 0,n for n ≥ 10 is not a Mori dream space: Castravet, Tevelev, 2013, for n ≥ 134. Gonzáles, Karu, 2016, for n ≥ 13. Hausen, Keicher, Laface, 2016, for n ≥ 10.

Janko Boehm (TU-KL)


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Example: GIT-fan for G(2, 5) Example Cox ring of M 0,5 is isomorphic to Z5 -graded coordinate ring R = K[T1 , . . . , T10 ]/a of affine cone over G(2, 5).

Janko Boehm (TU-KL)


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Example: GIT-fan for G(2, 5) Example Cox ring of M 0,5 is isomorphic to Z5 -graded coordinate ring R = K[T1 , . . . , T10 ]/a of affine cone over G(2, 5). Symmetry group action of S5 ∼ = G = h(2, 3)(5, 6)(9, 10), (1, 5, 9, 10, 3)(2, 7, 8, 4, 6)i ⊂ S10

Janko Boehm (TU-KL)


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monomial containment tests a-faces

Janko Boehm (TU-KL)

number 210 = 1024 172


number of orbits 34 14

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number 210 = 1024 172


172 = (1 + 1) + (5 + 5) + (10 + 10 + 10 + 10 + 10) + (15 + 15) + 20 + (30 + 30)

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number 210 = 1024 172


172 = (1 + 1) + (5 + 5) + (10 + 10 + 10 + 10 + 10) + (15 + 15) + 20 + (30 + 30)

|Ω(5)/G | = 4

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number 210 = 1024 172


172 = (1 + 1) + (5 + 5) + (10 + 10 + 10 + 10 + 10) + (15 + 15) + 20 + (30 + 30)

|Ω(5)/G | = 4 |Λ(5)| = 76 = 1 + 10 + 30 + 10 + 20 + 5 Janko Boehm (TU-KL)


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Example: GIT-fan for G(2, 5) Adjacency graph of the maximal-dimensional GIT-cones and their orbits:

Janko Boehm (TU-KL)


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Mori Chamber Decomposition of Mov(M 0,6 ) The moving cone Mov(M 0,6 ) classifies all small modifications (rational maps which are isomorphisms on open subsets which have a complement of codimension ≥ 2).

Example Cox ring is Z16 -graded, has 40 generators (Castravet, 2009),

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Example Cox ring is Z16 -graded, has 40 generators (Castravet, 2009), and 225 relations (Bernal Guillen, 2012),

Janko Boehm (TU-KL)


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Example Cox ring is Z16 -graded, has 40 generators (Castravet, 2009), and 225 relations (Bernal Guillen, 2012), and natural G = S6 –action.

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Example Cox ring is Z16 -graded, has 40 generators (Castravet, 2009), and 225 relations (Bernal Guillen, 2012), and natural G = S6 –action. The moving cone Mov(M 0,6 ) has 176 512 225 GIT-cones of maximal dimension 16, which decompose into 249 605 orbits under the S6 -action:

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Example Cox ring is Z16 -graded, has 40 generators (Castravet, 2009), and 225 relations (Bernal Guillen, 2012), and natural G = S6 –action. The moving cone Mov(M 0,6 ) has 176 512 225 GIT-cones of maximal dimension 16, which decompose into 249 605 orbits under the S6 -action: cardinality no. of orbits

1 1

cardinality no. of orbits

72 4

6 1

10 1

90 46

15 4

120 32

20 1 180 488

30 1 240 4

45 10 360 7934

60 27 720 241051

The cone with orbit length one is the semiample cone (dual of Mori cone). Janko Boehm (TU-KL)


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References J. Boehm, S. Keicher, Y. Ren. Computing GIT-fans with symmetry and the Mori chamber decomposition of M 0,6 , arXiv:1603.09241 (2016). S. Keicher. Computing the GIT-fan, Int. J. Algebra Comput. (2012). D. Mumford, J. Fogarty, F. Kirwan. Geometric invariant theory. Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer (1994). I. V. Dolgachev and Y. Hu. Variation of geometric invariant theory quotients. Publ. Math., Inst. Hautes Etud. Sci. (1998). F. Berchtold, J. Hausen. GIT equivalence beyond the ample cone. Michigan Math. J. (2006). I. Arzhantsev, U. Derenthal, J. Hausen, A. Laface. Cox Rings, Cambridge studies in advanced mathematics (2014). A.-M. Castravet. The Cox ring of M 0,6 . Trans. Amer. Math. Soc. (2009). M. M. Bernal Guillen. Relations in the Cox Ring of M 0,6 . PhD (2012). Janko Boehm (TU-KL)


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Mirror symmetry For Calabi-Yau variety X (elliptic curve, quintic in P4 ,...) and g ∈ N0 :

Janko Boehm (TU-KL)


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Mirror symmetry For Calabi-Yau variety X (elliptic curve, quintic in P4 ,...) and g ∈ N0 : X

mirror construction

A-model

B-model Q=Q(q)

∑d∞=1 Ng ,d · q d = Ag (q ) Ng ,d = Gromov-Witten invariants Intersection numbers on moduli space of stable maps M g ,n (X , d )

Janko Boehm (TU-KL)

X∗


Bg ( Q ) Integrals on M(X ∗ )

???

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mirror construction

A-model

X∗ B-model

Q=Q(q)



???

Mirror constructions: Greene-Plesser ’90, Batyrev ’93,...

Janko Boehm (TU-KL)


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mirror construction

A-model

X∗ B-model

Q=Q(q)



???

Mirror constructions: Greene-Plesser ’90, Batyrev ’93,... String theory: Candelas-Horowitz-Strominger-Witten ’85, Candelasde la Ossa-Green-Parkes ’91,...

Janko Boehm (TU-KL)


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mirror construction

A-model

X∗ B-model

Q=Q(q)



???

Mirror constructions: Greene-Plesser ’90, Batyrev ’93,... String theory: Candelas-Horowitz-Strominger-Witten ’85, Candelasde la Ossa-Green-Parkes ’91,... Algebraic/symplectic geometry: Fulton-Pandharipande ’95, Kontsevich ’95, Behrend-Fantechi ’97,... Janko Boehm (TU-KL)


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Mirror theorems Theorem (Givental ’96, Lian-Liu-Yau ’97, Gathmann ’03) A0 = B0 for quintic hypersurface in P4 .

Janko Boehm (TU-KL)


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Mirror theorems Theorem (Givental ’96, Lian-Liu-Yau ’97, Gathmann ’03) A0 = B0 for quintic hypersurface in P4 . ⇒ A0 (q ) = 23 · 53 + (4874 · 53 +

Janko Boehm (TU-KL)

23·53 ) · q 23

+ (2537651 · 53 +


23·53 ) · q 2 33

+ ...

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23·53 ) · q 23

+ (2537651 · 53 +

23·53 ) · q 2 33

+ ...

Is enumerative geometry result on X : number of lines, conics, cubics,... (number of genus 0 curves on X of degree d, delicate counting).

Janko Boehm (TU-KL)


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23·53 ) · q 23

+ (2537651 · 53 +

23·53 ) · q 2 33

+ ...

Is enumerative geometry result on X : number of lines, conics, cubics,... (number of genus 0 curves on X of degree d, delicate counting). Similar theorems for g = 0, 1 in case of degree n + 1 hypersurfaces in Pn (Klemm-Pandharipande ’07, Zinger ’07)

Janko Boehm (TU-KL)


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23·53 ) · q 23

+ (2537651 · 53 +

23·53 ) · q 2 33

+ ...

Is enumerative geometry result on X : number of lines, conics, cubics,... (number of genus 0 curves on X of degree d, delicate counting). Similar theorems for g = 0, 1 in case of degree n + 1 hypersurfaces in Pn (Klemm-Pandharipande ’07, Zinger ’07) Questions: Mirror theorems for other Calabi-Yau varieties and g ≥ 2?

Janko Boehm (TU-KL)


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23·53 ) · q 23

+ (2537651 · 53 +

23·53 ) · q 2 33

+ ...

Is enumerative geometry result on X : number of lines, conics, cubics,... (number of genus 0 curves on X of degree d, delicate counting). Similar theorems for g = 0, 1 in case of degree n + 1 hypersurfaces in Pn (Klemm-Pandharipande ’07, Zinger ’07) Questions: Mirror theorems for other Calabi-Yau varieties and g ≥ 2? Geometric understanding of mirror theorem beyond combinatorics?

Janko Boehm (TU-KL)


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23·53 ) · q 23

+ (2537651 · 53 +

23·53 ) · q 2 33

+ ...

Is enumerative geometry result on X : number of lines, conics, cubics,... (number of genus 0 curves on X of degree d, delicate counting). Similar theorems for g = 0, 1 in case of degree n + 1 hypersurfaces in Pn (Klemm-Pandharipande ’07, Zinger ’07) Questions: Mirror theorems for other Calabi-Yau varieties and g ≥ 2? Geometric understanding of mirror theorem beyond combinatorics? What are the B-model integrals? Janko Boehm (TU-KL)


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Elliptic curves

Start with easiest Calabi-Yau: elliptic curve E (e.g. smooth plane cubic). Here, Gromov-Witten numbers are numbers of covers:

Janko Boehm (TU-KL)


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Elliptic curves


Definition (Hurwitz numbers) Nd,g = |Aut1(f )| -weighted number of degree d covers f : C → E , where C is smooth of genus g and f has 2g − 2 simple ramifications points. according to Riemann-Hurwitz formula 2g (C ) − 2 = d · (2g (E ) − 2) + ∑P ∈C (e (P ) − 1)

Janko Boehm (TU-KL)


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Elliptic curves


Definition (Hurwitz numbers) Nd,g = |Aut1(f )| -weighted number of degree d covers f : C → E , where C is smooth of genus g and f has 2g − 2 simple ramifications points. according to Riemann-Hurwitz formula 2g (C ) − 2 = d · (2g (E ) − 2) + ∑P ∈C (e (P ) − 1)

Nd,0 = 0, so have to look at g ≥ 1 invariants!

Janko Boehm (TU-KL)


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Tropical point of view How to understand all Ng ,d ? Pass to tropical geometry: E

Janko Boehm (TU-KL)

7→


trop(E )

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Gromov-Witten invariants

7→

trop(E )

Mirror Theorem

Correspondence Theorem

B-model Tropical Mirror Theorem

tropical Gromov-Witten invariants

Janko Boehm (TU-KL)


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7→

trop(E )

Mirror Theorem




For X = P2 (building block of C-Y) and g = 0: tropical mirror theorem (Gross ’10) Janko Boehm (TU-KL)


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7→

trop(E )

Mirror Theorem




For X = P2 (building block of C-Y) and g = 0: tropical mirror theorem (Gross ’10) partial correspondence theorem (Markwig-Rau ’09, Mikhalkin ’05) Janko Boehm (TU-KL)


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Tropical Mirror Symmetry

Hurwitz

Univariate B as

Multivariate B as

numbers

Feynman integral

Feynman integral


⇐

tropical Hurwitz numbers

Janko Boehm (TU-KL)

Refined Tropical Mirror Theorem

numbers of labeled tropical covers


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Hurwitz

Univariate B as

Multivariate B as

numbers

Feynman integral

Feynman integral


⇐




Correspondence theorem for all g and d.

Janko Boehm (TU-KL)


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Hurwitz

Univariate B as

Multivariate B as

numbers

Feynman integral

Feynman integral


⇐




Correspondence theorem for all g and d. Tropical mirror theorem for all g as corollary to

Janko Boehm (TU-KL)


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Hurwitz

Univariate B as

Multivariate B as

numbers

Feynman integral

Feynman integral


⇐




Correspondence theorem for all g and d. Tropical mirror theorem for all g as corollary to refined tropical mirror theorem for each trivalent connected graph of genus g and branch type. Computation of refined Feynman integrals.

Janko Boehm (TU-KL)


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Feynman integrals (B-side) Definition A Feynman graph is a 3-valent, connected graph Γ of genus g .

Janko Boehm (TU-KL)


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Feynman integrals (B-side) Definition A Feynman graph is a 3-valent, connected graph Γ of genus g . By g (Γ) = 1 − |vert(Γ)| + |edges(Γ)| and 3 |vert(Γ)| = 2 |edges(Γ)|

|vert(Γ)| = 2g − 2

Janko Boehm (TU-KL)

|edges(Γ)| = 3g − 3


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|vert(Γ)| = 2g − 2

|edges(Γ)| = 3g − 3

Fix labeling zi for vertices and qi for edges.

Janko Boehm (TU-KL)


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|vert(Γ)| = 2g − 2

|edges(Γ)| = 3g − 3

Fix labeling zi for vertices and qi for edges.

Example z1 q2

z3

q3

q5

z2 Janko Boehm (TU-KL)

q1

q4

q6

z4


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Feynman integrals (B-side) Definition (Propagator) P (z, q ) = −

Janko Boehm (TU-KL)

1 1 ℘(z, q ) − E2 (q ) 2 4π 12


for z ∈ E = C/Λ

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1 1 ℘(z, q ) − E2 (q ) 2 4π 12

for z ∈ E = C/Λ

with Weierstraß-℘-function ℘ = z12 + ... and the Eisenstein series E2 = 1 − 24 ∑d∞=1 σ1 (d )q 2d = 1 − 24q 2 − 72q 4 − ...

Janko Boehm (TU-KL)


σ1 (d ) = ∑m|d m

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1 1 ℘(z, q ) − E2 (q ) 2 4π 12

for z ∈ E = C/Λ

with Weierstraß-℘-function ℘ = z12 + ... and the Eisenstein series E2 = 1 − 24 ∑d∞=1 σ1 (d )q 2d = 1 − 24q 2 − 72q 4 − ...

σ1 (d ) = ∑m|d m

Definition (Feynman integral) For ordering Ω ∈ S2g −2 of integration paths on E

IΓ,Ω =

Z

Z

... γ2g −2

Janko Boehm (TU-KL)

γ1

∏

e ∈edges(Γ)

! P (ze+ − ze− , q ) dzΩ(1) ...dzΩ(2g −2)


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Correspondence Theorem As a direct generalization of (Cavalieri-Johnson-Markwig ’10) and (Bertrand-Brugallé-Mikhalkin ’11) obtain correspondence theorem:

Janko Boehm (TU-KL)


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Theorem (BBBM ’15) trop Nd,g = Nd,g by correspondence of tropical and algebraic covers.

Janko Boehm (TU-KL)


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Theorem (BBBM ’15) trop Nd,g = Nd,g by correspondence of tropical and algebraic covers.

Janko Boehm (TU-KL)


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Tropical Hurwitz numbers – Example trop N3,3 =?

Janko Boehm (TU-KL)


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Two trivalent, connected combinatorial types (non-metric graphs)

of genus g = 3 with 2g − 2 = 4 vertices 3g − 3 = 6 edges no bridges

Janko Boehm (TU-KL)


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Two trivalent, connected combinatorial types (non-metric graphs)

of genus g = 3 with 2g − 2 = 4 vertices 3g − 3 = 6 edges no bridges (weight 0 edges would be contracted):

Janko Boehm (TU-KL)


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Tropical Hurwitz numbers – Example trop N3,3 =

Janko Boehm (TU-KL)


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2·

4·

4·

mult(π ) = 22 · 32 = 36

Janko Boehm (TU-KL)


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2·

4·

mult(π ) = 22 · 32 = 36

Janko Boehm (TU-KL)

mult(π ) =

4·

1 2

· 22 · 3 = 6


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2·

4·

mult(π ) = 22 · 32 = 36

Janko Boehm (TU-KL)

mult(π ) =

4·

1 2

· 22 · 3 = 6


mult(π ) = 22 · 3 = 12

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2·

4·

mult(π ) = 22 · 32 = 36

4·

mult(π ) =

4·

1 2

· 22 · 3 = 6

mult(π ) = 22 · 3 = 12

2·

mult(π ) =

1 2

·2·2 = 2

Janko Boehm (TU-KL)


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2·

4·

mult(π ) = 22 · 32 = 36

4·

mult(π ) =

4·

1 2

· 22 · 3 = 6

mult(π ) = 22 · 3 = 12

2·

mult(π ) =

1 2

·2·2 = 2

Janko Boehm (TU-KL)

mult(π ) = 22 = 4 Algorithms in Singular

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Tropical Hurwitz numbers – Example trop N3,3 = 112 + 48 = 160

2·

4·

mult(π ) = 22 · 32 = 36

4·

mult(π ) =

4·

1 2

· 22 · 3 = 6

mult(π ) = 22 · 3 = 12

2·

mult(π ) =

1 2

·2·2 = 2

Janko Boehm (TU-KL)

mult(π ) = 22 = 4 Algorithms in Singular

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Refined Feynman integrals Definition (Refined Feynman integrals) IΓ,Ω (q1 , ..., q3g −3 ) =

Janko Boehm (TU-KL)

Z

... γ2g −2

!

3g −3

Z γ1

∏

P (zk+

− zk− , qk )

dzΩ(1) ...dzΩ(2g −2)

k =1


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Refined Feynman integrals Definition (Refined Feynman integrals) IΓ,Ω (q1 , ..., q3g −3 ) =

Z

... γ2g −2

!

3g −3

Z γ1

∏

P (zk+

q1

z3

− zk− , qk )

dzΩ(1) ...dzΩ(2g −2)

k =1

Example For

z1 q2

q3

q5

z2 we have to integrate

q4

q6

z4

P (z1 − z2 , q1 ) · P (z1 − z2 , q2 ) · P (z1 − z3 , q3 ) · P (z2 − z4 , q4 ) · P (z3 − z4 , q5 ) · P (z3 − z4 , q6 ) Janko Boehm (TU-KL)


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Tropical mirror theorem Theorem (Multivariate tropical mirror theorem, BBBM ’13) trop q 2a = IΓ,Ω (q1 , ..., q3g −3 ) ∑ Na,Γ,Ω a

Janko Boehm (TU-KL)


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Setting qi = q we get (using the action of Aut(Γ) on labeled covers):

Corollary (Tropical mirror theorem) 1

trop 2d q =∑ IΓ,Ω (q ) ∑ Nd,g |Aut(Γ)| ∑ d

Janko Boehm (TU-KL)

Γ


Ω

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Setting qi = q we get (using the action of Aut(Γ) on labeled covers):

Corollary (Tropical mirror theorem) 1

trop 2d q =∑ IΓ,Ω (q ) ∑ Nd,g |Aut(Γ)| ∑ d

Γ

Ω

Together with the correspondence theorem this proves:

Corollary (Mirror symmetry for elliptic curves) For elliptic curves Ag = Bg for all g . Janko Boehm (TU-KL)


April 18, 2017

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Computing Feynman integrals By coordinate change xk = exp(i πzk ),

Janko Boehm (TU-KL)


April 18, 2017

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Computing Feynman integrals By coordinate change xk = exp(i πzk ), path γk becomes circle around 0,

Janko Boehm (TU-KL)


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Computing Feynman integrals By coordinate change xk = exp(i πzk ), path γk becomes circle around 0, factor x1k ,

Janko Boehm (TU-KL)


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Computing Feynman integrals By coordinate change xk = exp(i πzk ), path γk becomes circle around 0, factor x1k , integral becomes residue,

Janko Boehm (TU-KL)


April 18, 2017

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Computing Feynman integrals By coordinate change xk = exp(i πzk ), path γk becomes circle around 0, factor x1k , integral becomes residue, difference becomes quotient.

Proposition (BBBM ’15) P (x, q ) =

Janko Boehm (TU-KL)

∞ x2 + w (x 2w + x −2w )q 2a ∑ (x 2 − 1)2 a=1 w∑ |a


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∞ x2 + w (x 2w + x −2w )q 2a ∑ (x 2 − 1)2 a=1 w∑ |a

( Pa (x, y ) :=

Janko Boehm (TU-KL)

x 2y 2 (x 2 −y 2 )2 x 4w +y 4w ∑w |a w (xy )2w


for a = 0 for a > 0

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∞ x2 + w (x 2w + x −2w )q 2a ∑ (x 2 − 1)2 a=1 w∑ |a

( Pa (x, y ) :=

x 2y 2 (x 2 −y 2 )2 x 4w +y 4w ∑w |a w (xy )2w

for a = 0 for a > 0

Theorem (BBBM ’15) trop Na,Γ,Ω = constxΩ(2g −2) ... constxΩ(1)

3g −3

∏

Pak (xk+ , xk− )

k =1

Janko Boehm (TU-KL)


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Sketch of Proof 1:1

{labeled tropical covers} {constant products of Laurent monomials}

Janko Boehm (TU-KL)


April 18, 2017

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Sketch of Proof 1:1

{labeled tropical covers} {constant products of Laurent monomials} order the vertices according to Ω,

Janko Boehm (TU-KL)


April 18, 2017

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Sketch of Proof 1:1

{labeled tropical covers} {constant products of Laurent monomials} order the vertices according to Ω, associate edges of weight ai to Laurent monomials.

Janko Boehm (TU-KL)


April 18, 2017

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Sketch of Proof 1:1


Example x1 q2

q1

q3 x2

x3 q5

q4

x1 < x3 < x4 < x2 q6

x4

Janko Boehm (TU-KL)


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Sketch of Proof 1:1


Example x1 q2

q1

q3 x2

x3 q5

q4

x1 < x3 < x4 < x2

a = (0, 2, 2, 0, 1, 0)

q6

x4

Janko Boehm (TU-KL)


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Sketch of Proof 1:1


Example x1 q2

q1

q3 x2

x3 q5

q4

q6

x1 < x3 < x4 < x2 a = (0, 2, 2, 0, 1, 0) 2 2·2 2 2 2 2·2 x1 · 2 · xx21 · xx12 · xx42 · xx34 · 2 · xx43 x3

x4

Janko Boehm (TU-KL)


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Sketch of Proof 1:1


Example x1 q2

q1

q3 x2

x3 q5

q4

q6

x1 < x3 < x4 < x2 a = (0, 2, 2, 0, 1, 0) 2 2·2 2 2 2 2·2 x1 · 2 · xx21 · xx12 · xx42 · xx34 · 2 · xx43 x3

x4

x1

q3 , 1 x3

q2 , 2

x4

q1 , 1 q5 , 1

q6 , 2

x2

q3 , 1 q3 , 1 Janko Boehm (TU-KL)

q5 , 1

q4 , 1

q2 , 2

q3 , 1 Algorithms in Singular

April 18, 2017

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References J. Böhm, K. Bringmann, A. Buchholz, H. Markwig, Tropical mirror symmetry for elliptic curves, J. Reine Angew. Math. (2015). J. Böhm, K. Bringmann, A. Buchholz, H. Markwig, ellipticcovers.lib. A Singular 4 library for Gromov-Witten invariants of elliptic curves, Singular distribution. A. Okounkov, R. Pandharipande, Gromov-Witten theory, Hurwitz theory and completed cycles, Ann. Math. 163 (2006). R. Dijkgraaf, Mirror symmetry and elliptic curves, in Progr. Math. 129 (1995). M. Gross, Mirror symmetry for P2 and tropical geometry, Adv. Math. 224 (2010). B. Bertrand, E. Brugallé, G. Mikhalkin, Tropical open Hurwitz numbers, Rend. Semin. Mat. Univ. Padova 125 (2011). Janko Boehm (TU-KL)


April 18, 2017

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Fundamental and Advanced Algorithms in Singular

Fundamental and Advanced Algorithms in Singular

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