Tropical Severi Varieties and Applications

Jihyeon Jessie YangMcMaster University

October 20, 2012

Background/Motivation

“Asymptotic behavior of a hypersurface, X = V (f ) in (C∗)n, f ∈ C[Zn]

is described by Newton(f ).”

Ex. f = 1 + x + y2 + z2

x(t) = x0tu + l .o.t .,

y(t) = y0tv + l .o.t .,

z(t) = z0tw + l .o.t .

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Background/Motivation

“Asymptotic behavior of a hypersurface, X = V (f ) in (C∗)n, f ∈ C[Zn]

is described by Newton(f ).”

Ex. f = 1 + x + y2 + z2

x(t) = x0tu + l .o.t .,

y(t) = y0tv + l .o.t .,

z(t) = z0tw + l .o.t .

(t →∞)

f(u,v ,w) = y2 + z2: two (C∗)2

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Background/Motivation

“Asymptotic behavior of a hypersurface, X = V (f ) in (C∗)n, f ∈ C[Zn]

is described by Newton(f ).”

Ex. f = 1 + x + y2 + z2

x(t) = x0tu + l .o.t .,

y(t) = y0tv + l .o.t .,

z(t) = z0tw + l .o.t .

(t →∞)

f(u,v ,w) = y2 + z2: two (C∗)2

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R3

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Background/Motivation

“Asymptotic behavior of a hypersurface, X = V (f ) in (C∗)n, f ∈ C[Zn]

is described by Newton(f ).”

• An application: (BKK counts) number of common roots of polynomials

as mixed volume of polytopes

What if X is not a hypersurface?

• X = V (f ) −→ X = V (I), Iideal⊂ C[Zn]

• “Asymptotic behavior of X

is described by Trop(X ), weighted fan with dim = dim(X )”

• intersections in (C∗)n ↔ intersections of fans in Rn

• algebro-geometric −→ combinatorial/polytopal

Background/Motivation

“Asymptotic behavior of a hypersurface, X = V (f ) in (C∗)n, f ∈ C[Zn]

is described by Newton(f ).”

• An application: (BKK counts) number of common roots of polynomials

as mixed volume of polytopes

What if X is not a hypersurface?

• X = V (f ) −→ X = V (I), Iideal⊂ C[Zn]

• “Asymptotic behavior of X

is described by Trop(X ), weighted fan with dim = dim(X )”

• intersections in (C∗)n ↔ intersections of fans in Rn

• algebro-geometric −→ combinatorial/polytopal

Background/Motivation

“Asymptotic behavior of a hypersurface, X = V (f ) in (C∗)n, f ∈ C[Zn]

is described by Newton(f ).”

• An application: (BKK counts) number of common roots of polynomials

as mixed volume of polytopes

What if X is not a hypersurface?

• X = V (f ) −→ X = V (I), Iideal⊂ C[Zn]

• “Asymptotic behavior of X

is described by Trop(X ), weighted fan with dim = dim(X )”

• intersections in (C∗)n ↔ intersections of fans in Rn

• algebro-geometric −→ combinatorial/polytopal

Background/Motivation

“Asymptotic behavior of a hypersurface, X = V (f ) in (C∗)n, f ∈ C[Zn]

is described by Newton(f ).”

• An application: (BKK counts) number of common roots of polynomials

as mixed volume of polytopes

What if X is not a hypersurface?

• X = V (f ) −→ X = V (I), Iideal⊂ C[Zn]

• “Asymptotic behavior of X

is described by Trop(X ), weighted fan with dim = dim(X )”

• intersections in (C∗)n ↔ intersections of fans in Rn

• algebro-geometric −→ combinatorial/polytopal

Background/Motivation

“Asymptotic behavior of a hypersurface, X = V (f ) in (C∗)n, f ∈ C[Zn]

is described by Newton(f ).”

• An application: (BKK counts) number of common roots of polynomials

as mixed volume of polytopes

What if X is not a hypersurface?

• X = V (f ) −→ X = V (I), Iideal⊂ C[Zn]

• “Asymptotic behavior of X

is described by Trop(X ), weighted fan with dim = dim(X )”

• intersections in (C∗)n ↔ intersections of fans in Rn

• algebro-geometric −→ combinatorial/polytopal

Background/Motivation

“Asymptotic behavior of a hypersurface, X = V (f ) in (C∗)n, f ∈ C[Zn]

is described by Newton(f ).”

• An application: (BKK counts) number of common roots of polynomials

as mixed volume of polytopes

What if X is not a hypersurface?

• X = V (f ) −→ X = V (I), Iideal⊂ C[Zn]

• “Asymptotic behavior of X

is described by Trop(X ), weighted fan with dim = dim(X )”

• intersections in (C∗)n ↔ intersections of fans in Rn

• algebro-geometric −→ combinatorial/polytopal

Definition of Trop(X )

1. The support of Trop(X ):

2. The weighting function:

Definition of Trop(X )

1. The support of Trop(X ):• (algebraic) {ω ∈ Qn : inωIX 6= C[Zn]} =• (geometric){ω ∈ Qn : ∃ a germ of a curve z(t) = z̄tω + l .o.t . in X}

2. The weighting function: mX (ω) = the sum of themultiplicities of minimal associate prime ideals of inωIX

ExampleX = V ((1 + x + y)2)...

Definition of Trop(X )

1. The support of Trop(X ):• (algebraic) {ω ∈ Qn : inωIX 6= C[Zn]} =• (geometric){ω ∈ Qn : ∃ a germ of a curve z(t) = z̄tω + l .o.t . in X}

2. The weighting function: mX (ω) = the sum of themultiplicities of minimal associate prime ideals of inωIX

ExampleX = V ((1 + x + y)2)...

Definition of Trop(X )

1. The support of Trop(X ):• (algebraic) {ω ∈ Qn : inωIX 6= C[Zn]} =• (geometric){ω ∈ Qn : ∃ a germ of a curve z(t) = z̄tω + l .o.t . in X}

2. The weighting function: mX (ω) = the sum of themultiplicities of minimal associate prime ideals of inωIX

ExampleX = V ((1 + x + y)2)...

Definition of Trop(X )

1. The support of Trop(X ):• (algebraic) {ω ∈ Qn : inωIX 6= C[Zn]} =• (geometric){ω ∈ Qn : ∃ a germ of a curve z(t) = z̄tω + l .o.t . in X}

2. The weighting function: mX (ω) = the sum of themultiplicities of minimal associate prime ideals of inωIX

ExampleX = V ((1 + x + y)2)...

Definition of Trop(X )

1. The support of Trop(X ):• (algebraic) {ω ∈ Qn : inωIX 6= C[Zn]} =• (geometric){ω ∈ Qn : ∃ a germ of a curve z(t) = z̄tω + l .o.t . in X}

2. The weighting function: mX (ω) = the sum of themultiplicities of minimal associate prime ideals of inωIX

ExampleX = V ((1 + x + y)2)...

Tropical Severi Varieties

• Sev(∆, δ) ={ complex plane curves V (f ) with δ nodes, Newton(f ) =∆}, (∆ : a polygon , δ ∈ N)

• Found a partial description of Trop(Sev(∆, δ)) in terms ofthe polygon, ∆.

• Applications:1. Mikhalkin’s Correspondence theorem in terms of Tropical

Intersection Theory2. Relation with Secondary Fans

Tropical Severi Varieties

• Sev(∆, δ) ={ complex plane curves V (f ) with δ nodes, Newton(f ) =∆}, (∆ : a polygon , δ ∈ N)

• Found a partial description of Trop(Sev(∆, δ)) in terms ofthe polygon, ∆.

• Applications:1. Mikhalkin’s Correspondence theorem in terms of Tropical

Intersection Theory2. Relation with Secondary Fans

Tropical Severi Varieties

• Sev(∆, δ) ={ complex plane curves V (f ) with δ nodes, Newton(f ) =∆}, (∆ : a polygon , δ ∈ N)

• Found a partial description of Trop(Sev(∆, δ)) in terms ofthe polygon, ∆.

• Applications:1. Mikhalkin’s Correspondence theorem in terms of Tropical

Intersection Theory2. Relation with Secondary Fans

Tropical Severi Varieties• Sev(∆, δ) ={ complex plane curves V (f ) with δ nodes,Newton(f ) =∆}, (∆ : a polygon , δ ∈ N)

• Found a partial description of Trop(Sev(∆, δ)) in terms ofthe polygon, ∆.

• Applications:1. Mikhalkin’s Correspondence theorem:

• a major work in tropical mathematics(2005).• “Counting complex curves (GW invariants) is equal to

counting tropical curves.”• Direct counting (Purely combinatorial)→ Intersection number

of Trop(Sev(∆, δ))

2. Secondary Fans:• Gelfand, Kapranov, Zelevinsky (1994)• complete fans in real vector spaces• Rich connections to algebraic geometry• Found a criteria when Trop(Sev(∆, δ)) fails to be a subfan of

a Secondary fan.

Tropical Severi Varieties• Sev(∆, δ) ={ complex plane curves V (f ) with δ nodes,Newton(f ) =∆}, (∆ : a polygon , δ ∈ N)

• Found a partial description of Trop(Sev(∆, δ)) in terms ofthe polygon, ∆.

• Applications:1. Mikhalkin’s Correspondence theorem:

• a major work in tropical mathematics(2005).• “Counting complex curves (GW invariants) is equal to

counting tropical curves.”• Direct counting (Purely combinatorial)→ Intersection number

of Trop(Sev(∆, δ))

2. Secondary Fans:• Gelfand, Kapranov, Zelevinsky (1994)• complete fans in real vector spaces• Rich connections to algebraic geometry• Found a criteria when Trop(Sev(∆, δ)) fails to be a subfan of

a Secondary fan.

Main TheoremsLet r = dim(Sev(∆, δ)).

1. If rank(ω) > r , ω /∈ Trop(Sev(∆, δ)).

2. If rank(ω) = r and ω ∈ Trop(Sev(∆, δ)), ∆ω is simple-nodal.

3. If ω ∈ Trop(Sev(∆, δ)) is regular with the maximal rank r ,

inωSev(∆, δ) ↔set V∆ω

mSev(∆,δ)(ω) = l(V) ·∏̃

length(Edges(∆ω)).

4. Let p = {p1, . . . , pr} ⊂ ((K∗)2)r , generic (K = ∪n≥1C(t1/n)).

Trop(L(p))∩Trop(Sev(∆, δ)) ↔ {tropical curves passing points inVal (p)};

m(ω; Trop(L(p)), Trop(Sev(∆, δ))) =∏

2area(Triangles).

5. If ∃ω ∈ Trop(Sev(∆, δ)) with maximal rank, not extending to aconcave function on ∆, then Trop(Sev(∆, δ)) cannot be a subfanof SecFan(∆,∆ ∩ Z2).

Main TheoremsLet r = dim(Sev(∆, δ)).

1. If rank(ω) > r , ω /∈ Trop(Sev(∆, δ)).

2. If rank(ω) = r and ω ∈ Trop(Sev(∆, δ)), ∆ω is simple-nodal.

3. If ω ∈ Trop(Sev(∆, δ)) is regular with the maximal rank r ,

inωSev(∆, δ) ↔set V∆ω

mSev(∆,δ)(ω) = l(V) ·∏̃

length(Edges(∆ω)).

4. Let p = {p1, . . . , pr} ⊂ ((K∗)2)r , generic (K = ∪n≥1C(t1/n)).

Trop(L(p))∩Trop(Sev(∆, δ)) ↔ {tropical curves passing points inVal (p)};

m(ω; Trop(L(p)), Trop(Sev(∆, δ))) =∏

2area(Triangles).

5. If ∃ω ∈ Trop(Sev(∆, δ)) with maximal rank, not extending to aconcave function on ∆, then Trop(Sev(∆, δ)) cannot be a subfanof SecFan(∆,∆ ∩ Z2).

Main TheoremsLet r = dim(Sev(∆, δ)).

1. If rank(ω) > r , ω /∈ Trop(Sev(∆, δ)).

2. If rank(ω) = r and ω ∈ Trop(Sev(∆, δ)), ∆ω is simple-nodal.

3. If ω ∈ Trop(Sev(∆, δ)) is regular with the maximal rank r ,

inωSev(∆, δ) ↔set V∆ω

mSev(∆,δ)(ω) = l(V) ·∏̃

length(Edges(∆ω)).

4. Let p = {p1, . . . , pr} ⊂ ((K∗)2)r , generic (K = ∪n≥1C(t1/n)).

Trop(L(p))∩Trop(Sev(∆, δ)) ↔ {tropical curves passing points inVal (p)};

m(ω; Trop(L(p)), Trop(Sev(∆, δ))) =∏

2area(Triangles).

5. If ∃ω ∈ Trop(Sev(∆, δ)) with maximal rank, not extending to aconcave function on ∆, then Trop(Sev(∆, δ)) cannot be a subfanof SecFan(∆,∆ ∩ Z2).

Main TheoremsLet r = dim(Sev(∆, δ)).

1. If rank(ω) > r , ω /∈ Trop(Sev(∆, δ)).

2. If rank(ω) = r and ω ∈ Trop(Sev(∆, δ)), ∆ω is simple-nodal.

3. If ω ∈ Trop(Sev(∆, δ)) is regular with the maximal rank r ,

inωSev(∆, δ) ↔set V∆ω

mSev(∆,δ)(ω) = l(V) ·∏̃

length(Edges(∆ω)).

4. Let p = {p1, . . . , pr} ⊂ ((K∗)2)r , generic (K = ∪n≥1C(t1/n)).

Trop(L(p))∩Trop(Sev(∆, δ)) ↔ {tropical curves passing points inVal (p)};

m(ω; Trop(L(p)), Trop(Sev(∆, δ))) =∏

2area(Triangles).

5. If ∃ω ∈ Trop(Sev(∆, δ)) with maximal rank, not extending to aconcave function on ∆, then Trop(Sev(∆, δ)) cannot be a subfanof SecFan(∆,∆ ∩ Z2).

Main TheoremsLet r = dim(Sev(∆, δ)).

1. If rank(ω) > r , ω /∈ Trop(Sev(∆, δ)).

2. If rank(ω) = r and ω ∈ Trop(Sev(∆, δ)), ∆ω is simple-nodal.

3. If ω ∈ Trop(Sev(∆, δ)) is regular with the maximal rank r ,

inωSev(∆, δ) ↔set V∆ω

mSev(∆,δ)(ω) = l(V) ·∏̃

length(Edges(∆ω)).

4. Let p = {p1, . . . , pr} ⊂ ((K∗)2)r , generic (K = ∪n≥1C(t1/n)).

Trop(L(p))∩Trop(Sev(∆, δ)) ↔ {tropical curves passing points inVal (p)};

m(ω; Trop(L(p)), Trop(Sev(∆, δ))) =∏

2area(Triangles).

5. If ∃ω ∈ Trop(Sev(∆, δ)) with maximal rank, not extending to aconcave function on ∆, then Trop(Sev(∆, δ)) cannot be a subfanof SecFan(∆,∆ ∩ Z2).

Main TheoremsLet r = dim(Sev(∆, δ)).

1. If rank(ω) > r , ω /∈ Trop(Sev(∆, δ)).

2. If rank(ω) = r and ω ∈ Trop(Sev(∆, δ)), ∆ω is simple-nodal.

3. If ω ∈ Trop(Sev(∆, δ)) is regular with the maximal rank r ,

inωSev(∆, δ) ↔set V∆ω

mSev(∆,δ)(ω) = l(V) ·∏̃

length(Edges(∆ω)).

4. Let p = {p1, . . . , pr} ⊂ ((K∗)2)r , generic (K = ∪n≥1C(t1/n)).

Trop(L(p))∩Trop(Sev(∆, δ)) ↔ {tropical curves passing points inVal (p)};

m(ω; Trop(L(p)), Trop(Sev(∆, δ))) =∏

2area(Triangles).

5. If ∃ω ∈ Trop(Sev(∆, δ)) with maximal rank, not extending to aconcave function on ∆, then Trop(Sev(∆, δ)) cannot be a subfanof SecFan(∆,∆ ∩ Z2).

Example of Trop(Sev(∆, δ))

∆ :

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f = ay2 + bx2y + cxy + dy + e ∈ P4[a:···:e]

Sev(∆, 1)

= {f ∈ P4 : f defines a curve with one node}

= V (16b2d2 − 8bc2d + c4 − 64ab2e)

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Example of Trop(Sev(∆, δ))

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Example of Trop(Sev(∆, δ))

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=

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a

c b

f = ay2 + bx2y + cxy + dy + e ∈ P4[a:···:e]

Sev(∆, 1)

= {f ∈ P4 : f defines a curve with one node}

= V (16b2d2 − 8bc2d + c4 − 64ab2e)

=

. ......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..................................................................................................................................................................................................... .

........................................................................................................................................................................................................................................................................................................................................................•−8bc2d

•16b2d2

•c4

•−64ab2e

R5 ?

BB

BB

BBBM

����*

R5∗2

1

1

Trop(Sev(∆, 1))

?ω

. ..........................................................................................................................................................................................................................................................................

............................................................................................................................................................................................................................................................................................................................................................................................................................

. ......................................................................................................................................................................................................................................................................... .

..............

..............

..............

..............

..............

..............

..............

..............

..............

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..............

..............

..............

..............

..............

..............

..............

..............

..............

..............

..............

..............

..............

..............

..............

..............

..............

..............

..............

......

∆ω:

∏Edges(∆ω) = 2

.

.........

.........

.........

.........

.........

.........

.........

.........

.........

.........

.........

.........

.........

.

.

..........................................................................................................................................

.

..........................................................................................................................................

•

•

•• •. ......................................................................................................................

Example of Trop(Sev(∆, δ))

∆ :

.

...........................................................................................................................................

......................................................................................................................

.

........................................

........................................

........................................

..................

-

6

•

•

•

• •R2

e

d

a

c b

f = ay2 + bx2y + cxy + dy + e ∈ P4[a:···:e]

Sev(∆, 1)

= {f ∈ P4 : f defines a curve with one node}

= V (16b2d2 − 8bc2d + c4 − 64ab2e)

=

. ......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..................................................................................................................................................................................................... .

........................................................................................................................................................................................................................................................................................................................................................•−8bc2d

•16b2d2

•c4

•−64ab2e

R5 ?

BB

BB

BBBM

����*

R5∗2

1

1

Trop(Sev(∆, 1))

?ω

. ..........................................................................................................................................................................................................................................................................

............................................................................................................................................................................................................................................................................................................................................................................................................................

. ......................................................................................................................................................................................................................................................................... .

..............

..............

..............

..............

..............

..............

..............

..............

..............

..............

..............

..............

..............

..............

..............

..............

..............

..............

..............

..............

..............

..............

..............

..............

..............

..............

..............

..............

..............

......

∆ω:

∏Edges(∆ω) = 2

.

.........

.........

.........

.........

.........

.........

.........

.........

.........

.........

.........

.........

.........

.

.

..........................................................................................................................................

.

..........................................................................................................................................

•

•

•• •. ......................................................................................................................

Example of Trop(Sev(∆, δ))

∆ :

.

...........................................................................................................................................

......................................................................................................................

.

........................................

........................................

........................................

..................

-

6

•

•

•

• •R2

e

d

a

c b

f = ay2 + bx2y + cxy + dy + e ∈ P4[a:···:e]

Sev(∆, 1)

= {f ∈ P4 : f defines a curve with one node}

= V (16b2d2 − 8bc2d + c4 − 64ab2e)

=

. ......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..................................................................................................................................................................................................... .

........................................................................................................................................................................................................................................................................................................................................................•−8bc2d

•16b2d2

•c4

•−64ab2e

R5 ?

BB

BB

BBBM

����*

R5∗2

1

1

Trop(Sev(∆, 1))

?ω

. ..........................................................................................................................................................................................................................................................................

............................................................................................................................................................................................................................................................................................................................................................................................................................

. ......................................................................................................................................................................................................................................................................... .

..............

..............

..............

..............

..............

..............

..............

..............

..............

..............

..............

..............

..............

..............

..............

..............

..............

..............

..............

..............

..............

..............

..............

..............

..............

..............

..............

..............

..............

......

∆ω:

∏Edges(∆ω) = 2

.

.........

.........

.........

.........

.........

.........

.........

.........

.........

.........

.........

.........

.........

.

.

..........................................................................................................................................

.

..........................................................................................................................................

•

•

•• •. ......................................................................................................................

Trop(Sev(∆, δ)) vs. Subdivisions of ∆

?

BB

BBBM

�����

�*

Sev(∆, δ = 1) :

.

............................................................

.....................................................................

.

.....................................................................

•

••

rk = 2

.

......................................................................

...........................................................

.

.....................................................................

••

••. ...........................................................

rk = 3 .

......................................................................

...........................................................

.

.....................................................................

•••

••. ...........................................................

.

........................................

.

........................

................

rk = 4

.

..........................................................................................

.......................................................................................................

.

.......................................................................................................

•

•••

rk = 2

.

..........................................................................................

.......................................................................................................

.

.......................................................................................................

•

•••

rk = 2

.

.........

.........

.........

.........

.........

.........

.........

.........

.........

.........

.........

.........

.........

.

.

..........................................................................................................................................

.

..........................................................................................................................................

•

•

•• •. ......................................................................................................................

rk = 3

Trop(Sev(∆, δ)) vs. Subdivisions of ∆

?

BB

BBBM

�����

�*

Sev(∆, δ = 1) :

.

............................................................

.....................................................................

.

.....................................................................

•

••

rk = 2

.

......................................................................

...........................................................

.

.....................................................................

••

••. ...........................................................

rk = 3 .

......................................................................

...........................................................

.

.....................................................................

•••

••. ...........................................................

.

........................................

.

........................

................

rk = 4

.

..........................................................................................

.......................................................................................................

.

.......................................................................................................

•

•••

rk = 2

.

..........................................................................................

.......................................................................................................

.

.......................................................................................................

•

•••

rk = 2

.

.........

.........

.........

.........

.........

.........

.........

.........

.........

.........

.........

.........

.........

.

.

..........................................................................................................................................

.

..........................................................................................................................................

•

•

•• •. ......................................................................................................................

rk = 3

Trop(Sev(∆, δ)) vs. Subdivisions of ∆

?

BB

BBBM

�����

�*

Sev(∆, δ = 1) :

.

............................................................

.....................................................................

.

.....................................................................

•

••

rk = 2

.

......................................................................

...........................................................

.

.....................................................................

••

••. ...........................................................

rk = 3 .

......................................................................

...........................................................

.

.....................................................................

•••

••. ...........................................................

.

........................................

.

........................

................

rk = 4

.

..........................................................................................

.......................................................................................................

.

.......................................................................................................

•

•••

rk = 2

.

..........................................................................................

.......................................................................................................

.

.......................................................................................................

•

•••

rk = 2

.

.........

.........

.........

.........

.........

.........

.........

.........

.........

.........

.........

.........

.........

.

.

..........................................................................................................................................

.

..........................................................................................................................................

•

•

•• •. ......................................................................................................................

rk = 3

First Application: Degree of Sev(∆, δ)

?

BB

BBBM

����

��*

.

.........

.........

.........

.........

.........

.........

.........

.........

.........

.........

.........

.........

.........

.

.

..........................................................................................................................................

.

..........................................................................................................................................

•

•

•• •. ......................................................................................................................

..................................

.....................................

..................................................................

.

.......................................................

...................

...................

.....................

....................................Trop(L(p))

m(ω; Trop(L(p)), Trop(Sev(∆, δ)))

= mL(p)(ω) ·mSev(∆,1)(ω) · ξ(ω; T1, T2)

= 1 · 2 · 2 = 4

(combinatorial formula for ξ

is also found in (Y,11))

Mikhalkin’s multiplicity of τω

:=∏

∆ω2area(Triangles)

= 2 · 2 = 4

First Application: Degree of Sev(∆, δ)

?

BB

BBBM

����

��*

.

.........

.........

.........

.........

.........

.........

.........

.........

.........

.........

.........

.........

.........

.

.

..........................................................................................................................................

.

..........................................................................................................................................

•

•

•• •. ......................................................................................................................

..................................

.....................................

..................................................................

.

.......................................................

...................

...................

.....................

....................................Trop(L(p))

m(ω; Trop(L(p)), Trop(Sev(∆, δ)))

= mL(p)(ω) ·mSev(∆,1)(ω) · ξ(ω; T1, T2)

= 1 · 2 · 2 = 4

(combinatorial formula for ξ

is also found in (Y,11))

Mikhalkin’s multiplicity of τω

:=∏

∆ω2area(Triangles)

= 2 · 2 = 4

First Application: Degree of Sev(∆, δ)

?

BB

BBBM

����

��*

.

.........

.........

.........

.........

.........

.........

.........

.........

.........

.........

.........

.........

.........

.

.

..........................................................................................................................................

.

..........................................................................................................................................

•

•

•• •. ......................................................................................................................

..................................

.....................................

..................................................................

.

.......................................................

...................

...................

.....................

....................................Trop(L(p))

m(ω; Trop(L(p)), Trop(Sev(∆, δ)))

= mL(p)(ω) ·mSev(∆,1)(ω) · ξ(ω; T1, T2)

= 1 · 2 · 2 = 4

(combinatorial formula for ξ

is also found in (Y,11))

Mikhalkin’s multiplicity of τω

:=∏

∆ω2area(Triangles)

= 2 · 2 = 4

First Application: Degree of Sev(∆, δ)

?

BB

BBBM

����

��*

.

.........

.........

.........

.........

.........

.........

.........

.........

.........

.........

.........

.........

.........

.

.

..........................................................................................................................................

.

..........................................................................................................................................

•

•

•• •. ......................................................................................................................

..................................

.....................................

..................................................................

.

.......................................................

...................

...................

.....................

....................................Trop(L(p))

m(ω; Trop(L(p)), Trop(Sev(∆, δ)))

= mL(p)(ω) ·mSev(∆,1)(ω) · ξ(ω; T1, T2)

= 1 · 2 · 2 = 4

(combinatorial formula for ξ

is also found in (Y,11))

Mikhalkin’s multiplicity of τω

:=∏

∆ω2area(Triangles)

= 2 · 2 = 4

Second Application: Secondary Fans

• SecFan(∆,A) is a complete fan in RA parameterizingregular marked subdivisions of (∆,A). (Discriminantalvariety, Chow quotient, etc)

• Trop(Sev(∆, 1)) is a subfan of SecFan(∆,∆ ∩ Z2)

• What about Sev(∆, δ) for general δ?

• A counterexample is found by E.Katz(2008).

Theorem (Y,11)If ∃ ω ∈ Trop(Sev(∆, δ)) with maximal rank which does notextend to a concave function on ∆, then Trop(Sev(∆, δ)) cannotbe a subfan of SecFan(∆,∆ ∩ Z2).

. ............................................................

..........................................................................................................................................

. ........................................................... .

..........................................................................................................................................

• •

• •

•

Second Application: Secondary Fans

• SecFan(∆,A) is a complete fan in RA parameterizingregular marked subdivisions of (∆,A). (Discriminantalvariety, Chow quotient, etc)

• Trop(Sev(∆, 1)) is a subfan of SecFan(∆,∆ ∩ Z2)

• What about Sev(∆, δ) for general δ?

• A counterexample is found by E.Katz(2008).

Theorem (Y,11)If ∃ ω ∈ Trop(Sev(∆, δ)) with maximal rank which does notextend to a concave function on ∆, then Trop(Sev(∆, δ)) cannotbe a subfan of SecFan(∆,∆ ∩ Z2).

. ............................................................

..........................................................................................................................................

. ........................................................... .

..........................................................................................................................................

• •

• •

•

Second Application: Secondary Fans

• SecFan(∆,A) is a complete fan in RA parameterizingregular marked subdivisions of (∆,A). (Discriminantalvariety, Chow quotient, etc)

• Trop(Sev(∆, 1)) is a subfan of SecFan(∆,∆ ∩ Z2)

• What about Sev(∆, δ) for general δ?

• A counterexample is found by E.Katz(2008).

Theorem (Y,11)If ∃ ω ∈ Trop(Sev(∆, δ)) with maximal rank which does notextend to a concave function on ∆, then Trop(Sev(∆, δ)) cannotbe a subfan of SecFan(∆,∆ ∩ Z2).

. ............................................................

..........................................................................................................................................

. ........................................................... .

..........................................................................................................................................

• •

• •

•

[Yang1] J.Yang, Initial Schemes of Very Affine Severi Varieties.arXiv:1108.5839[Yang2] J.Yang, Some Parameter Spaces of Curves on ToricSurfaces, Their tropicalizations and Degrees. In preparation.

Thank you!

Tropical Product

...........................................................................................

.........

.........

.........

.........

.........

2

.

.....................................................................

......................................................................................................................................................

...............................................................................2

.

............................................................

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. ...........................................................................

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.......................................................................................................................

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.........

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.........

.........

.........

.........

.........

.........

.........

.........

.........

.........

.........

.........

.........

.........

.........

.........

.........

........

T1

.............................................

..............

..............

..............

..

................................................................................................

T2

...........................................................................................

.........

.........

.........

.........

.........

2

.

.....................................................................

......................................................................................................................................................

...............................................................................2

.

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. ...........................................................................

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.......................................................................................................................

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.........

........

supp(T1) ∩ supp(T2)

•ω2

.............................................

..............

..............

..............

..............

..............

...

................................................................................................•ω1

Tropical Product

...........................................................................................

.........

.........

.........

.........

.........

2

.

.....................................................................

......................................................................................................................................................

...............................................................................2

.

............................................................

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................................................................................................................................................................................................................

...........................................................

. ...........................................................................

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.........

.........

.........

.........

.........

.........

.........

.........

.........

.........

.........

........

T1

.............................................

..............

..............

..............

..

................................................................................................

T2

...........................................................................................

.........

.........

.........

.........

.........

2

.

.....................................................................

......................................................................................................................................................

...............................................................................2

.

............................................................

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................................................................................................................................................................................................................

...........................................................

. ...........................................................................

...................................................................................................................... .

.......................................................................................................................

.........

.........

.........

.........

.........

.........

.........

.........

.........

.........

.........

.........

.........

.........

.........

.........

.........

.........

.........

.........

.........

.........

........

supp(T1) ∩ supp(T2)

•ω2

.............................................

..............

..............

..............

..............

..............

...

................................................................................................•ω1

� ���

........................................

..............................

The tropical intersection multiplicity of T1 and T2 at ω is

m(ω) = m(ω; T1, T2) := mT1(ω) ·mT2(ω) · ξ(ω; T1, T2),

where ξ(ω; T1, T2), is the volume of the parallelepipedconstructed by the fundamental cells of the latticesLi ∩ Zn, (i = 1, 2).

Algebraic Geometry Tropical Geometryvery affine varieties tropical varieties: balanced weighted

rational polyhedral complex ⊂ Rn

X ⊂ (C∗)n Trop(X ), tropicalization of X• dim(X ) = dim(Trop(X ))• Trop(X1 ∪ X2) = Trop(X1) + Trop(X2)• Trop(X1 ∩ gX2) = Trop(X1) · Trop(X2)for generic g ∈ (C∗)n

−→

inωSev(∆, δ) vs V∂∆ω,nodal

DefinitionLet S(∆) : ∆1 ∪ · · · ∪∆m be a nodal subdivision of ∆.f ∈ V∂∆ω,nodal ⊂ P∆ ⇔• s ∈ Edges(S(∆)) ⇒ fs is a pure power of a binomial,

xayb(αxc + βyd)|s|;

• ∆i is a trangle ⇒ f∆i defines a rational curve which isunibranch at each intersection point with the boundarydivisors of the toric surface X∆i ;

• ∆j is a parallelogram ⇒ f∆j has the formxky l(αxa + βyb)p(γxc + δyd)q.

inωSev(∆, δ) vs V∂∆ω,nodal

DefinitionLet S(∆) : ∆1 ∪ · · · ∪∆m be a nodal subdivision of ∆.f ∈ VS(∆)ω,nodal

⊂ P∆ ⇔• s ∈ Edges(S(∆)) ⇒ fs is a pure power of a binomial,

xayb(αxc + βyd)|s|;

• ∆i is a trangle ⇒ f∆i defines a rational curve which has aunibranch at each intersection point with the boundarydivisors of the toric surface X∆i ;

• ∆j is a parallelogram ⇒ f∆j has the formxky l(αxa + βyb)p(γxc + δyd)q.

TheoremVS(∆)ω,nodal

is a translation of a closed subgroup GS(∆)ω,nodalof

an algebraic torus.

M∂S(∆) =

ξ1 ξ2 ξ3 ξ4 ξ5 ξ6

F1 −2 1 1 0 0 0F2 0 −1 0 4 1 0F3 0 0 −1 0 −1 2

M∂∆,P1 =

α1 β1 α2 β2 α3 β3

s12 1 2 −1 −2 0 0s13 1 0 0 0 −1 0s23 0 0 −1 2 1 −2

The Smith Normal Forms of M∂S(∆) and M∂S(∆),P1 coincide toeach other as : 1 0 0 0 0 0

0 1 0 0 0 00 0 2 0 0 0

.

Thus VS(∆) is a union of two translations of 3-dimensionalsubtorus of T∆.

.

.............................................................................................................................................................................................................................................

...........................................................................................................................................

.............................................................................................................................................................

. ............................................................

.......................................................................................................................................... .

.............................................................................................................................................................

•

•

•

•

•

•

•

•

•F1

F2 F3