TROPICAL LIMIT OF LOG-INFLECTION POINTS FOR
PLANAR CURVES
GRIGORY MIKHALKIN AND ARTHUR RENAUDINEAU
Abstract. The paper describes behavior of log-inflection points
of curves in (C× )2 under passing to the tropical limit. We show
that such points accumulate by pairs at the midpoints of bounded
edges in the limiting tropical curve. Log-inflection points are points
of inflection with respect to the parallelization of (C× )2 given by
the multiplicative group law.
1. Introduction
Let f : (C× )2 → C, C× = C \ {0}, be a Laurent polynomial in two
variables with the Newton polygon ∆. The zero locus
Vf = {(z, w) ∈ (C× )2 | f (z, w) = 0} ⊂ (C× )2
is a non-compact complex curve. For non-singular Vf the logarithmic
Gauss map (see [Kap91])
(1)
γf : Vf → CP 1
∂f
: w ∂w
). This map depends only on Vf
is defined by γf (z, w) = (z ∂f
∂z
and not on f itself. It is the map that takes a point of Vf to the slope
of its tangent plane with respect to the parallelization of (C× )2 given
by the multiplicative translations (z, w) 7→ (αz, βw), α, β ∈ C× .
Definition 1. The critical points of γf are called the log-inflection
points of Vf .
Denote the set of critical points with ρVf ⊂ Vf .
If all coefficients of the Laurent polynomial f are real we define
RVf = {(z, w) ∈ (C× )2 | f (z, w) = 0} ⊂ (R× )2 . We have
γf |RVf : RVf → RP 1 ,
and we denote by ρRVf ⊂ RVf the set of critical points of γf |RVf (called
real log-inflection points). Recall the definition of tropical limits (see
e.g. [IKMZ], [BIMS15] as well as [Mik06]). Let
X
{ft =
ajk,t z j wk }t∈A
j,k∈Z
be a family of polynomials parameterized by an (infinite) set A. Suppose that the degree of ft is universally bounded in z, z −1 , w and w−1 .
Let α : A → R be a map whose image is unbounded from above.
1
2
GRIGORY MIKHALKIN AND ARTHUR RENAUDINEAU
The pair (ft , α) is called a scaled sequence of polynomials ft (α is the
scaling).
The scaled sequence of polynomials is said to converge tropically if
the limit
atrop
jk = lim logα(t) |ajk,t | ∈ R ∪ {−∞}
t∈A
exists, and is a finite number or −∞ for all j, k ∈ Z. Here the limit is
taken with respect to the directed set A. where the order is given by
α. Define the limiting Newton polygon
2
∆ = ConvexHull{(j, k) ∈ Z2 | atrop
jk 6= −∞} ⊂ R .
It is a bounded convex lattice polygon.
The set R ∪ {−∞} is called the set of tropical numbers and denoted
with T. If a scaled sequence of polynomials (ft , α) converges tropically,
then the limit
C = lim Logα(t) (Vt )
t∈A
exists in the Hausdorff sense, i.e. as the limit of closed subsets of R2
with the topology given by the Hausdorff distance. It is called the
tropical limit of {Vt ⊂ (C× )2 }t∈A . Here Log : (C× )2 → R2 is defined by
Log(z, w) = (log |z|, log |w|).
More generally, given a scaled sequence (Vt , α) of curves Vt ⊂ (C∗ )n we
say that it converges tropically to C ⊂ Rn if C is the limit of Logα(t) (Vt )
in the Hausdorff sense.
It can be proved (see [Mik05], and [IKMZ] for a more general statement) that the limit C is a rectilinear graph in Rn . The edges E of
C are straight intervals with rational slopes, and can be prescribed
integer weights w(E) (coming from Vt ) so that the balancing condition
X
w(E)u(E) = 0
E3v
holds for every vertex v ∈ C. Here the sum runs over all edges E
adjacent to v, and u(E) ∈ Zn is the primitive vector parallel to E in
the direction away from v.
Such a graph C ⊂ Rn is called a tropical curve. If n = 2 then C
defines a lattice subdivision of the Newton polygon ∆. If each polygon
of the subdivision is a triangle of area 21 (the minimal possible area
for a lattice polygon) then C is called smooth. For details we refer to
[IMS09] and [BIMS15].
Definition 2. Let C be a smooth plane tropical curve. The tropical
parabolic locus of C is the set ρC ⊂ C formed by the midpoints of all
bounded edges of C.
The main result of this paper is the following.
TROPICAL LIMIT OF LOG-INFLECTION POINTS FOR PLANAR CURVES 3
Theorem 3. Let C ⊂ R2 be a smooth tropical curve, and {Vt ⊂
(C× )2 }t∈A be a family of complex curves parameterised by the scaled
sequence α : A → R. Suppose that Vt tropically converges to C and
that the limiting Newton polygon ∆ coincides with the Newton polygons of Vt for large α(t). Then Logα(t) (ρVt ) converges to ρC in the
Hausdorff metric when α(t) → +∞. Furthermore, a small neighborhood of a point from ρC contains exactly two points of Logα(t) (ρVt ) for
large α(t).
In other words, 2ρC is the tropical limit of ρVt .
We refer to [BIMS15], Section 3 for the notion of twisted edge and the
notion of twist admissible set of edges of a plane tropical smooth curve.
The next result is a straightforward corollary of Theorem 3.
Corollary 4. Let C be a tropical smooth plane curve and let T be a
twist admissible set of edges of C. Let {Vt ⊂ (C× )2 }t∈A be a family
of real curves parameterised by the scaled sequence α : A → R and
tropically converging to C, so that the corresponding set of twisted edges
is T . Then Logα(t) (ρRVt ) converges to the midpoints of the edges in T
and a small neighborhood of a midpoint from an edge in T contains
exactly two points of Logα(t) (ρRVt ) for large α(t).
Acknowledgements. Research is supported in part by the grants
159240, 159581 and the NCCR SwissMAP project of the Swiss National Science Foundation. The first author is thankful to Ernesto
Lupercio for useful conversations about tropical Gauss maps.
2. Tropical limit of logarithmic Gauss map
Let Vt = {ft = 0}t∈A ⊂ (C× )2 be a family of complex curves parameterised by the scaled sequence α : A → R. Let Vet , t ∈ A, be the graph
of the Gauss map γft (which we denote with γt for brevity) given by
Vet = {(zt , γt (zt )) |zt ∈ Vt } ⊂ (C× )2 × CP1 .
Denote by π1 : Vet → Vt ⊂ (C× )2 the projection on the first factor
and by π2 : Vet → CP1 the projection on the second factor. The map
π1 is an isomorphism and the map π2 is of degree 2Vol(∆) (by the
Bernstein-Kouchnirenko formula [Kou76], see also [Mik00]). Denote
by π1trop : R2 × TP1 → R2 the projection on the first factor and by
π2trop : R2 × TP1 → TP1 the projection on the second factor. The
following proposition follows from compactness theorem (see [IKMZ],
Section 3.4).
Proposition 5. There exists a subfamily A0 ⊂ A with unbounded
α(A0 ) ⊂ R such that the subsequence (Vet )t∈A0 which tropically converges
e in R2 × TP1 . We have π1trop (C)
e = C.
to a tropical curve C
4
GRIGORY MIKHALKIN AND ARTHUR RENAUDINEAU
Proof. The proposition is a special case of Theorem 38 of [IKMZ]. We
apply this theorem to the projective curves V̄t , t ∈ A, obtained as the
e to be the closure in R2 × TP1
closures of Vet ∩ (C× )3 in CP3 . We set C
of C̄ ∩ R3 , where C̄ ⊂ TP3 is the tropical limit of a subfamily from
Theorem 38 of [IKMZ]. By the uniqueness of tropical limit we have
e
C = π1trop (C).
e → TP1 as a tropical
Definition 6. We refer to the map π2trop |Ce : C
limit of the family γt .
Remark 7. The same construction works in the case of hypersurfaces
of higher dimensions. However in this text, we will focus on the case
of curves.
e not only depends on
Remark 8. By construction, the tropical curve C
C but also on an approximation (Vt )t∈A of C, and even the choice of
subfamily A0 ⊂ A. Different choices may result in different models of
e Nevertheless, the tropical limit of the ramification locus of γt is a
C.
well-defined subset of C and does not depend on these choices as shown
in this paper.
e
If x ∈ C is a generic point, then the fiber (π1trop )−1 (x) intersects C
transversally (meaning that any point of intersection is in the relative
e at finitely many points. Denote by e the
interior of an edge of C)
edge of C containing x and denote by Ze the sublattice of Z3 generated
−
−
by (0, 0, 1) and (→
v , 0), where →
v is a primitive vector contained in e.
trop
−1
e
e containing
Let z ∈ C ∩ (π1 ) (x), and denote by ee the edge of C
trop
e (π1 )−1 (x)) as
z. Define the local tropical intersection number ιz (C,
the index of the sublattice in Ze generated by (0, 0, 1) and a primitive
vector contained in ee (multiplied by the weight of ee). The degree of
π1trop is defined as the sum
X
e (π1trop )−1 (x)).
d1 :=
ιz (C,
e
z∈(π1trop )−1 (x)∩C
Similarly, if y ∈ R is a generic point, then the 2-dimensionnal fiber
e transversally. Let z ∈ C
e ∩ (π2trop )−1 (x), and
(π2trop )−1 (y) intersects C
e containing z. The local tropical intersection
denote by ee the edge of C
e (π2trop )−1 (y)) denote in this case the index of the sublattice
number ιz (C,
in Z3 generated by (π2trop )−1 (y) ∩ Z3 and a primitive vector contained
in ee (multiplied by its weight). Define the degree d2 as the sum
X
e (π2trop )−1 (y)).
d2 :=
ιz (C,
e
z∈(π2trop )−1 (y)∩C
Proposition 9. One has d1 = 1 and d2 = 2Vol(∆).
TROPICAL LIMIT OF LOG-INFLECTION POINTS FOR PLANAR CURVES 5
Proof. It follows from the proof of Proposition 42 of [IKMZ] (the degree
of π1 is 1 and the degree of π2 is 2Vol(∆)).
e
One can define similarly local degree of π1trop (resp., π2trop ). Let s ∈ C
and let Us be a small neighborhood of s only containing points from
edges adjacent to s. Let x ∈ C ∩ π1trop (Us ) (resp., y ∈ π2trop (Us )) be
a generic point. Define the local degree of π1trop (resp., π2trop ) at s as
e (π1trop )−1 (x)) (resp.,
the sum of local tropical intersection number ιz (C,
e (π2trop )−1 (y))) over all z ∈ Us .
ιz (C,
3. First properties of tropical Gauss map
Proposition 10. Let {Vt ⊂ (C× )2 }t∈A be a family of complex curves
parameterised by the scaled sequence α : A → R which tropically converges to a smooth plane tropical curve C. Let e be an edge of C of
slope p ∈ QP1 , and let x belongs to the relative interior of e. If xt ∈ Vt
satisfies limt∈A Logα(t) (xt ) = x, then
lim γt (xt ) = p.
t∈A
Proof. After
amultiplicative changes of coordinates (z, w) 7→ (z a wb , z c wd ),
a b
where
∈ SL2 (Z), and a homothetic change of coordinates
c d
(z, w) 7→ (f (t)z, g(t)w), one sees that it is enough to prove the statement for the point (0, 0) and the edge of slope ∞ in the tropical limit of
the family of complex curves (Vt )t>1 given by the family of polynomials
Ft (z, w), where
Ft (z, w) = 1 + z + t−1 Rt (z, w),
where all coefficients rjk,t of a Laurent polynomial Rt (z, w) satisfies
limt→+∞ logt (rjk,t ) ≤ 0. One has then
∂Rt
−1
−1 ∂Rt
γt (zt , wt ) = zt (1 + t
) : t wt
.
∂z
∂w
If xt = (zt , wt ) ∈ Vt satisfies limt→+∞ logt (xt ) = (0, 0), then limt→+∞ zt =
∂Rt
∂Rt
−1 and limt→+∞ (zt t−1
) = limt→+∞ (wt t−1
) = 0, which proves
∂z
∂w
the proposition.
Taking the logarithm with base α(t), we obtain the following corollary.
Corollary 11. Let {Vt ⊂ (C× )2 }t∈A be a family of complex curves
parametrised by the scaled sequence α : A → R which tropically converges to a smooth plane tropical curve C. Let e be an edge of C of
slope p ∈ QP1 , x be a point of the relative interior of e, and xt ∈ Ct
be a sequence such that limt∈A Logα(t) (xt ) = x. Then up to passing to
a subsequence one has
• if p 6= 0 or ∞, then limt∈A Logα(t) (γt (xt )) = [1T : 1T ] ∈ TP1 ,
6
GRIGORY MIKHALKIN AND ARTHUR RENAUDINEAU
• if p = 0, then limt∈A Logα(t) (γt (xt )) = [a : 1T ] ∈ TP1 , for some
a ∈ [−∞, 0],
• if p = ∞, then limt∈A Logα(t) (γt (xt )) = [1T : a] ∈ TP1 , for some
a ∈ [−∞, 0].
Since the tropical map π1trop is of degree 1, one has a continuous
e such that π1trop ◦ iC = id.
inclusion iC : C ,→ C
Proposition 12. We have π2trop (s) = [1T : 1T ] ∈ TP1 for any s ∈
iC (Vert(C)).
Proof. At least one of the edges adjacent to s must have the slope
different from 0 and ∞. The proposition follows then immediately
from Corollary 11.
Proposition 13. For any s ∈ iC (Vert(C)), the tropical map π2trop is
of local degree 1 at s.
Proof. Similarly to the proof of Proposition 10, we can assume that
s = (0, 0) and that a polynomial defining an approximation of C is
given by 1 + z + w + t−1 Rt (z, w), where all coefficients rjk,t of Rt (z, w)
satisfies limt∈A logα(t) (rjk,t ) ≤ 0. The graph of the logarithmic Gauss
map over Vt is then given by
−1 ∂Rt
−1 ∂Rt
e
Vt = z, w, z(1 + t
) : w(1 + t
) | (z, w) ∈ Vt
∂z
∂w
and the graph of the logarithmic Gauss map on the line L = {1 + z + w = 0}
is given by
Le = {z, w, [z : w] | (z, w) ∈ L} .
e the tropical limit of L.
e It follows from the description of
Denote by L
e
e
Vt and L that there exists a small neighborhood U of the point (0, 0, 0)
e∩U = L
e ∩ U . But the map π2trop : L
e → TP1 is of
in R3 such that C
degree one, which proves the proposition.
Example 14. In Figure 1, we draw the affine part of the tropical logarithmic Gauss map for the tropical curve given by “1 + x + y + (−1)xy”,
and for the tropical curve given by “1 + x + y + (−1)x2 ”.
4. Proof of Theorem 3
Denote by m1 , · · · , ml the midpoints of bounded edges of C, and
by zt1 , · · · , ztl some points in Vt such that limt∈A Logα(t) (zti ) = mi , for
1 ≤ i ≤ l. Put γt (zti ) = [xit : yti ], for 1 ≤ i ≤ l and denote by
Φt :
CP1
→ CPl+1
,
[z1 : z2 ] 7→ z1 : z2 : ϕ1t (z1 , z2 ) : · · · : ϕlt (z1 , z2 )
where ϕit (z1 , z2 ) = z1 yti − z2 xit are linear maps.
TROPICAL LIMIT OF LOG-INFLECTION POINTS FOR PLANAR CURVES 7
T1 ⊂ TP1
T1 ⊂ TP1
T2
(−∞, −∞, −∞)
T2
(−∞, −∞, −∞)
Figure 1. Tropical Gauss map for the tropical curve
given by “1 + x + y + (−1)xy” and for the tropical curve
given by “1 + x + y + (−1)x2 ”.
Consider the line Lt = Φt (CP1 ), and denote by L its tropical limit
in TPl+1 (after passing to a subsequence), which exists by compactness
theorem (see [IKMZ], Section 3.4). Introduce also
Vbt = {(zt , Φt ◦ γt (zt )) | zt ∈ Vt } ⊂ (C× )2 × CPl+1 ,
b ⊂ R2 × TPl+1 its tropical limit (after passing to a
and denote by C
subsequence). One has isomorphisms Vbt ' Vet and Lt ' CP1 given by
projections, and one obtains tropical maps π trop , p and q such that the
degree of π trop is 2Area(∆), the degrees of p and q are 1, and such that
the following diagram commutes.
trop
b −π−−→
C

p
y
L

q
y
trop
e −π−2−→ TP1
C

 trop
yπ1
C
b the continuous inclusion that is right-inverse to
Denote by ip : C → C
trop
the projection π1 ◦ p, and denote by sL the image in L of [1T : 1T ] ∈
TP1 by the continuous inclusion TP1 ,→ L that is right-inverse to
q. It follows from Proposition 13 that for any s ∈ Vert(C), one has
8
GRIGORY MIKHALKIN AND ARTHUR RENAUDINEAU
π trop (ip (s)) = sL . In fact, if π trop (ip (s)) 6= sL , a small neighborhood
of iC (s) would be contracted to [1T : 1T ] by π2trop , contradicting Proposition 13. Moreover, since π2trop is locally of degree 1 at iC (s) for any
s ∈ Vert(C) and since the projections p and q are both of degree 1, we
deduce that the tropical map π trop is locally of degree 1 at ip (s) for any
s ∈ Vert(C).
Proposition 15. Let s be a vertex of C and let eb be a bounded edge of
ip (C) adjacent to ip (s). Then eb is not contracted by π trop .
Proof. Assume that eb is contracted by π trop , and denote by e the edge
of C containing π1trop ◦ p(b
e). Let us first prove that ip (e) = eb.
b
Assume that eb ip (e), and denote by s0 ∈
/ ip (V ert(C)) the vertex of C
trop
trop 0
trop
adjacent to eb. Since eb is contracted by π , one has π (s ) = π (s),
and the local degree of π trop at s0 is at least 1 (if the local degree would
be 0, the point s0 would not be a vertex of eb). But then the degree of
π trop would be at least 2Area(∆) + 1, which impossible. Then one has
ip (e) = eb.
b that for any bounded edge eC of
It follows from the construction of C
C, the set ip (eC ) consists of at least two edges (if mC is the middle point
b in a boundary divisor of TPl+1 projecting
of eC , there is a point of C
to mC ). Then ip (eC ) 6= e and e cannot be contracted by π trop .
We call a tropical map f finite over a subset F if it does not contract
any edge in F .
Proposition 16. Let e be a bounded edge of C, and let m be the midpoint of e. The map π trop is finite over ip (e) and is of local degree 1 at
any point of ip (e) except at ip (m), where it is of local degree 2.
Proof. Denote by T the image of ip (e) by the map π trop . The set T is a
tree since T ⊂ L. Moreover, sL is a vertex of valence 1 of T . In fact, if
sL would have at least two adjacent edges in T , it would mean since T
is a tree that there are at least 3 vertices of ip (e) mapped to sL , which
would mean that the degree of π trop would be at least 2Area(∆) + 1,
which is impossible. Denote by eL the edge of T adjacent to sL .
Assume first that the edges e11 and e21 of ip (e) adjacent to the vertices
over sL join over the edge eL , see Figure 2.
In this case, one has ip (e) = e11 ∪ e21 . Since the local degree of π trop
on the edges e11 and e21 is equal to one, the vertex adjacent to e11 and
e21 is the vertex ip (m). Moreover, the map π trop is of local degree 2 at
ip (m). In fact, if the local degree would be more than 2, then sL would
have at least 3 preimages by π trop over the set (π1trop ◦ p)−1 (e), which is
impossible.
Assume now that the two edges e11 and e21 do not join over eL , and
denote by s1 the other vertex of T adjacent to eL , see Figure 3.
In this case, the preimage of s1 consists of 2 vertices at which the map
TROPICAL LIMIT OF LOG-INFLECTION POINTS FOR PLANAR CURVES 9
Figure 2. The two edges e11 and e21 join over the edge eL .
Figure 3. The two edges e11 and e21 do not join over the
edge eL .
π trop is of local degree one. As in the proof of Proposition 15, one see
that every edge of if (C) adjacent to if (s1 ) is not contracted by π trop .
One conclude then easily by induction.
We are ready to prove Theorem 3.
Proof of Theorem 3. Let e be a bounded edge of C, and denote by v1
and v2 the vertices adjacent to e. Consider the fibration λt : Vt → C
as defined in [Mik04]. It follows from Proposition 13 that there exists
a small neighborhood Ni of vi , i = 1, 2 in C such that for t big enough,
the map γt |λ−1
is of degree one on its image, see Figure 4. It follows
t (Ni )
from Proposition 10 that the image by γt of a boundary component of
λ−1
t (Ni ) converges to the slope of the edge of C supporting this boundary component. Consider a small circle Γ centered at the slope of e and
10
GRIGORY MIKHALKIN AND ARTHUR RENAUDINEAU
v1
e
v2
Figure 4. The sets λ−1
t (Ni ).
of radius ε. The image by γt of the boundary component of λ−1
t (Ni )
supported above e is contained in the same connected component of
CP1 \ Γ as the slope se of e. Then for t big enough, Γ is contained
−1
in γt λ−1
(Γ) is a circle Γi ⊂ λ−1
t (Ni ) , and the set γt |λ−1
t (Ni )
t (Ni )
(see Figure 5). Note that one of the two connected components of
Γ1
Γ2
Figure 5. The circles Γi .
−1
λ−1
t (Ni ) \ Γi only contains the boundary component of λt (Ni ) supported on e. If not, then there would exist a path in Vt \Γi with ends at
and bt , such that the tropical limit of at lie in the interior of e and the
tropical limit of bt lie in the interior of another edge adjacent to vi (see
Figure 6). But it follows from Proposition 10 that for t big enough,
bt
at
Figure 6. The path with ends at and bt .
such a path should intersect Γi .
TROPICAL LIMIT OF LOG-INFLECTION POINTS FOR PLANAR CURVES 11
The restriction of γt to the annulus A supported on e with boundary Γ1 ∪ Γ2 gives us a map of degree 2 to the disc containing se with
boundary Γ. In fact, the preimage of Γ by γt in A consists exactly of
Γ1 ∪ Γ2 since the map γt is of degree 2Area(∆) and since from Proposition 13 for any vertex v of C there exists a small neighborhood N
of v such that the map γt restricted to λ−1
t (N ) is of degree 1. By the
Riemann-Hurwitz formula, such a map has two critical points. The
tropical limit of a scaled sequence of critical points must be a point
with the local degree of π trop greater than one. Proposition 16 now
implies the theorem.
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Grigory Mikhalkin, Université de Genève, Mathématiques, Villa
Battelle, 1227 Carouge, Suisse
E-mail address: [email protected]
Arthur Renaudineau, Eberhard Karls Universität Tübingen, Fachbereich Mathematik, Auf der Morgenstelle 10, 72076 Tübingen, Deutschland
E-mail address: [email protected]