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 .
(t →∞)
f(u,v ,w) = y2 + z2: two (C∗)2
PPPPPPq
6
�����)
R3
••
•
••
••
q
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq��
����
(u, v , w)
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
PPPPPPq
6
�����)
R3
••
•
••
••
q
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq��
����
(u, v , w)
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
PPPPPPq
6
�����)
R3
••
•
••
••
q
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq��
����
(u, v , w)
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))
∆ :
.
...........................................................................................................................................
......................................................................................................................
.
........................................
........................................
........................................
..................
-
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
.
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.
.
..........................................................................................................................................
.
..........................................................................................................................................
•
•
•• •. ......................................................................................................................
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
.
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.
.
..........................................................................................................................................
.
..........................................................................................................................................
•
•
•• •. ......................................................................................................................
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
.
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.
.
..........................................................................................................................................
.
..........................................................................................................................................
•
•
•• •. ......................................................................................................................
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
.
............................................................
.....................................................................
................................................................................................................................................................................................................
...........................................................
. ...........................................................................
...................................................................................................................... .
.......................................................................................................................
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
........
T1
.............................................
..............
..............
..............
..
................................................................................................
T2
...........................................................................................
.........
.........
.........
.........
.........
2
.
.....................................................................
......................................................................................................................................................
...............................................................................2
.
............................................................
.....................................................................
................................................................................................................................................................................................................
...........................................................
. ...........................................................................
...................................................................................................................... .
.......................................................................................................................
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
........
supp(T1) ∩ supp(T2)
•ω2
.............................................
..............
..............
..............
..............
..............
...
................................................................................................•ω1
Tropical Product
...........................................................................................
.........
.........
.........
.........
.........
2
.
.....................................................................
......................................................................................................................................................
...............................................................................2
.
............................................................
.....................................................................
................................................................................................................................................................................................................
...........................................................
. ...........................................................................
...................................................................................................................... .
.......................................................................................................................
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
........
T1
.............................................
..............
..............
..............
..
................................................................................................
T2
...........................................................................................
.........
.........
.........
.........
.........
2
.
.....................................................................
......................................................................................................................................................
...............................................................................2
.
............................................................
.....................................................................
................................................................................................................................................................................................................
...........................................................
. ...........................................................................
...................................................................................................................... .
.......................................................................................................................
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
.........
........
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