toric varieties and combinatorial algebraic geometry · 1. toric varieties deﬁnition a toric...

Toric Varieties

and

Combinatorial Algebraic Geometry

David A. Cox

Department of Mathematics and StatisticsAmherst College

[email protected]

Beijing, 2015

David A. Cox (Amherst College) Toric Varieties & Combinat. Algebraic Geom. Beijing, 2015 1 / 38

Outline

Themes of the Course

Toric geometry is a well-established part of algebraic geometry.

Combinatorial algebraic geometry is an emerging area.

1 Toric Varieties

1.1 Toric Bézier Patches

1.2 The Quotient Construction

1.3 Cones and Fans

1.4 A Quantum Computer

2 An Interesting Secant Variety

2.1 Veronese Maps and Secant Varieties

2.2 16,051 Formulas

3 Moduli and Tropical Geometry

3.1. A Classical Moduli Space

3.2 The Tree Connection

3.3 The Tropical Connection

1. Toric Varieties

Definition

A toric variety is an irreducible algebraic variety X that contains a torus

(C∗)n as a Zariski open subset in such a way that the action of thetorus on itself extends to an action X .

We will work over the field of complex numbers C. Examples of toric

varieties include Cn,Pn,Pn × Pm, etc,

There are four ways to think about toric varieties:

Monomials

Cones and Fans

Polytopes

Quotients

Given our limited time, I will use toric Bézier patches to introduce some

of these ideas.

1. Toric Varieties

Definition

Monomials

Cones and Fans

Polytopes

Quotients

of these ideas.

1. Toric Varieties

Definition

Monomials

Cones and Fans

Polytopes

Quotients

of these ideas.

1. Toric Varieties

Definition

Monomials

Cones and Fans

Polytopes

Quotients

of these ideas.

1. Toric Varieties

Definition

Monomials

Cones and Fans

Polytopes

Quotients

of these ideas.

1. Toric Varieties

Definition

Monomials

Cones and Fans

Polytopes

Quotients

of these ideas.

1.1. Toric Bézier Functions

Let ∆ ⊆ R2 be the convex hull of finitely many points in Z2. Assumedim∆ = 2. Hence ∆ is a lattice polygon.

If u1, . . . ,uℓ are the (primitive, integral, inward pointing) normal vectors tothe edges of ∆, then we can write

∆ = {p ∈ R2 | 〈p,ui〉 ≥ −ai ∀ i}

To simplify notation, set hi(p) = 〈p,ui〉+ ai .

Toric Bézier Functions (Krasauskas, 1996)

For m ∈ ∆ ∩ Z2 and p ∈ ∆, define

βm(p) = h1(p)h1(m) · · · hℓ(p)

hℓ(m)

Note that βm(p) ≥ 0 since p ∈ ∆.

1.1. Toric Bézier Functions

Let ∆ ⊆ R2 be the convex hull of finitely many points in Z2. Assumedim∆ = 2. Hence ∆ is a lattice polygon.

If u1, . . . ,uℓ are the (primitive, integral, inward pointing) normal vectors tothe edges of ∆, then we can write

∆ = {p ∈ R2 | 〈p,ui〉 ≥ −ai ∀ i}

To simplify notation, set hi(p) = 〈p,ui〉+ ai .

Toric Bézier Functions (Krasauskas, 1996)

For m ∈ ∆ ∩ Z2 and p ∈ ∆, define

βm(p) = h1(p)h1(m) · · · hℓ(p)

hℓ(m)

Note that βm(p) ≥ 0 since p ∈ ∆.

1.1. Toric Bézier Patches

Given control points Pm ∈ R3, m ∈ ∆ ∩ Z2, the toric Bézier patch is

defined by

p ∈ ∆ 7−→1

∑

m∈∆∩Z2 βm(p)

∑

m∈∆∩Z2

βm(p)Pm ∈ R3.

1.1. Toric Bézier Patches

Given control points Pm ∈ R3, m ∈ ∆ ∩ Z2, the toric Bézier patch is

defined by

p ∈ ∆ 7−→1

∑

m∈∆∩Z2 βm(p)

∑

m∈∆∩Z2

βm(p)Pm ∈ R3.

∆

−→

The control point for the center of the hexagon determines the “bulge”

of the surface patch. Picture on the right created by F. Sottile.

1.2. The Quotient Construction I

Let ∆ be a lattice polygon. The edges give edge variables xi fori = 1, . . . , ℓ. Then a lattice point m ∈ ∆ ∩ Z2 gives the ∆-monomial

xm = xh1(m)1 · · · x

hℓ(m)ℓ .

← sameexponents!

Let Z = V(xm | m is a vertex) ⊆ Cℓ, and define the group

G = {(t1, . . . , tℓ) ∈ (C∗)ℓ |

∏ℓi=1t

〈m,ui 〉i = 1 ∀m ∈ Z

2}.

The toric variety of ∆ ⊆ R2 is the quotient

X∆ =(

Cℓ \ Z )/G.

In this case, X∆ is a surface since dim∆ = 2.

Example

When ∆ is the unit simplex in R2, this gives P2 = (C3 \ {0})/C∗.

xm = xh1(m)1 · · · x

hℓ(m)ℓ .

← sameexponents!

G = {(t1, . . . , tℓ) ∈ (C∗)ℓ |

∏ℓi=1t

〈m,ui 〉i = 1 ∀m ∈ Z

2}.

X∆ =(

Cℓ \ Z )/G.

Example

xm = xh1(m)1 · · · x

hℓ(m)ℓ .

← sameexponents!

G = {(t1, . . . , tℓ) ∈ (C∗)ℓ |

∏ℓi=1t

〈m,ui 〉i = 1 ∀m ∈ Z

2}.

X∆ =(

Cℓ \ Z )/G.

Example

xm = xh1(m)1 · · · x

hℓ(m)ℓ .

← sameexponents!

G = {(t1, . . . , tℓ) ∈ (C∗)ℓ |

∏ℓi=1t

〈m,ui 〉i = 1 ∀m ∈ Z

2}.

X∆ =(

Cℓ \ Z )/G.

Example

1.2. The Quotient Construction II

The quotient construction has a nice relation to the ∆-monomials.Let ∆ ∩ Z2 = {m1, . . . ,mN}.

The map x 7→ (xm1 , . . . , xmN ) gives a well-defined map

Φ : Cℓ \ Z −→ PN−1.

The image of Φ is the toric variety X∆ (special to surface case).

The ∆-monomials transform the same way under G, so that Φinduces a commutative diagram

Cℓ \ Z

��

Φ

((PPP

PPPP

P

X∆� // P

N−1

(Cℓ \ Z )/Gφ

∼ 66♥♥♥♥♥♥♥♥

Φ : Cℓ \ Z −→ PN−1.

Cℓ \ Z

��

Φ

((PPP

PPPP

P

X∆� // P

N−1

(Cℓ \ Z )/Gφ

∼ 66♥♥♥♥♥♥♥♥

Φ : Cℓ \ Z −→ PN−1.

Cℓ \ Z

��

Φ

((PPP

PPPP

P

X∆� // P

N−1

(Cℓ \ Z )/Gφ

∼ 66♥♥♥♥♥♥♥♥

1.2. Back to Toric Bézier Patches I

Recall that hi(p) = 〈p,ui〉+ ai . A point p ∈ ∆ gives

γ(p) = (h1(p), . . . ,hℓ(p)) ∈ Cℓ.

Then:

γ(p) /∈ Z = V(xm | m is a vertex). EXERCISE!

The toric Bézier function of m ∈ ∆ ∩ Z2 is γ(p)m. EXERCISE!

If zm, m ∈ ∆ ∩ Z2, are homogeoneous coordinates for PN−1, then

control points Pm ∈ R3 determine a projection map

PN−1

99K P3

where zm maps to (1,Pm) ∈ P3.

The notation (1,Pm) ∈ P3 refers to the inclusion

C3 →֒ P3

defined by (a,b, c) ∈ C3 7→ (1,a,b, c) ∈ P3.

γ(p) = (h1(p), . . . ,hℓ(p)) ∈ Cℓ.

Then:

PN−1

99K P3

C3 →֒ P3

γ(p) = (h1(p), . . . ,hℓ(p)) ∈ Cℓ.

Then:

PN−1

99K P3

C3 →֒ P3

γ(p) = (h1(p), . . . ,hℓ(p)) ∈ Cℓ.

Then:

PN−1

99K P3

C3 →֒ P3

1.2. Back to Toric Bézier Patches II

Putting this all together, we see that the toric Bézier patch ∆→ R3 fitsinto the commutative diagram

∆

γ��

toric Bézier patch// R

3 _

��

Cℓ \ Z

��

Φ

&&▼▼▼

▼▼▼▼

▼▼P

3

X∆� //

;;✈✈

✈✈

PN−1

control points

OO✤✤✤

(Cℓ \ Z )/G

φ

∼88qqqqqqqqq

EXERCISE: PROVE THIS!

Corollary

A toric Bézier patch for ∆ is a projection of the toric surface X∆.

1.2. Back to Toric Bézier Patches II

Putting this all together, we see that the toric Bézier patch ∆→ R3 fitsinto the commutative diagram

∆

γ��

toric Bézier patch// R

3 _

��

Cℓ \ Z

��

Φ

&&▼▼▼

▼▼▼▼

▼▼P

3

X∆� //

;;✈✈

✈✈

PN−1

control points

OO✤✤✤

(Cℓ \ Z )/G

φ

∼88qqqqqqqqq

EXERCISE: PROVE THIS!

Corollary

A toric Bézier patch for ∆ is a projection of the toric surface X∆.

1.3. Cones and Fans I

A classic approach to toric varieties uses cones and fans.

For later purposes, I will say a few words about this. We begin with a

polytope and its normal fan:

∆v1 v2

v3

v4v5

v6σ1

σ4 σ6

σ3σ2

σ5Σ∆

Inward normals at a vertex of ∆ generate a maximal cone of Σ∆.

Variables x1, . . . , xℓ ←→ edges (facets) of ∆←→ rays of Σ∆.

In general, a fan consists of cones that fit together nicely.

∆v1 v2

v3

v4v5

v6σ1

σ4 σ6

σ3σ2

σ5Σ∆

∆v1 v2

v3

v4v5

v6σ1

σ4 σ6

σ3σ2

σ5Σ∆

1.3. Cones and Fans II

Intuition Behind Cones and Fans

A cone σ gives an affine toric variety Uσ.

A fan Σ consists of finitely many cones, each of which gives anaffine toric variety.

The affine toric varietiesUσ, σ ∈ Σ, glue together to give the toricvariety XΣ since the cones of Σ fit together nicely.

Example

The toric variety X∆ is the toric variety of the normal fan Σ∆ of ∆.

Example

1.4. A Quantum Computer

Here is a diagram from a D-Wave adiabatic quantum computer:

1.4. Polytopes are Involved

A Polytope

The complete bipartite graph Kn,n has vertices q1, . . . ,q2n and edgesqiqj , i = 1, . . . ,n, j = n + 1, . . . ,2n. Consider the 2

2n points

(q1, ...,q2n,q1qn+1, ...,qnq2n) ∈ R2n+n2 = Rn(n+2)

as q1, . . . ,q2n independently range over {0,1}. The convex hull ofthese points is a polytope with 22n vertices.

When n = 4, each facet of this polytope gives an Ising model that canbe implemented on the D-Wave computer. The idea is to sample from

states associated to the Ising model and check if the states returned

are ground states of roughly comparble probability.

How Many Facets?

36,391,264, as computed by Mathieu Dutour

1.4. Polytopes are Involved

A Polytope

The complete bipartite graph Kn,n has vertices q1, . . . ,q2n and edgesqiqj , i = 1, . . . ,n, j = n + 1, . . . ,2n. Consider the 2

2n points

(q1, ...,q2n,q1qn+1, ...,qnq2n) ∈ R2n+n2 = Rn(n+2)

as q1, . . . ,q2n independently range over {0,1}. The convex hull ofthese points is a polytope with 22n vertices.

When n = 4, each facet of this polytope gives an Ising model that canbe implemented on the D-Wave computer. The idea is to sample from

states associated to the Ising model and check if the states returned

are ground states of roughly comparble probability.

How Many Facets?

36,391,264, as computed by Mathieu Dutour

1.4. A Toric Connection?

Denny Dahl, a physicist at D-Wave, knew about toric varieties from my

book with Little and Schenck. So he asked me about these polytopes.

With help from Greg Smith, I was able to connect Denny with people

like Mathieu Dutour, Komei Fukuda and David Avis.

Email from Denny Dahl to David Avis

Our most pressing problem concerns how to construct facets of

similar systems in a way that captures logical constraints arising

from combinatorial optimization problems.

Two Observations

This example has little to do with toric varieties – polytopes are the

key players.

However, toric varieties have links to mirror symmetry in physics,

which is why the physics community knows about toric varieties.

Two Observations

key players.

Two Observations

key players.

1. References

References

D. Cox, What is a toric variety?, in Topics in algebraic geometry

and geometric modeling, Contemp. Math. 334, AMS, 2003,

203–223.

D. Cox, J. Little, H. Schenck, Toric Varieties, AMS, 2011.

R. Krasauskas, Toric Surface Patches, Adv. Comput. Math. 17

(2002), 89–113.

B. Sturmfels, Gröbner Bases and Convex Polytopes, AMS, 1996.

2. An Interesting Secant Variety

We begin the transition to combinatorial algebraic geometry.

Provisional Definition

Combinatorial algebraic geometry is the study of varieties with an

explicit combinatorial structure.

The toric varieties defined in Section 1 provide an interesting class

of examples.

In Section 2, we will study an interesting example that involves a

secant variety.

In Section 3, we will explore a wonderful example of a

combinatorial moduli space.

of examples.

secant variety.

of examples.

secant variety.

of examples.

secant variety.

2.1. Veronese Maps and Secant Varieties

The Veronese Map

The d -th Veronese map vd : Pn → PN−1 is the projective embedding

given by all N =(

n+dn

)

monomials of degree d .

Remarks

The curve vd (P1) ⊆ Pd is the rational normal curve of degree d .

vd (Pn) ⊆ PN−1 is the toric variety of the polytope d∆n, where

∆n ⊆ Rn is the unit simplex in Rn.

The Secant Variety of X ⊆ Pm

The secant variety σ2(X ) is the Zariski closure of the union of alllines through two distinct points of X .

The k-th secant variety σk (X ) is the Zariski closure of the union ofall (k−1)-planes formed by k points of X .

The Veronese Map

given by all N =(

n+dn

)

Remarks

The Veronese Map

given by all N =(

n+dn

)

Remarks

The Veronese Map

given by all N =(

n+dn

)

Remarks

The Veronese Map

given by all N =(

n+dn

)

Remarks

2.1. An Example from Probability

Consider two biased coins with probability s1 and s2 of heads.

Let the probability of picking each coin be π1 and π2.

Then the probability of getting i heads in d coin tosses is

pi =(

di

)

π1si1(1− s1)

d−i +(

di

)

π2si2(1− s2)

d−i .

Up to projective equivalence, the point for the i th coin

Ai =((

d0

)

s0i (1− si)d , . . . ,

(

dd

)

sni (1− si)0)

∈ Pd

lies on the rational normal curve vd (P1).

Thus the probability distribution for two coins satisfies

(p0, . . . ,pn) = π1A1 + π2A2 ∈ σ2(vn(P1).

Using k coins gives a point in the k-secant variety σk (vd (P1).

pi =(

di

)

π1si1(1− s1)

d−i +(

di

)

π2si2(1− s2)

d−i .

Ai =((

d0

)

s0i (1− si)d , . . . ,

(

dd

)

sni (1− si)0)

∈ Pd

(p0, . . . ,pn) = π1A1 + π2A2 ∈ σ2(vn(P1).

pi =(

di

)

π1si1(1− s1)

d−i +(

di

)

π2si2(1− s2)

d−i .

Ai =((

d0

)

s0i (1− si)d , . . . ,

(

dd

)

sni (1− si)0)

∈ Pd

(p0, . . . ,pn) = π1A1 + π2A2 ∈ σ2(vn(P1).

2.1. The Dimension of a Secant Variety

For X ⊆ Pm, we have the inequality

dimσk (X ) ≤ min(m, k dim(X ) + k − 1)

since σk (X ) ⊆ Pm and points in σk (X ) are determined by points in

X k × Pk−1.

For the secant variety σk (vd (Pn)), note that

vd (Pn) ⊆ PN−1 for N =

(

n+dn

)

and dim vd (Pn) = dimPn = n.

It follows that

(1) dimσk (vd (Pn)) ≤ min(

(

n+dn

)

− 1, kn + k − 1).

In 1995, Alexander and Hirschowitz computed dimσk (vd (Pn)),

including the deficient case when (1) is a strict inequality.

X k × Pk−1.

(

n+dn

)

It follows that

(

n+dn

)

− 1, kn + k − 1).

X k × Pk−1.

(

n+dn

)

It follows that

(

n+dn

)

− 1, kn + k − 1).

X k × Pk−1.

(

n+dn

)

It follows that

(

n+dn

)

− 1, kn + k − 1).

2.2. A Particular Secant Variety

One deficient case identified by Alexander and Hirschowitz is when

k = 7, d = 3, n = 4, i.e., σ7(v3(P4)). Here,

min((

4+34

)

− 1,7 · 4 + 7− 1) = min(34,34) = 34,

yet

σ7(v3(P4)) ⊆ P34

has dimension 33 and degree 15.

Thus σ7(v3(P4)) is given by a single equation of degree 15. There are

(

34+15

15

)

= 1,575,580,702,584

monomials of degree 15 in 35 variables. Is it possible to write down the

defining equation of this secant variety?

k = 7, d = 3, n = 4, i.e., σ7(v3(P4)). Here,

min((

4+34

)

− 1,7 · 4 + 7− 1) = min(34,34) = 34,

yet

σ7(v3(P4)) ⊆ P34

(

34+15

15

)

= 1,575,580,702,584

k = 7, d = 3, n = 4, i.e., σ7(v3(P4)). Here,

min((

4+34

)

− 1,7 · 4 + 7− 1) = min(34,34) = 34,

yet

σ7(v3(P4)) ⊆ P34

(

34+15

15

)

= 1,575,580,702,584

2.2. 16,051 Formulas

Ottaviani, 2009

The defining equation F = 0 of σ7(v3(P4)) ⊆ P34 is SL5-invariant,

so the number of monomials reduces to

317,881,154.

F 3 is the determinant of a 45× 45 determinant with linear entries.

A Recent Preprint

In 16,051 formulas for Ottaviani’s invariant of cubic threefolds,

Abdesselam, Ikenmeyer and Royle give many descriptions of F .

They use certain bipartite graphs to encode the information

needed to compute the polynomial F .

Ottaviani, 2009

317,881,154.

A Recent Preprint

Ottaviani, 2009

317,881,154.

A Recent Preprint

Ottaviani, 2009

317,881,154.

A Recent Preprint

2.2. The Bipartite Graph

The Veronese map v3 : P4 → P34 uses the 35 monomials

yα = xα = xa11 · · · x

a55 with |α| = 3. Fix a bipartite graph

15 vertices, all degree 3

9 vertices, all degree 5

with oriented edges on the bottom. Color the edges using 1, . . . ,5. Avertex • with edges colored i1, i2, i3 gives xα(•) = xi1xi2xi3 , hence acoordinate yα(•). A vertex • with edges colored i1, i2, i3, i4, i5 gives

ǫ(•) =

{

0 ij not distinct

±1 ij distinct

2.2. The Formula

Then a 5-coloring of the edges gives a term of degree 15

∏

9 •

ǫ(•) ·∏

15 •

yα(•)

The sum of these terms for all 5-colorings of the edges gives a

polynomial F of degree 15 in the 35 variables yα.

Theorem (Abdesselam, Ikenmeyer and Royle)

Fix a bipartite graph as above with no 7-coloring of the vertices and

F 6= 0. Then:

F = 0 is the defining equation of σ7(v3(P4)).

There are 16,051 such bipartite graphs.

2.2. The Formula

∏

9 •

ǫ(•) ·∏

15 •

yα(•)

F 6= 0. Then:

2.2. The Formula

∏

9 •

ǫ(•) ·∏

15 •

yα(•)

F 6= 0. Then:

2.2. Final Comments

Question

σ7(v3(P4)) is combinatorial (it is a secant variety of the toric variety of

the polytope 3∆4). The same is true for the bipartite graphs used tocreate the defining equation F = 0. How are these related?

Two Remarks:

For a parametrized hypersurface, methods from tropical geometry

can be used to describe the Newton polytope (i.e., exponent

vectors) of the defining equation.

The methods used by Abdesselam, Ikenmeyer and Royle involve

the representation theory of SL5. The bipartite graphs encode

contractions of tensors from⊗45

C5 to

⊗15(Sym3(C5)).

2.2. Final Comments

Question

Two Remarks:

C5 to

⊗15(Sym3(C5)).

2.2. Final Comments

Question

Two Remarks:

C5 to

⊗15(Sym3(C5)).

2. References

References

A. Abdesselam, C. Ikenmeyer, G. Royle, 16,051 formulas for

Ottaviani’s invariant of cubic threefolds, arXiv:1402.2669

[math.AG].

G. Ottaviani, An invariant regarding Waring’s problem for cubic

polynomials, Nagoya Math. J. 193 (2009), 95–110.

We will now take a 10 minute break before resuming the course.

2. References

References

A. Abdesselam, C. Ikenmeyer, G. Royle, 16,051 formulas for

Ottaviani’s invariant of cubic threefolds, arXiv:1402.2669

[math.AG].

G. Ottaviani, An invariant regarding Waring’s problem for cubic

polynomials, Nagoya Math. J. 193 (2009), 95–110.

We will now take a 10 minute break before resuming the course.

3. Moduli and Tropical Geometry

In algebraic geometry, the study of moduli spaces goes back to

the 19th century when Riemann showed that moduli space of

curves of genus g has dimension 3g − 3 when g ≥ 2.

For genus g = 1, the moduli space has dimension 1. Here, we arestudying elliptic curves.

For genus g = 0, there is only one smooth projective curve of thisgenus, namely P1. However, when we add n-marked points, a

remarkable picture emerges.Recall the projective linear group PGL(2,C):

PGL(2,C) acts on P1 via(

a bc d

)

· (x , y) = (ax + by , cx + dy).

In terms of P1 = C ∪ {∞}, PGL(2,C) acts via the linear fractionaltransformation

(

a bc d

)

· x = ax+bcx+d .

Given distinct points p, q, r ∈ P1 = C ∪ {∞}, there is a uniqueγ ∈ PGL(2,C) such that

γ(p) = 0, γ(q) = 1, γ(r) =∞.

a bc d

)

· (x , y) = (ax + by , cx + dy).

(

a bc d

)

· x = ax+bcx+d .

γ(p) = 0, γ(q) = 1, γ(r) =∞.

a bc d

)

· (x , y) = (ax + by , cx + dy).

(

a bc d

)

· x = ax+bcx+d .

γ(p) = 0, γ(q) = 1, γ(r) =∞.

a bc d

)

· (x , y) = (ax + by , cx + dy).

(

a bc d

)

· x = ax+bcx+d .

γ(p) = 0, γ(q) = 1, γ(r) =∞.

a bc d

)

· (x , y) = (ax + by , cx + dy).

(

a bc d

)

· x = ax+bcx+d .

γ(p) = 0, γ(q) = 1, γ(r) =∞.

a bc d

)

· (x , y) = (ax + by , cx + dy).

(

a bc d

)

· x = ax+bcx+d .

γ(p) = 0, γ(q) = 1, γ(r) =∞.

a bc d

)

· (x , y) = (ax + by , cx + dy).

(

a bc d

)

· x = ax+bcx+d .

γ(p) = 0, γ(q) = 1, γ(r) =∞.

3.1. M0,n and M0,n

M0,n

M0,n is the moduli space of rational curves with n ≥ 3 marked points.This is an irreducible variety of dimension n − 3. It is not compact.

We compactify M0,n using the following curves:

Stable Curves of Genus 0 with n Marked Points

Such a curve consists of a tree of P1’s such that

No three P1’s intersect.

When two P1’s meet, they intersect transversely.

For each P1, the number of marked points plus the number of

intersection points is ≥ 3

This gives the compact moduli space M0,n. The boundary M0,n \M0,nhas a wonderful combinatorial stratification.

3.1. M0,n and M0,n

M0,n

3.1. M0,n and M0,n

M0,n

3.1. M0,n and M0,n

M0,n

3.1. M0,n and M0,n

M0,n

3.1. The Case n = 4

Given distinct points x1, x2, x3, x4 ∈ P1 = C ∪ {∞}, we can use PGL to

move them to 0,1,∞, λ, where

λ =(x1 − x3)(x2 − x4)

(x2 − x3)(x1 − x4)

is the cross-ratio of x1, x2, x3, x4. Since 0,1,∞, λ are distinct, we get

M0,4 = P1 \ {0,1,∞}.

The “missing points” 0,1,∞ correspond to the stable curves

Note that(

42

)

= 3. Thus M0,4 = P1.

3.1. The Case n = 4

Given distinct points x1, x2, x3, x4 ∈ P1 = C ∪ {∞}, we can use PGL to

move them to 0,1,∞, λ, where

λ =(x1 − x3)(x2 − x4)

(x2 − x3)(x1 − x4)

is the cross-ratio of x1, x2, x3, x4. Since 0,1,∞, λ are distinct, we get

M0,4 = P1 \ {0,1,∞}.

The “missing points” 0,1,∞ correspond to the stable curves

1

2 4

3

1

3 4

2

1

4 3

2

Note that(

42

)

= 3. Thus M0,4 = P1.

3.1. Two Examples when n = 6

Here are two genus 0 stable curves with n = 6, together with their dualgraphs:

dual to

1

2

3

6

5

4

1

2

3

4

5

6

1

2

3 4

5

6

1 3

2

5

4 6

3.2. Metric Trees and a Fan

Consider a tree with n labeled leaves and internal vertices have degree

≥ 3. Edges are ends if connected to a leaf and bounded otherwise.

Metric Trees

In a metric tree C, the n ends have length∞ and bounded edges havelength in R>0. T

trop0,n is the set of such trees up to isomorphism.

Let C be a metric tree. Given leaves i , j , let

dij = distance between vertices adjacent to i , j .

Then C gives a point (dij) ∈ R(n2)/L, L = {(xi + xj) : (x1, . . . , xn) ∈ R

n}.

Ttrop0,n is a Fan

Via the map C 7→ (dij), we can regard Ttrop0,n as a fan ∆ in R

(n2)/L. Eachcone of ∆ consists of combinatorially equivalent trees.

3.2. Metric Trees and a Fan

Consider a tree with n labeled leaves and internal vertices have degree

≥ 3. Edges are ends if connected to a leaf and bounded otherwise.

Metric Trees

In a metric tree C, the n ends have length∞ and bounded edges havelength in R>0. T

trop0,n is the set of such trees up to isomorphism.

Let C be a metric tree. Given leaves i , j , let

dij = distance between vertices adjacent to i , j .

Then C gives a point (dij) ∈ R(n2)/L, L = {(xi + xj) : (x1, . . . , xn) ∈ R

n}.

Ttrop0,n is a Fan

Via the map C 7→ (dij), we can regard Ttrop0,n as a fan ∆ in R

(n2)/L. Eachcone of ∆ consists of combinatorially equivalent trees.

3.2. Constructing M0,n

The fan ∆ in R(n2)/L ≃ R(

n2)−n gives the toric variety X∆ with torus

(C∗)(n2)−n ⊆ X∆.

Torus orbits stratify X∆ and correspond to cones of ∆.

Theorem (Gibney-Maclagan)

There is an embedding M0,n ⊆ (C∗)(

n2)−n such that:

1 M0,n is the Zariski closure of M0,n in X∆.

2 The stratification of M0,n \M0,n is induced by the torus orbitstratification of X∆.

Furthermore, if a stratum Y ⊆ M0,n \M0,n is induced by the torus orbitcorresponding to σ ∈ ∆, then:

C ∈ Y ⇒ trees ∈ σ are combinatorially equivalent to dual graph of C.

(C∗)(n2)−n ⊆ X∆.

n2)−n such that:

(C∗)(n2)−n ⊆ X∆.

n2)−n such that:

(C∗)(n2)−n ⊆ X∆.

n2)−n such that:

3.2. Consequences

The toric variety X∆ has global coordinates determined by rays of the

fan ∆. Using these coordinates, Gibney and Maclagan give explicitequations for M0,n ⊆ X∆.

However, there are lots of variables since

variables←→ rays of ∆←→ graphs

Taking labels into account, there are 2n−1 − n − 1 variables.

One can also work locally, since X∆ is covered by affine toric varieties

Uσ for σ ∈ ∆ a maximal cone. However,

maximal cones of ∆←→ phylogenetic trees

Taking labels into account, there are

(2n − 5)!! = (2n − 5)(2n − 3) · · · 5 · 3 · 1 maximal cones.

3.3. Tropical Geometry

In tropical geometry, one uses R ∪ {∞} with operations

addition : a⊕ b = min(a,b)

multiplication : a⊙ b = a + b.

R ∪ {∞} is a semiring under these operations with∞ as additiveidentity and 0 as multiplicative identity

Example

If p = a⊙ x2 ⊕ b⊙ xy ⊕ c ⊙ y2 ⊕ d ⊙ x ⊕ e⊙ y ⊕ f , then p = R2 → R isgiven by

p = min(a + 2x ,b + x + y , c + 2y ,d + x ,e + y , f ).

The tropical curve V(p) ⊆ R2 is the locus of all points in R2 where pfails to be linear.

3.3. Tropical Varieties

For various choices of coefficients a, . . . , f in the previous example, thetropical curve V(p) ⊆ R2 looks like

The edges ending in red go to infinity, i.e., have infinite length, and the

other edges are bounded. Look familiar?

3.3. Tropicalization

Given a field K with a non-archimedean valuation and a variety

Y ⊆ (K ∗)n, one gets trop(V ) ⊆ Rn.

If Y ⊆ (C∗)n and C has the trivial valuation, then trop(V ) ⊆ Rn isthe underlying space of a fan.

M0,n ⊆ (C∗)(

n2)−n tropicalizes to

trop(M0,n) = Ttrop0,n , (1)

where Ttrop0,n is the set of metric trees with n ends defined earlier.

Ttrop0,n is the moduli space of abstract rational tropical curves with n

marked infinite edges.

The fan ∆ coming from (1) satisfies

M0,n = Zariski closure of M0,n in X∆.

This is an example of tropical compactification, a.k.a geometric

tropicalization.

M0,n ⊆ (C∗)(

tropicalization.

M0,n ⊆ (C∗)(

tropicalization.

M0,n ⊆ (C∗)(

tropicalization.

M0,n ⊆ (C∗)(

tropicalization.

M0,n ⊆ (C∗)(

tropicalization.

3.3. Lots of Papers!

M0,n and Tropical Curves:

Gibney and Maclagan, 2007: Equations for Chow and Hilbert Quotients.Gathmann, Kerber and Markwig, 2007: Tropical fans and the moduli

spaces of tropical curves.

Tropical Compactification/Geometric Tropicalization:

Cueto, 2009: Implicitization of surfaces via geometric tropicalization.

Katz and Payne, 2009: Realization spaces for tropical fans.

Combinatorial Moduli Spaces:

Ren, Sam and Sturmfels, 2013: Tropicalization of classical moduli spaces.Cavalieri, Hampe, Markwig, and Ranganathan, 2014: Moduli spaces of

rational weighted stable curves and tropical geometry.

Ulirsch, 2014: Tropical geometry of moduli spaces of weighted stablecurves.

Gross, 2014: Correspondence Theorems via Tropicalizations of Moduli

Spaces.Cavalieri, Markwig, and Ranganathan, 2014: Tropical compactification and

the Gromov–Witten theory of P1.

Conclusion

This definition is incomplete (what is a “combinatorial structure”?). The

whole subject is a work-in-progress. In June 2013, Bernd Sturmfels

lectured on Likelihood Geometry at the CIME-CIRM summer course

on Combinatorial Algebraic Geometry. The lecture notes, written with

June Huh, are available on arXiv.

From the Abstract

We present an introduction to this theory and its statistical motivations.

Many favorite objects from combinatorial algebraic geometry are

featured: toric varieties, A-discriminants, hyperplane arrangements,

Grassmannians, and determinantal varieties.

Conclusion

From the Abstract

Conclusion

From the Abstract

The End!

In this course, we explored toric varieties and two topics not on

Sturmfels’ list: secant varieties and moduli spaces. I would summarize

the course by saying that we have seen:

some rich examples, and

some themes that run deep.

Upcoming Event

Thematic Program on Combinatorial Algebraic Geometry

July–December 2016

Fields Institute, Toronto

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