cones of divisors and simplicial complexes · 2019-03-08 · of simplicial complexes. based on...

Cones of Divisors and Simplicial Complexes

Elijah Gunther Olivia Zhang

Yale University

MathFest, August 7, 2015

Elijah Gunther, Olivia Zhang Yale University

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.

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.

Divisors

a1D1 + a2D2 + · · ·+ arDr

Divisors

a1D1 + a2D2 + · · ·+ arDr

Divisors

a1D1 + a2D2 + · · ·+ arDr

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.

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.

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.

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.

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.

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.

Visualizing Simplicial Complexes0-complexes are points (or vertices):

{{1}, {2}}

1-complexes can be represented as graphs, with each simplex beingan edge:

{{1, 2}, {1, 3}} {{1, 2}, {1, 4}, {2, 3}, {3, 4}} {{1, 1}, {1, 2}, {2, 2}}

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}

Star Construction at a VertexGiven ∆ with vertex i, we define:

∆∗i = {σ \ {i} : σ ∈ ∆ s.t. i ∈ σ}

Example

∆∗4

Weighting

We weight a complex over a ring R by assigning each σi ∈ ∆ anon-zero value wi.

Example

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

balanced balanced

Minimality

A balanced complex is minimal if and only if no proper subcomplexof it is balanceable.

Example

non-minimal minimal

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

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.

Results

k is balanceable.

Results

k is balanceable.

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.

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.

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

)This implies that there are no nonsingular balanceable d-complexeson ≤ 2d+ 1 vertices.

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

)This implies that there are no nonsingular balanceable d-complexeson ≤ 2d+ 1 vertices.

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.

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.

Thank You!

I Jose Gonzalez and Jeremy Usatine

I Nathan Kaplan, Sam Payne, and the Yale Math Department

cones of divisors and simplicial complexes · 2019-03-08 · of simplicial complexes. based on...

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