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Cones of Divisors and Simplicial Complexes
Elijah Gunther Olivia Zhang
Yale University
MathFest, August 7, 2015
Elijah Gunther, Olivia Zhang Yale University
Cones of Divisors and Simplicial Complexes
Our Ambient Space: M0,n
M0,n is a space parametrizing all possible configurations of ndistinct points on P1.
M0,n is a space parametrizing all these and all their limits.
Elijah Gunther, Olivia Zhang Yale University
Cones of Divisors and Simplicial Complexes
Divisors
I Divisors are formal linear combinations of codimension-1objects D1, D2, . . . , Dr with real coefficients a1, a2, . . . , ar:
a1D1 + a2D2 + · · ·+ arDr
I These form a vector space.
I We have an equivalence relation on divisors based on theirintersection with curves.
I After taking the quotient of the space with respect to ourequivalence relation, we get a finite-dimensional vector space,N1, which is isomorphic to Rn.
Elijah Gunther, Olivia Zhang Yale University
Cones of Divisors and Simplicial Complexes
Divisors
I Divisors are formal linear combinations of codimension-1objects D1, D2, . . . , Dr with real coefficients a1, a2, . . . , ar:
a1D1 + a2D2 + · · ·+ arDr
I These form a vector space.
I We have an equivalence relation on divisors based on theirintersection with curves.
I After taking the quotient of the space with respect to ourequivalence relation, we get a finite-dimensional vector space,N1, which is isomorphic to Rn.
Elijah Gunther, Olivia Zhang Yale University
Cones of Divisors and Simplicial Complexes
Divisors
I Divisors are formal linear combinations of codimension-1objects D1, D2, . . . , Dr with real coefficients a1, a2, . . . , ar:
a1D1 + a2D2 + · · ·+ arDr
I These form a vector space.
I We have an equivalence relation on divisors based on theirintersection with curves.
I After taking the quotient of the space with respect to ourequivalence relation, we get a finite-dimensional vector space,N1, which is isomorphic to Rn.
Elijah Gunther, Olivia Zhang Yale University
Cones of Divisors and Simplicial Complexes
Divisors
I Divisors are formal linear combinations of codimension-1objects D1, D2, . . . , Dr with real coefficients a1, a2, . . . , ar:
a1D1 + a2D2 + · · ·+ arDr
I These form a vector space.
I We have an equivalence relation on divisors based on theirintersection with curves.
I After taking the quotient of the space with respect to ourequivalence relation, we get a finite-dimensional vector space,N1, which is isomorphic to Rn.
Elijah Gunther, Olivia Zhang Yale University
Cones of Divisors and Simplicial Complexes
Effective Divisors
I Effective divisors are linear combinations with onlynon-negative coefficients.
a1D1 + a2D2 + · · ·+ arDr , ai ≥ 0
I If D ∈ N1 is equivalent to an effective divisor, we also saythat it is effective.
Elijah Gunther, Olivia Zhang Yale University
Cones of Divisors and Simplicial Complexes
Effective Divisors
I Effective divisors are linear combinations with onlynon-negative coefficients.
a1D1 + a2D2 + · · ·+ arDr , ai ≥ 0
I If D ∈ N1 is equivalent to an effective divisor, we also saythat it is effective.
Elijah Gunther, Olivia Zhang Yale University
Cones of Divisors and Simplicial Complexes
Cone of Effective Divisors
Problem: Find a minimalgenerating set for the cone ofeffective divisors of M0,n.
Doran-Giansiracusa-Jensenrecently connected the studyof effective divisors on M0,n
to the study of certain typesof simplicial complexes.
Based on this, we have foundnew minimal generators of theeffective cone of M0,7 and ingeneral on M0,n.
Elijah Gunther, Olivia Zhang Yale University
Cones of Divisors and Simplicial Complexes
Cone of Effective Divisors
Problem: Find a minimalgenerating set for the cone ofeffective divisors of M0,n.
Doran-Giansiracusa-Jensenrecently connected the studyof effective divisors on M0,n
to the study of certain typesof simplicial complexes.
Based on this, we have foundnew minimal generators of theeffective cone of M0,7 and ingeneral on M0,n.
Elijah Gunther, Olivia Zhang Yale University
Cones of Divisors and Simplicial Complexes
Simplicial Complexes
I Take d ∈ Z≥0.
I A d-simplex σ is a multiset on {1, 2, . . . , n} such that|σ| = d+ 1. It is nonsingular if there are no repetitions, and issingular otherwise.
I A d-simplicial complex ∆ is a set of d-simplices:∆ = {σ1, . . . , σk}. It is nonsingular if every simplex itcontains is nonsingular, and is singular otherwise.
Elijah Gunther, Olivia Zhang Yale University
Cones of Divisors and Simplicial Complexes
Simplicial Complexes
I Take d ∈ Z≥0.
I A d-simplex σ is a multiset on {1, 2, . . . , n} such that|σ| = d+ 1. It is nonsingular if there are no repetitions, and issingular otherwise.
I A d-simplicial complex ∆ is a set of d-simplices:∆ = {σ1, . . . , σk}. It is nonsingular if every simplex itcontains is nonsingular, and is singular otherwise.
Elijah Gunther, Olivia Zhang Yale University
Cones of Divisors and Simplicial Complexes
Visualizing Simplicial Complexes0-complexes are points (or vertices):
1 2
{{1}, {2}}
1-complexes can be represented as graphs, with each simplex beingan edge:
1
2 3
1 2
341 2
{{1, 2}, {1, 3}} {{1, 2}, {1, 4}, {2, 3}, {3, 4}} {{1, 1}, {1, 2}, {2, 2}}
Elijah Gunther, Olivia Zhang Yale University
Cones of Divisors and Simplicial Complexes
Product ConstructionGiven ∆1 = {σi} and ∆2 = {σj}, d1 and d2 complex, we definetheir product as the (d1 + d2 + 1)-complex
∆1 ·∆2 = {σi ∪ σj : σi ∈ ∆1, σj ∈ ∆2}
6
5
Elijah Gunther, Olivia Zhang Yale University
Cones of Divisors and Simplicial Complexes
Star Construction at a VertexGiven ∆ with vertex i, we define:
∆∗i = {σ \ {i} : σ ∈ ∆ s.t. i ∈ σ}
Example
1
2
3
4
∆∗4
1
2 3
Elijah Gunther, Olivia Zhang Yale University
Cones of Divisors and Simplicial Complexes
Weighting
We weight a complex over a ring R by assigning each σi ∈ ∆ anon-zero value wi.
Example
1
-1
1
-1-2
1 1
Elijah Gunther, Olivia Zhang Yale University
Cones of Divisors and Simplicial Complexes
BalancingA weighted d-complex is balanced in degree j if for each multisetS with |S| = j we have:∑
σi⊇Swi ·m(S ⊆ σi) = 0
If it is balanced in all j ∈ {0, 1, ..., d}, then it is balanced overall.
Example
1
-1
1
-1-2
1 1
balanced balanced
Elijah Gunther, Olivia Zhang Yale University
Cones of Divisors and Simplicial Complexes
Minimality
A balanced complex is minimal if and only if no proper subcomplexof it is balanceable.
Example
non-minimal minimal
Elijah Gunther, Olivia Zhang Yale University
Cones of Divisors and Simplicial Complexes
Divisors from Simplicial Complexes
Theorem (DGJ)
If ∆ is a balanceable, minimal, nonsingular, nonproduct d-complexon n ≥ d+ 5 vertices, it corresponds to a minimal generatingdivisor D∆ ofM0,n+1.
Example
Elijah Gunther, Olivia Zhang Yale University
Cones of Divisors and Simplicial Complexes
Results
I Let ∆ be a d-complex, weighted on a ring R where (d+ 1)! isnot a zero divisor. If ∆ is balanced in degree d, then it isbalanced overall.
I Let ∆ with a weighting {wi} be a balanced d-complex on nvertices. Then ∀k ∈ {1, ..., n}, ∆∗
k is balanceable.
I Let ∆ be weighted {wi} on any ring where (d+ 1)! is not azero-divisor. If for each k ∈ {1, ..., n}, ∆∗
k is balanced with aweighting based on wi, ∆ is balanced.
Elijah Gunther, Olivia Zhang Yale University
Cones of Divisors and Simplicial Complexes
Results
I Let ∆ be a d-complex, weighted on a ring R where (d+ 1)! isnot a zero divisor. If ∆ is balanced in degree d, then it isbalanced overall.
I Let ∆ with a weighting {wi} be a balanced d-complex on nvertices. Then ∀k ∈ {1, ..., n}, ∆∗
k is balanceable.
I Let ∆ be weighted {wi} on any ring where (d+ 1)! is not azero-divisor. If for each k ∈ {1, ..., n}, ∆∗
k is balanced with aweighting based on wi, ∆ is balanced.
Elijah Gunther, Olivia Zhang Yale University
Cones of Divisors and Simplicial Complexes
Results
I Let ∆ be a d-complex, weighted on a ring R where (d+ 1)! isnot a zero divisor. If ∆ is balanced in degree d, then it isbalanced overall.
I Let ∆ with a weighting {wi} be a balanced d-complex on nvertices. Then ∀k ∈ {1, ..., n}, ∆∗
k is balanceable.
I Let ∆ be weighted {wi} on any ring where (d+ 1)! is not azero-divisor. If for each k ∈ {1, ..., n}, ∆∗
k is balanced with aweighting based on wi, ∆ is balanced.
Elijah Gunther, Olivia Zhang Yale University
Cones of Divisors and Simplicial Complexes
Computational Results
For indexed multisets Si ⊂ {1, ..., n}, i ∈ {1, ..., r}, and indexedsimplices σj , we define the r × |∆| multiplicity matrix:
M(∆)ij = m(Si ⊂ σj).
M(∆) =
σ1 ... σk
S1 m(S1 ⊂ σ1) · · · m(S1 ⊂ σk)...
.... . .
...Sr m(Sr ⊂ σ1) · · · m(Sr ⊂ σk)
Any vector in its kernel corresponds to a balancing weighting of ∆,allowing zeros.
Elijah Gunther, Olivia Zhang Yale University
Cones of Divisors and Simplicial Complexes
Computational Results
For indexed multisets Si ⊂ {1, ..., n}, i ∈ {1, ..., r}, and indexedsimplices σj , we define the r × |∆| multiplicity matrix:
M(∆)ij = m(Si ⊂ σj).
M(∆) =
σ1 ... σk
S1 m(S1 ⊂ σ1) · · · m(S1 ⊂ σk)...
.... . .
...Sr m(Sr ⊂ σ1) · · · m(Sr ⊂ σk)
Any vector in its kernel corresponds to a balancing weighting of ∆,allowing zeros.
Elijah Gunther, Olivia Zhang Yale University
Cones of Divisors and Simplicial Complexes
Vector Space of Balancings
The complete singular complex ∆K,s contains all d-simplicies on nvertices. We have shown:
nul(M(∆K,s)) =
(n+ d− 1
d− 1
)
The complete nonsingular complex ∆K,n contains all nonsingulard-simplices on n vertices. We have shown:
nul(M(∆K,n)) =
(n
d+ 1
)−(n
d
)This implies that there are no nonsingular balanceable d-complexeson ≤ 2d+ 1 vertices.
Elijah Gunther, Olivia Zhang Yale University
Cones of Divisors and Simplicial Complexes
Vector Space of Balancings
The complete singular complex ∆K,s contains all d-simplicies on nvertices. We have shown:
nul(M(∆K,s)) =
(n+ d− 1
d− 1
)
The complete nonsingular complex ∆K,n contains all nonsingulard-simplices on n vertices. We have shown:
nul(M(∆K,n)) =
(n
d+ 1
)−(n
d
)This implies that there are no nonsingular balanceable d-complexeson ≤ 2d+ 1 vertices.
Elijah Gunther, Olivia Zhang Yale University
Cones of Divisors and Simplicial Complexes
Results
We have used these results, as well as some others, to find classesof complexes that correspond to minimal generating divisors of theeffective cone.
I Exhaustively search for them on a given number of vertices
I Triangulating closed manifolds, in particular d-tori.
We have found:
I New minimal generators of the effective cones of M0,7.
I Infinitely many new minimal generators of the effective conesof M0,n for large n.
Elijah Gunther, Olivia Zhang Yale University
Cones of Divisors and Simplicial Complexes
Results
We have used these results, as well as some others, to find classesof complexes that correspond to minimal generating divisors of theeffective cone.
I Exhaustively search for them on a given number of vertices
I Triangulating closed manifolds, in particular d-tori.
We have found:
I New minimal generators of the effective cones of M0,7.
I Infinitely many new minimal generators of the effective conesof M0,n for large n.
Elijah Gunther, Olivia Zhang Yale University
Cones of Divisors and Simplicial Complexes
Thank You!
I Jose Gonzalez and Jeremy Usatine
I Nathan Kaplan, Sam Payne, and the Yale Math Department
Elijah Gunther, Olivia Zhang Yale University
Cones of Divisors and Simplicial Complexes