Tropical Geometry of genus two curves

Web page for the paper by María Angélica Cueto and Hannah Markwig

The computational results in this web page were obtained using the software packages SAGE 6.9, Macaulay 2, and Singular, including the special library tropical.lib. The sage outputs are stored as .sobj files and plain text files (as commends in the scripts). The Macaulay 2 outputs are recorded as comments on the scripts.

All scripts and outputs described in this page are available as a zip file here. CAVEAT: Some of the .sobj files may not load when working with earlier versions of SAGE. You can run the scripts to obtained readable .sobj files.


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§0. Naive tropicalization of plane hyperelliptic curves and tropical modifications inducing faithful tropicalizations

The following scripts allow us to produce faithful tropicalizations on the minimal skeleton (or extended, whenever indicated) of all genus 2 smooth curves defined over a non-Archimedean field K with a valuation val: K*R. Our input data for the curve is the hyperelliptic equation, i.e. its six ramification points α1,…,α6 in P1. Furthermore, we may assume α1= (0,1) and α6=(0,1) (with -val(α1)="-Infinity", -val(α1)="+Infinity"), and the remaining α2,…,α5 lie in K*. Their negative valuations

ωi = -val(αi) for i = 1,…,6,    d34 = -val(α3 - α4)

and the initial forms in(αi) satisfy the required conditions indicated in Table 1.



Table 1: Classification of skeleta of smooth genus 2 curves.
CellExtended skeleta Defining conditions Lengths
(I) ω2 < ω3 < ω45 L0 = (ω4 - ω3)/2
L1 = 2(ω5 - ω4)
L2 = (ω3 - ω2)/2
(II) ω2 < ω3 = ω45
d343
in(α3)=in(α4)
L0 = (ω4 - d34)/2
L1 = 2(ω5 - ω3)
L2 = (ω3 - ω2)/2
(III) ω2 < ω3 = ω45
d34 = ω3
in(α3) ≠ in(α4)
L0 = 0
L1 = 2(ω5 - ω3)
L2 = (ω3 - ω2)/2
(IV) ω2 < ω3 < ω4
ω4 = ω5
in(α4) ≠ in(α5)
L0 = (ω4 - ω3)/2
L1 = 0
L2 = (ω3 - ω2)/2
(V) ω2 < ω4
ω2 = ω3
ω4 = ω5
in(α2) ≠ in(α3)
in(α4) ≠ in(α5)
L0 = (ω4 - ω3)/2
L1 = 0
L2 = 0
(VI) ω2 < ω3
ω3 = ω4 = ω5
in(α3) ≠ in(α4)
in(α3) ≠ in(α5)
in(α4) ≠ in(α5)
L0 = 0
L1 = 0
L2 = (ω3 - ω2)/2
(VII) ω2 = ω3 = ω4 = ω5
in(αi) ≠ in(αj) (for i≠ j)
L0 = 0
L1 = 0
L2 = 0

The naive tropicalization induced by the hyperelliptic equation

g(x,y) = y2 - x (x- α2)(x- α3)(x- α4)(x- α5)

is never faithful. We use the tropical modification of R2 along the tropical polynomial

F(X,Y) = max{Y, A + X, B + 2X}   for A = (ω5 + ω4 + ω3) / 2 and B = ω5 / 2.

A lifting of this tropical polynomial yields a faithful tropicalization on the minimal skeleta via the ideal Ig,f = ⟨ g(x,y), z-f(x,y) ⟩ ⊂ K[x±, y±, z±], i.e. trop(f) = F where

f(x,y) = y - (- α5 α4 α3)1/2 x + ( - α5)1/2 x2.

Since the lifting f(x,y) of the modification involves square roots of the branch points, we replace the 4 varying branch points αi by squares of 4 new variables bi, i.e.

α2 = b22 , α3 = b32 , α4 = b42 and α5 = - b52,     so     ωi = - 2 val(bi)   for i = 2, 3, 4, 5 .

Therefore the hyperelliptic equation becomes

g(x,y) = y2 - x (x - b22)(x - b32)(x - b42)(x + b52)

The combinatorial types of the tropical curves of Type II defined by the ideal Ig,f are determined by means of the Newton subdivisions of the 3 coordinate projections. In order to characterize them, we compute the coefficients of each of the 3 defining equations in terms of the parameters b5, b4, b3, b2 in Singular (in a second step we replace b3 = b4 + b34. In all cases, the (x,y)-projection is well understood: it looks like the Type III case except that the x3 point is marked.

Furthermore, if we add a tail (i.e., an element in K with valuation -A + ε and -B + ε', respectively, with 0<ε, ε'<<1) to the coefficients of f(x,y) we obtain a faithful tropicalization of the extended skeleta for types (I) and (III). We called such choice a "refined modification."

In the following sections, we describe the resulting faithful tropical curve in R3 by means the three coordinate projections and the Newton subdivisions of the resulting plane curves. Numerical examples for K = Q[t] illustrate each of the various cases that arise. To simplify the expressions, all our branch points are (up to sign) squares of polynomials in t with integer coefficients.

We provide Singular scripts as well as the output tropical curve for each projection computed with the Singular library tropical.lib. Each Singular file contains the defining equations for each plane equation.

Whenever ωi = ωj for i ≠ j, the legs li, and lj in the extended Berkovich skeleta of the curve X marked with the branch points αi and αj map to the same leg in the naive tropicalization. If k of them meet, we separate them by means of k-1 vertical modifications along max{X,ωi1} with liftings zij = x - αij for j=1, …, s-1.

In Types (IV) and (V) we show that each vertex in the xy- and yz-projections dual to a lattice triangle with a unique interior vertex is the image of the corresponding genus 1 vertex in the Berkovich skeleton. We do so by means of a j-invariant computation.

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§0.1 Type (I) Cone (Dumbbell):

Table 0.1: Dumbbell Graph Files and Projections.
Branch points Singular Scripts (x,y)-subdivision (x,z)-subdivision (y,z)-subdivision
 α2 = (t10+t12)2
 α3 = (t5+t9)2
 α4 = (3 t3+t8)2
 α5 = -(1+t5)2
Standard Script
(no refinement)


tex output
ps output


tex output
ps output


tex output
ps output
 α2 = (t10+t12)2
 α3 = (t5+t9)2
 α4 = (3 t3+t8)2
 α5 = -(1+t5)2
Refinement Script
(tε and tε')
ε = 4
ε'= 2


tex output
ps output


tex output
ps output


tex output
ps output

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§0.2 Type (II) Cone (Theta Graph):

This is the most challenging cell from the combinatorics perspective, since its the only case for which the chart σ4 contains points of the tropical curve in its relative interior. Assuming the initial terms of the branch points to be generic, there will be exactly eight combinatorial types for the modified tropical curve in R3, coming from a subdivision of the Type II Cone into three pieces, obtained by adding the sum of the three extremal rays, and subdividing the cone accordingly. More types will arise in the non-generic case. Our proof by explicit computation determines these conditions in a precise fashion. Examples for each case are provided in the tables below.

Since the lifting f(x,y) of the modification involves square roots of the branch points, we replace the 4 varying branch points αi by squares of 4 new variables bi for i = 2, 3, 4, 5. Furthermore, since the valuation of α3 - α4 is relevant in this cell, we will replace b3 by introducing a new variable b34 = b3-b4, i.e.

α2 = b22 , α3 = (b4 + b34)2 , α4 = b42 and α5 = - b52,     so     ωi = - 2 val(bi)   for i = 2, 4, 5   and   d34 = - val(b34) - val(b4).

Therefore the hyperelliptic equation becomes

g(x,y) = y2 - x (x - b22)(x - (b4+b34)2)(x - b42)(x + b52)

To avoid working with non-real expressions, we sometimes replace α2 with - b22 (it will be indicated accordingly). In the new coordinates

(v5, v4, v34, v2) := (-val(b5), -val(b4), -val(b34), -val(b2)).

the Type II Cone is defined by the following inequalities:

v5 > v4 ,   v4 > v34 ,   and   v4 > v2.

We record the 17 relevant coefficients for the (x,z)- and (y,z)-projections in terms of the parameters b5, b4, b34 and b2:

In order to find all possible combinatorial types of resulting modified 3D-tropical curves with Theta graphs as minimal skeletons, we must compute the Gröbner fans of all the relevant coefficients, to decide what is the height function for each corresponding lattice point in the Newton polytope of the defining equations.

Once the Gröbner fan of the product of all coefficients is computed, we intersect the resulting fan with the Type II Cone, check for possible subdivisions of this cone induced by the fan, and compute the leading terms for each piece in the subdivision. The Type II Cone gets subdivided into 3 and 4 maximal pieces, respectively, as shown in Figure 1. The blue cones have maximal dimension 4, the red cones have dimension 3 and the purple cones have dimension 2. Since we are interested in the relative interior of the Type II Cone, we ignore components of the subdivision lying on the 3 facets.


                     

Figure 1: From left to right: subdivision of the Type II Cone induced by the Gröbner fans of the coefficients of the (x,z)- and (y,z)-projections, with the coordinates of each extremal ray, after moding out by the all-ones vector.

The following scripts were used to compute the subvisions from Figure 1 and group together the cones yielding the same leading term expressions. In order to determine the leading terms of each of the 17 relevant coefficients, we pick a sample point on each cone. For each cone we always take the sum of its extremal rays, moding out by the all-ones vector to make all entries non-positive and taking the corresponding primitive representative. This will facilitate the construction of examples to determine the possible Newton subdivisions for each projection.

Tables 1 and 2 collect the expressions of the leading terms, factored to indicate situations where the expected leading terms have higher valuation than expected. The cone labels agree with those in Figure 1. Caveat: The computations are valid for any characteristic of the residue field, except 2 and 3.

Table 1: Expected leading terms for each coefficient of the (x,z)-projections and each of the 6 relevant cones in the leftmost subdivision of the Type II Cone in Figure 1.
Monomials Leading TermsCones
x4 (-2) * b4^2  [0, 1, 2, 3, 4, 5, 6] 

x3
(-1) * b2^2 * b5^2
(-1) * b34^2 * b5^2
(-1) * b5^2 * (b34^2 + b2^2)
b4^4
(-1) * (-b4^2 + b5*b2) * (b4^2 + b5*b2)
(-1) * (-b4^2 + b5*b34) * (b4^2 + b5*b34)
(-1) * (-b4^4 + b5^2*b34^2 + b5^2*b2^2)
[1]
[2]
[4]
[0]
[3]
[5]
[6]
x2 (2) * b2^2 * b4^2 * b5^2  [0, 1, 2, 3, 4, 5, 6] 


Table 2: Expected leading terms for each coefficient of the (y,z)-projections and each of the nine relevant cones in the rightmost subdivision of the Type II Cone in Figure 1.
Monomials Leading TermsCones
y 4 (2) * b2^2 * b5^3
(2) * b34^2 * b5^3
(2) * b5^3 * (b34^2 + b2^2)
[0, 2, 7]
[1, 3, 5]
[4, 6, 8]
y3 z (-2) * b4^2 * b5^3  [0, 1, 2, 3, 4, 5, 6, 7, 8] 
y2 z2 (4) * b5^5  [0, 1, 2, 3, 4, 5, 6, 7, 8] 
y z3 (-4) * b5^5  [0, 1, 2, 3, 4, 5, 6, 7, 8] 
z4 b5^5  [0, 1, 2, 3, 4, 5, 6, 7, 8] 
y3 b2^4 * b5^6
(-1) * b2^2 * b4^4 * b5^4
b2^2 * b5^4 * (-b4^2 + b5*b2) * (b4^2 + b5*b2)
b34^4 * b5^6
b5^6 * (b34^2 + b2^2)^2
(-1) * b34^2 * b4^4 * b5^4
(-1) * b4^4 * b5^4 * (b34^2 + b2^2)
b34^2 * b5^4 * (-b4^2 + b5*b34) * (b4^2 + b5*b34)
b5^4 * (b34^2 + b2^2) * (-b4^4 + b5^2*b34^2 + b5^2*b2^2)
[2]
[0]
[7]
[3]
[6]
[1]
[4]
[5]
[8]
y2 z (2) * b2^2 * b4^2 * b5^6
(-6) * b34^2 * b4^2 * b5^6
(-2) * b4^2 * b5^6 * (3*b34^2 - b2^2)
[0, 2, 7]
[1, 3, 5]
[4, 6, 8]
y z2 (-3) * b2^2 * b4^2 * b5^6
(9) * b34^2 * b4^2 * b5^6
(3) * b4^2 * b5^6 * (3*b34^2 - b2^2)
[0, 2, 7]
[1, 3, 5]
[4, 6, 8]
z3 b2^2 * b4^2 * b5^6
(-3) * b34^2 * b4^2 * b5^6
(-1) * b4^2 * b5^6 * (3*b34^2 - b2^2)
[0, 2, 7]
[1, 3, 5]
[4, 6, 8]
y2 (-4) * b2^2 * b34^2 * b4^4 * b5^7  [0, 1, 2, 3, 4, 5, 6, 7, 8] 
y z (2) * b34^2 * b4^6 * b5^7  [0, 1, 2, 3, 4, 5, 6, 7, 8] 
z2 (-1) * b34^2 * b4^6 * b5^7  [0, 1, 2, 3, 4, 5, 6, 7, 8] 
y b2^2 * b34^2 * b4^8 * b5^8  [0, 1, 2, 3, 4, 5, 6, 7, 8] 
z (-1) * b2^2 * b34^2 * b4^8 * b5^8  [0, 1, 2, 3, 4, 5, 6, 7, 8] 

The previous scripts allow us to predict the expected height of each lattice point in the truncated Newton polytopes of the (x,z)- and (y,z)-projection (involving only the 17 relevant monomials, hence to determine the Newton subdivisions.) However, the combinatorial types of the (y,z)-Newton subdivision will vary within a given cone. The following scripts allow us to compute the corresponding Gröbner fans.

In order to determine the domains of lineality of each combinatorial type, we further subdivide each piece C0,…, C8 by the projection of the Gröbner fan in R6 of the (y,z)-equation equation in the variables [b5, b4, b34, b2, y, z], to the first 4 parameters. Figure 2 shows the induced refined subdivision: only 2 of the 4 maximal pieces get subdivided. The following scripts compute the subdivisions for each cell, the H-representation, and sample points for each piece, to determine the Newton subdivision by explicit computations on examples, assuming no cancellations on the expected leading terms.




Figure 2: Refined subdivision of the Type II Cone induced by all possible Newton subdivisions of the (y,z)-projection. The first index of each cone reflect the label within the subdivision by leading terms of coefficients, as in Figure 1. We use the same color coding for the dimensions of each cone.

Once the refined subdivision and sample points for each piece are computed, we build concrete examples with Puiseux series to determine the Newton subdivisions for each cone assuming the leading terms have the expected valuation. Whenever the leading terms are non-monomial, cancellations of initial terms might occur. The following tables show the combinatorics of each projection and each cone by means of representative examples associated to the chosen sample points, including the relevant Singular scripts and both '.tex' and '.ps' outputs.

Even though the domains of lineality determining the combinatorics of the (y,z)-projection are those given in Figure 2, the combinatorics of the tropicalization induced by the ideal Ig,f agree for most of them. The domains of linealy are given by the subdivision on the left of Figure 1, i.e. the one computed for the (x,z)-projection.

As stated earlier, our branch points are obtained from (b5,b4, b34, b5) via the formulas:

α2 = b22 , α3 = (b4 + b34)2 , α4 = b42 and α2 = - b52

The only exception to this rule is provided by the non-generic cases in cells C4 and C6 (determined by Tables 1 and 2), where we use α2 = -b22. This change turns the non-genericity conditions from in (b34)2 = -in (b2)2 to in (b34)2 = in (b2)2 (for C6) and from in (b4)2 = -in (b2)2 to in (b4)2 = in (b2)2 (for C4). In this way we can construct examples over Q[t] where these genericity conditions are not satisfied.

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Cell C0,0:

Table 0.2.0: Theta Graph Files and Projections for C0,0 with sample point [0,-2,-7,-5].
Parameters Singular Scripts (x,y)-subdivision (x,z)-subdivision (y,z)-subdivision
   b2 = 3 t5
 b34 = 5 t7
   b4 = 11 t2
   b5 = 1+t2
Standard Script
(no refinement)


tex output
ps output


tex output
ps output


tex output
ps output

Back to Table of Contents

Cell C0,1:

Table 0.2.1: Theta Graph Files and Projections for C0,1 with sample point [0,-1,-5,-4].
Parameters Singular Scripts (x,y)-subdivision (x,z)-subdivision (y,z)-subdivision
   b2 = 3 t4
 b34 = 5 t5
   b4 = 11 t
   b5 = 1+t2
Standard Script
(no refinement)


tex output
ps output


tex output
ps output


tex output
ps output

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Cell C0,2:

Table 0.2.2: Theta Graph Files and Projections for C0,2 with sample point [0,-1,-6,-3].
Parameters Singular Scripts (x,y)-subdivision (x,z)-subdivision (y,z)-subdivision
   b2 = 3 t3
 b34 = 5 t6
   b4 = 11 t
   b5 = 1+t2
Standard Script
(no refinement)


tex output
ps output


tex output
ps output


tex output
ps output

Back to Table of Contents

Cell C0,3:

Table 0.2.3: Theta Graph Files and Projections for C0,3 with sample point [0,-1,-4,-3].
Parameters Singular Scripts (x,y)-subdivision (x,z)-subdivision (y,z)-subdivision
   b2 = 3 t3
 b34 = 5 t4
   b4 = 11 t
   b5 = 1+t2
Standard Script
(no refinement)


tex output
ps output


tex output
ps output


tex output
ps output

Back to Table of Contents

Cell C0,4:

Table 0.2.4: Theta Graph Files and Projections for C0,4 with sample point [0,-1,-5,-3].
Parameters Singular Scripts (x,y)-subdivision (x,z)-subdivision (y,z)-subdivision
   b2 = 3 t3
 b34 = 5 t5
   b4 = 11 t
   b5 = 1+t2
Standard Script
(no refinement)


tex output
ps output


tex output
ps output


tex output
ps output

Back to Table of Contents


Cell C1:

Table 0.2.5: Theta Graph Files and Projections for C1 with sample point [0,-1,-3,-4].
Parameters Singular Scripts (x,y)-subdivision (x,z)-subdivision (y,z)-subdivision
   b2 = 3 t4
 b34 = 5 t3
   b4 = 11 t
   b5 = 1+t2
Standard Script
(no refinement)


tex output
ps output


tex output
ps output


tex output
ps output

Back to Table of Contents


Cell C2,0:

Table 0.2.6: Theta Graph Files and Projections for C2,0 with sample point [0,-3,-6,-5].
Parameters Singular Scripts (x,y)-subdivision (x,z)-subdivision (y,z)-subdivision
   b2 = 3 t5
 b34 = 5 t6
   b4 = 11 t3
   b5 = 1+t2
Standard Script
(no refinement)


tex output
ps output


tex output
ps output


tex output
ps output

Back to Table of Contents

Cell C2,1:

Table 0.2.7: Theta Graph Files and Projections for C2,1 with sample point [0,-2,-5,-3].
Parameters Singular Scripts (x,y)-subdivision (x,z)-subdivision (y,z)-subdivision
   b2 = 3 t3
 b34 = 5 t5
   b4 = 11 t2
   b5 = 1+t2
Standard Script
(no refinement)


tex output
ps output


tex output
ps output


tex output
ps output

Back to Table of Contents

Cell C2,2:

Table 0.2.8: Theta Graph Files and Projections for C2,2 with sample point [0,-2,-4,-3].
Parameters Singular Scripts (x,y)-subdivision (x,z)-subdivision (y,z)-subdivision
   b2 = 3 t3
 b34 = 5 t4
   b4 = 11 t2
   b5 = 1+t2
Standard Script
(no refinement)


tex output
ps output


tex output
ps output


tex output
ps output

Back to Table of Contents


Cell C3:

Table 0.2.9: Theta Graph Files and Projections for C3 with sample point [0,-2,-3,-4].
Parameters Singular Scripts (x,y)-subdivision (x,z)-subdivision (y,z)-subdivision
   b2 = 5 t4 - 17 t5
 b34 = 7 t3 + 11 t9
   b4 = t2+ t6
   b5 = 1+t2
Standard Script
(no refinement)


tex output
ps output


tex output
ps output


tex output
ps output

Back to Table of Contents

Cell C4:

In this case, cancellations in the expected initial terms can occur since the relevant leading terms are non-monomial (see Tables 1 and 2). In order to produce cancellations, we need in (b4)2 = -in (b2)2. To avoid computations with complex numbers, we replace b2 by i b2, i.e. α2 = - b2. With this reparameterization, cancellations will occur if (b4)2 = in (b2)2. As a result, we unmark the point (4,0) from the (y,z)-Newton subdivision. So the unmarked (x,z)-Newton subdivision is unaltered. The resulting plane tropical curves in the generic and special cases agree, as expected.

Table 0.2.10: Theta Graph Files and Projections for Generic C4 with sample point
[0,-1,-3,-3].
Parameters Singular Scripts (x,y)-subdivision (x,z)-subdivision (y,z)-subdivision
   b2 = 5 t3
 b34 = 5 t3 + t9
   b4 = 3 t + t7
   b5 = 1+t2
Standard Script
(no refinement)


tex output
ps output


tex output
ps output


tex output
ps output


Table 0.2.11: Theta Graph Files and Projections for Special C4 with sample point
[0,-1,-3,-3].
Parameters Singular Scripts (x,y)-subdivision (x,z)-subdivision (y,z)-subdivision
   b2 = 17 t3 + t6
 b34 = 5 t3 + t9
   b4 = 3 t + t7
   b5 = 1+t2
Standard Script
(no refinement)


tex output
ps output


tex output
ps output


tex output
ps output

Back to Table of Contents


Cell C5:

In this case, cancellations in the expected initial terms can occur since the relevant leading terms are non-monomial (see Tables 1 and 2). In order to produce cancellations in both projections, we need in (b4)2 = in (b3 b34). If that happens, the point y3 in the (y,z)-projection drops height and we will obtain a potentially subdivided triangle T with vertices (1,0), (4,0) and (1,1) in the (y,z)-Newton subdivision. Two things can occur: On the (x,z)-projection, the point x3 drops height and we obtain a potentially subdivided triangle T' with vertices (1,0), (4,0) and (1,1) in the (x,y)-Newton subdivision. Two things can occur:

Table 0.2.12: Theta Graph Files and Projections for Generic C5 with sample point
[0,-1,-2,-3].
Parameters Singular Scripts (x,y)-subdivision (x,z)-subdivision (y,z)-subdivision
   b2 = 17 t3 + 13 t4
 b34 = 5 t2 - t5
   b4 = 3 t + t7
   b5 = 1+t2
Standard Script
(no refinement)


tex output
ps output


tex output
ps output


tex output
ps output


Table 0.2.13: Theta Graph Files and Projections for Special C5 with sample points
[0,-1,-2,-3] and [0,-3,-6,-9], respectively.
Parameters Singular Scripts (x,y)-subdivision (x,z)-subdivision (y,z)-subdivision
   b2 = 17 t3 + 13 t4
 b34 = 4 t2 - t5
   b4 = 2 t + t7
   b5 = 1+t2
Standard Script
(no refinement)


tex output
ps output


tex output
ps output


tex output
ps output
   b2 = 17 t9 + 13 t12
 b34 = 4 t6 - t7
   b4 = 2 t3
   b5 = 1
Standard Script
(no refinement)


tex output
ps output


tex output
ps output


tex output
ps output

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Cell C6:

In this case, cancellations in the expected initial terms can occur since the relevant leading terms are non-monomial (see Tables 1 and 2). In order to produce cancellations, we need in (b34)2 = -in (b2)2. To avoid computations with complex numbers, we replace b2 by i b2, i.e. α2 = - b2. With this reparameterization, cancellations will occur if (b34)2 = in (b2)2. If that happens, the point y3 drops height in the (y,z)-projection and we will obtain a potentially subdivided triangle T with vertices (2,0), (5,0) and (2,2) in the (y,z)-Newton subdivision. Two things can occur: On the (x,z)-projection, the point x3 drops height and we obtain a potentially subdivided triangle T' with vertices (2,0), (5,0) and (2,1) in the (x,y)-Newton subdivision. Two things can occur:

Table 0.2.14: Theta Graph Files and Projections for Generic C6 with sample point
[0,-2,-3,-3].
Parameters Singular Scripts (x,y)-subdivision (x,z)-subdivision (y,z)-subdivision
   b2 = 17 t3 + t6
 b34 = 5 t3 + t10
   b4 = 4 t2 + t5
   b5 = 1+t2
Standard Script
(no refinement)


tex output
ps output


tex output
ps output


tex output
ps output


Table 0.2.15: Theta Graph Files and Projections for Special C6 with sample points
[0,-2,-3,-3] and [0,-6,-9,-9], respectively.
Parameters Singular Scripts (x,y)-subdivision (x,z)-subdivision (y,z)-subdivision
   b2 = t3 + t6
 b34 = t3 + t10
   b4 = 4 t2 + t5
   b5 = 1+t2
Standard Script
(no refinement)


tex output
ps output


tex output
ps output


tex output
ps output
   b2 = t9
 b34 = t9 + t10
   b4 = t6 + t15
   b5 = 1 + t6
Standard Script
(no refinement)


tex output
ps output


tex output
ps output


tex output
ps output

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Cell C7,0:

In this case, cancellations in the expected initial terms can occur since the relevant leading terms are non-monomial (see Tables 1 and 2). In order to produce cancellations in both projections, we need in (b4)2 = in (b2 b5). If that happens, the point y3 in the (y,z)-projection drops height and we will obtain a potentially subdivided triangle T with vertices (1,0), (4,0) and (1,1). Two things can occur: On the (x,z)-projection, the point x3 becomes unmarked and we obtain a trapezoid with vertices (1,0), (5,0), (2,1) and (1,1) that cannot be further subdivided.

Table 0.2.16: Theta Graph Files and Projections for Generic C7,0 with sample point
[0,-2,-5,-4].
Parameters Singular Scripts (x,y)-subdivision (x,z)-subdivision (y,z)-subdivision
   b2 = 3 t4
 b34 = 5 t5
   b4 = 11 t2
   b5 = 1+t2
Standard Script
(no refinement)


tex output
ps output


tex output
ps output


tex output
ps output


Table 0.2.17: Theta Graph Files and Projections for Special C7,0 with sample points
[0,-2,-5,-4] and [0,-6,-15,-12], respectively.
Parameters Singular Scripts (x,y)-subdivision (x,z)-subdivision (y,z)-subdivision
   b2 = 4 t4 + t16
 b34 = t5 + t10
   b4 = 2 t2 + t5
   b5 = 1 + t2
Standard Script
(no refinement)


tex output
ps output


tex output
ps output


tex output
ps output
   b2 = 4 t12 + t13
 b34 = t15
   b4 = 2 t6
   b5 = 1
Standard Script
(no refinement)


tex output
ps output


tex output
ps output


tex output
ps output

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Cell C7,1:

In this case, cancellations in the expected initial terms can occur since the relevant leading terms are non-monomial (see Tables 1 and 2). In order to produce cancellations in both projections, we need in (b4)2 = in (b2 b5). If that happens, the point y3 in the (y,z)-projection drops height and we will obtain a potentially subdivided triangle T with vertices (1,0), (4,0) and (1,1). Two things can occur: On the (x,z)-projection, the point x3 becomes unmarked and we obtain a trapezoid with vertices (1,0), (5,0), (2,1) and (1,1) that cannot be further subdivided.

Table 0.2.18: Theta Graph Files and Projections for Generic C7,1 with sample point
[0,-1,-4,-2].
Parameters Singular Scripts (x,y)-subdivision (x,z)-subdivision (y,z)-subdivision
   b2 = 3 t2
 b34 = 5 t4
   b4 = 11 t
   b5 = 1+t2
Standard Script
(no refinement)


tex output
ps output


tex output
ps output


tex output
ps output


Table 0.2.19: Theta Graph Files and Projections for Special C7,1 with sample point
[0,-1,-4,-2].
Parameters Singular Scripts (x,y)-subdivision (x,z)-subdivision (y,z)-subdivision
   b2 = 4 t2 + t16
 b34 = t4 + t10
   b4 = 2 t + t5
   b5 = 1 + t2
Standard Script
(no refinement)


tex output
ps output


tex output
ps output


tex output
ps output
   b2 = 4 t2 + t3
 b34 = t4
   b4 = 2 t
   b5 = 1
Standard Script
(no refinement)


tex output
ps output


tex output
ps output


tex output
ps output

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Cell C7,2:

In this case, cancellations in the expected initial terms can occur since the relevant leading terms are non-monomial (see Tables 1 and 2). In order to produce cancellations in both projections, we need in (b4)2 = in (b2 b5). If that happens, the point y3 in the (y,z)-projection drops in height and we obtain a potentially subdivided triangle T with vertices (1,0), (4,0) and (1,1). Two things can occur: On the (x,z)-projection, the point x3 becomes unmarked and we obtain a trapezoid with vertices (1,0), (5,0), (2,1) and (1,1) that cannot be further subdivided.

Table 0.2.20: Theta Graph Files and Projections for Generic C7,2 with sample point
[0,-1,-3,-2].
Parameters Singular Scripts (x,y)-subdivision (x,z)-subdivision (y,z)-subdivision
   b2 = 3 t2
 b34 = 5 t3
   b4 = 11 t
   b5 = 1+t2
Standard Script
(no refinement)


tex output
ps output


tex output
ps output


tex output
ps output


Table 0.2.21: Theta Graph Files and Projections for Special C7,2 with sample points
[0,-1,-3,-2] and [0,-3,-9,-6], respectively.
Parameters Singular Scripts (x,y)-subdivision (x,z)-subdivision (y,z)-subdivision
   b2 = 4 t2
 b34 = t3
   b4 = 2 t
   b5 = 1 + t2
Standard Script
(no refinement)


tex output
ps output


tex output
ps output


tex output
ps output
   b2 = 4 t6 + t7
 b34 = t9
   b4 = 2 t3
   b5 = 1 + t2
Standard Script
(no refinement)


tex output
ps output


tex output
ps output


tex output
ps output

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Cell C8:

In this case, cancellations in the expected initial terms can occur since the relevant leading terms are non-monomial (see Tables 1 and 2). Depending on our choice of factors to cancel (determined by Tables 1 and 2), the effect of cancellations will be to unmark the points y2 z or y3 and/or y4 on the (y,z)-Newton subdivision and unmark x3 on the (x,z)-Newton subdivision. This has no impact on the tropical world, so we omit including an example.

Table 0.2.22: Theta Graph Files and Projections for Generic C8 with sample point
[0,-1,-2,-2].
Parameters Singular Scripts (x,y)-subdivision (x,z)-subdivision (y,z)-subdivision
   b2 = 17 t2 + t6
 b34 = 5 t2 + t10
   b4 = 4 t + t5
   b5 = 1+t2
Standard Script
(no refinement)


tex output
ps output


tex output
ps output


tex output
ps output

Back to Table of Contents


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§0.3 Type (III) Cone (Figure Eight):

In addition to the modification along F and its lifting f, we compute a vertical modification along MAX{X,ω4} with lifting u = x - α4. This produces a faithful tropicalization at the level of the extended skeleta. We recover the resulting tropical curve in R4 by means of the planar (x,y), (x,z), (u,y) and (u,z)-projections.

Table 0.3: Figure Eight Graph Files and Projections.
Branch points Singular Scripts (x,y)-subdivision (x,z)-subdivision (u,y)-subdivision (u,z)-subdivision(y,z)-subdivision
 α2 = (t9+t12)2
 α3 = (t5+t9)2
 α4 = (3 t5+t8)2
 α5 = -(1+t5)2
Standard Script
(no refinement)


tex output
ps output


tex output
ps output


tex output
ps output


tex output
ps output


tex output
ps output
 α2 = (t9+t12)2
 α3 = (t5+t9)2
 α4 = (3 t5+t8)2
 α5 = -(1+t5)2
Refinement Script
(tε and tε')
ε = 3
ε'= 4


tex output
ps output


tex output
ps output


tex output
ps output


tex output
ps output


tex output
ps output

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§0.4 Type (IV) Cone:

The following files are used to compute the negative valuation of the j-invariant of the curve dual to the genus 1 vertex on the xy-projection.

In addition to the modification along F and its lifting f, we compute a vertical modification along MAX{X,ω4} with lifting u = x - α4. This produces a faithful tropicalization at the level of the extended skeleta. We recover the resulting tropical curve in R4 by means of the planar (x,y), (x,z) and (u,y)-projections. As the table shows, the (u,z)-projection is indistinguishable from (u,y) in the tropical world.

Table 0.4: Type IV Graph Files and Projections.
Branch points Singular Scripts (x,y)-subdivision (x,z)-subdivision (u,y)-subdivision (u,z)-subdivision(y,z)-subdivision
 α2 = (t14+t16)2
 α3 = (t5+t9)2
 α4 = (3 +t8)2
 α5 = - (1+t5)2
Standard Script
(no refinement)


tex output
ps output


tex output
ps output


tex output
ps output


tex output
ps output


tex output
ps output
 α2 = (t14+t16)2
 α3 = (t5+t9)2
 α4 = (3 +t8)2
 α5 = - (1+t5)2
Refinement Script
(tε only)
ε = 8


tex output
ps output


tex output
ps output


tex output
ps output


tex output
ps output


tex output
ps output

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§0.5 Type (V) Cone:

The following files are used to compute the negative valuation of the j-invariant of the curves dual to each of the genus 1 vertices on the xy-projection.


If we compute two successive vertical modifications along MAX{X,ω4} and MAX{X,ω2} (in that order), with liftings z2 = x - α2, and z4 = x - α4, we produce a faithful tropicalization at the level of the extended skeleta. We can compute the resulting tropical curve in R4 by means of the planar (x,y), (z2,y) and (z4,y)-projections.

Table 0.5: Type V Graph Files and Projections associated to two vertical modifications.
Branch points Singular Scripts (x,y)-subdivision (z2,y)-subdivision (z4,y)-subdivision
 α2 = (5 t14+t16)2
 α3 = (t14+t19)2
 α4 = (3 +t8)2
 α5 = - (1+t5)2
Standard Script
(no refinement)


tex output
ps output


tex output
ps output


tex output
ps output

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§0.6 Type (VI) Cone:

There are several possibilities for the (x,z)-projection, depending on the genericity of the initial forms of α3, α4, α5, β3 and β4 with respect to the expected coefficients for x3 and x4.

Generic for x3 (any condition for x4):

In addition to the modification along F and its lifting f, we compute two successive vertical modifications along MAX{X,ω4}, with liftings z3 = x - α3, and z4 = x - α4. This produces a faithful tropicalization at the level of the extended skeleta. We recover the resulting tropical curve in R5 by means of the planar (x,y), (z3,y) and (z4,y)-projections. Notice that the the last two agree with the (z4,y) and (z4,z)-projections so the latter ones need not be plotted.

Table 0.6.0: Type VI Graph Files and Projections in the generic case for x3.
Branch points Singular Scripts (x,y)-subdivision (x,z)-subdivision (z3,y)- and (z4,y)-subdivisions (z3,z)- and (z4,z)-subdivision (y,z)-subdivision
α2=(t14+t16)2
α3= (5+t9)2
α4= (3 +t8)2
α5= - (1+t5)2
Standard Script
(no ref.)


tex output
ps output


tex output
ps output


tex output (z3y)
ps output (z3y)
tex output (z4y)
ps output (z4y)


tex output (z3z)
ps output (z3y)
tex output (z4z)
ps output (z4y)


tex output
ps output
α2=(t14+t16)2
α3= (5+t9)2
α4= (3 +t8)2
α5= - (1+t5)2
Refinement Script
(tε only)
ε = 8


tex output
ps output


tex output
ps output


tex output (z3y)
ps output (z3y)
tex output (z4y)
ps output (z4y)


tex output (z3z)
ps output (z3y)
tex output (z4z)
ps output (z4y)


tex output
ps output

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Generic for x4, Non-Generic for x3 but x3 marked:

In addition to the modification along F and its lifting f, we compute two successive vertical modifications along MAX{X,ω4}, with liftings z3 = x - α3, and z4 = x - α4. This produces a faithful tropicalization at the level of the extended skeleta. We recover the resulting tropical curve in R5 by means of the planar (x,y), (z3,y) and (z3,z)-projections. Notice that the the last two agree with the (z4,y) and (z4,z)-projections so the latter ones need not be plotted.

Table 0.6.1: Type VI Graph Files and Projections in the non-generic case for x3 but x3 marked.
Branch points Singular Scripts (x,y)-subdivision (x,z)-subdivision (z3,y)- and (z4,y)-subdivisions (z3,z)- and (z4,z)-subdivision (y,z)-subdivision
α2=(t14+t16)2
α3= (2+t9)2
α4= (1 +t8)2
α5= - (2+t5)2
Standard Script
(no ref.)


tex output
ps output


tex output
ps output


tex output (z3y)
ps output (z3y)
tex output (z4y)
ps output (z4y)


tex output (z3z)
ps output (z3y)


tex output (z4z)
ps output (z4y)


tex output
ps output
α2=(t14+t16)2
α3= (2+t9)2
α4= (1 +t8)2
α5= - (2+t5)2
Refinement Script
(tε only)
ε = 13


tex output
ps output


tex output
ps output


tex output (z3y)
ps output (z3y)
tex output (z4y)
ps output (z4y)


tex output (z3z)
ps output (z3y)


tex output (z4z)
ps output (z4y)


tex output
ps output

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Generic for x4, Non-Generic for x3 and x3 unmarked:

In addition to the modification along F and its lifting f, we compute two successive vertical modifications along MAX{X,ω4}, with liftings z3 = x - α3, and z4 = x - α4. This produces a faithful tropicalization at the level of the extended skeleta. We recover the resulting tropical curve in R5 by means of the planar (x,y), (z3,y) and (z3,z)-projections. Notice that the the last two agree with the (z4,y) and (z4,z)-projections so the latter ones need not be plotted.

Table 0.6.2: Type VI Graph Files and Projections in the non-generic case for x3 and x3 unmarked.
Branch points Singular Scripts (x,y)-subdivision (x,z)-subdivision (z3,y)- and (z4,y)-subdivisions (z3,z)- and (z4,z)-subdivision (y,z)-subdivision
α2=(t14+t16)2
α3= (2)2
α4= (1)2
α5= - (2)2
Standard Script
(no ref.)


tex output
ps output


tex output
ps output


tex output (z3y)
ps output (z3y)
tex output (z4y)
ps output (z4y)


tex output (z3z)
ps output (z3y)


tex output (z4z)
ps output (z4y)


tex output
ps output
α2=(t14+t16)2
α3= (2)2
α4= (1)2
α5= - (2)2
Refinement Script
(tε only)
ε = 13


tex output
ps output


tex output
ps output


tex output (z3y)
ps output (z3y)
tex output (z4y)
ps output (z4y)


tex output (z3z)
ps output (z3y)


tex output (z4z)
ps output (z4y)


tex output
ps output

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Non-Generic for x4 and also for x3:

Recall that

α222, α3= β32, α4= β42 and α5= - β52.

Violating The genericity conditions for both x3 and x4 requires the use of non-real numbers, since we must have

(in (β3))2 = in (α3) = -in(α4) = - ( in (β4))2     and    in (β4) in (β4) = in (β5) (in (β3) - in (β4)).

We take in (β5) = 1, in (β3) = (ι - 1), and in (β4) = (1 + ι). We produce all the numerical examples by evaluating all the elimination ideals at the desired values of β2, β3, β4, and β5 with Singular, by working with the field extension associated to the polynomial x2 + 1. To plot the tropical plane curves with Singular, we replace each polynomial by one with coefficients in Q[t] with the same t-valuation (the result is obtained by specializing the Symbolic computations to a random value of ι1/2.) As was mentioned, we only treat the refined modifications.

Table 0.6.3: Type VI Graph Files and Projections in the non-generic case for x3 and x4 for refined modifications.
Branch points Singular Scripts (x,y)-subdivision (x,z)-subdivision (z3,y)- and (z4,y)-subdivisions (z3,z)- and (z4,z)-subdivision (y,z)-subdivision
α2= (5t14+t16)2
α3= ((ι)1/2-1+t18)2
α4= ((1+(ι)1/2)2
α5= - 1
Refinement Script with Symbolic ι1/2


Refinement Script with Numerical ι1/2

(tε and tε')
ε = 1
ε' = 2


tex output
ps output


tex output
ps output


tex output (z3y)
ps output (z3y)
tex output (z4y)
ps output (z4y)


tex output (z3z)
ps output (z3y)


tex output (z4z)
ps output (z4y)


tex output
ps output

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§0.7 Type (VII) Cone (Good Reduction):

If we compute three successive vertical modifications along MAX{X,ω2} with 3 different liftings

z2 = x - α2,  z3 = x - α3,   and   z4 = x - α4,

we produce a faithful tropicalization at the level of the extended skeleta. We can compute the resulting tropical curve in R5 by means of the planar (x,y), (z2,y), (z3,y), and (z4,y)-projections. Notice that these 3 projections are indistinguishable. Their sole purpose is to isolate the leg marked by the corresponding branch point.

Table 0.7.1: Type VII Graph Files and Projections associated to three vertical modifications.
Branch points Singular Scripts (x,y)-subdivision (z2,y)-subdivision (z3,y)-subdivision (z4,y)-subdivision
 α2 = ( t10+t12)2
 α3 = (4 t10+t19)2
 α4 = (3 t10+t18)2
 α5 = - (12 t10)2
Standard Script
(no refinement)


tex output
ps output


tex output
ps output


tex output
ps output


tex output
ps output

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§1. Igusa invariants and their tropicalizations

The following scripts allow to compute initial forms of the corresponding polynomials A, B, C, A', B', C', Q3 = AB - 3C, Q4 = AB - 4C, and Q4' = A'B' - 4C'. We refer to the last two sections of the paper for the definition of these polynomials.

We separate the Macaulay2 scripts into two types, depending on the maximal cone in M2trop. We assume the characteristic of the residue field to be different than 2 and 3. Our own implementation of the leading terms is given at the end of this section, and includes the computation for all lower dimensional cones.

§1.1 Dumbbell Cone

In all our scripts we input a weight vector ω=(ω654321) in R6, where -val(αi) = ωi for i=1,…, 6. The vector ω lies in the Dumbbell cone since it satisfies:

ω1 < ω2 < ω3 < ω4 < ω5 < ω6

In all the scripts below, we fix ω = (36,25,20,16,9,1). The scrpts where generated from the Sage script provided above. The outputs of the computations are commented after each command in the scripts.

§1.2 Theta Cone

In all our scripts we input a weight vector ω=(ω65,d34321) in R6, where -val(αi) = ωi for i=1,2,3, 5, 6, and -val(x34) = d34. Recall that x34 = α4 - α3.

The vector ω lies in the Theta cone since it satisfies:

ω1 < ω2 < ω35 < ω6       and       d34 < ω3.

The Sage script at the begining of this section computes the Gröbner fan of Q4' = A'B' - 4 C' by replacing Q4' by the sum of its extremal monomials. Its f-vector equals [1, 32, 174, 396, 420, 168]. The calculation yields a subdivision of the Theta cone into 3 pieces, which we call Θ0, Θ1 and Θ2. The Sage calculation gives the inequalities and extremal rays for each piece, recorded in Table 2 below. We compute a sample point on the relative interior of each piece by adding up the extremal rays and translating by the all ones vector to get non-negative weights for each variable. The variables are ordered as the tuple x = (a6, a5, x34, a3, a2, a1).

Table 2: Combinatorial data of the Subdivision of the Theta cone induced by the Gröbner fan of Q4'.
CellDefining inequalities Extremal rays Lines Sample point
Θ0 (0, 0, -1, 1, 0, 0) x + 0 >= 0,
(0, 0, 0, 0, 1, -1) x + 0 >= 0,
(1, -1, 0, 0, 0, 0) x + 0 >= 0,
(0, 0, 1, 0, -1, 0) x + 0 >= 0,
(0, 1, 1, -2, 0, 0) x + 0 >= 0
(0, 0, 0, 0, 0, -1)
(0, 0, 0, 0, -1, -1)
(0, 0, -2, -1, -2, -2)
(0, 0, -1, -1, -1, -1)
(0, -1, -1, -1, -1, -1)
(1, 1, 1, 1, 1, 1) ω(0)=(6, 5, 2, 3, 1, 0);
Θ1 (0, -1, -1, 2, 0, 0) x + 0 >= 0,
(0, -1, 0, 2, -1, 0) x + 0 >= 0,
(1, -1, 0, 0, 0, 0) x + 0 >= 0,
(0, 1, 0, -1, 0, 0) x + 0 >= 0,
(0, 0, 0, 0, 1, -1) x + 0 >= 0
(0, -1, -1, -1, -1, -1)
(0, 0, 0, 0, 0, -1)
(0, 0, 0, 0, -1, -1)
(0, 0, -1, 0, 0, 0)
(0, 0, -2, -1, -2, -2)
(1, 1, 1, 1, 1, 1) ω(1)=(6, 5, 2, 4, 2, 1);
Θ2 (0, 0, -1, 0, 1, 0) x + 0 >= 0,
(1, -1, 0, 0, 0, 0) x + 0 >= 0,
(0, 0, 0, 0, 1, -1) x + 0 >= 0,
(0, 0, 0, 1, -1, 0) x + 0 >= 0,
(0, 1, 0, -2, 1, 0) x + 0 >= 0
(0, 0, 0, 0, 0, -1)
(0, 0, -1, 0, 0, 0)
(0, 0, -2, -1, -2, -2)
(0, -1, -1, -1, -1, -1)
(0, 0, -1, -1, -1, -1)
(1, 1, 1, 1, 1, 1) ω(2)=(6, 5, 1, 3, 2, 1)


Macaulay2 scripts to confirm the calculation of the initial forms on the three pieces of the Theta cone. The outputs of the computations are commented after each command in the scripts. The calculations are also done at the end of the Sage script by computing the leading terms as the monomials maximizing the inner product between the weight vector and the exponent vectors.

§1.3 Initial terms and genericity conditions on M2trop

The following scripts compute the initial forms of all polynomials A, B, C, A', B', C', Q4 and Q4' and determine the exact genericity conditions to ensure the tropical Igusa invariants equal the negative valuation of the corresponding algebraic Igusa invariants. We always assume our residue field has characteristic different than 2 and 3. It relies on the subdivision computations done at the beginning of this section.

§1.4 Characteristic Issues

The calculations of the initial forms of A, B, C, D, Q3 and Q4 above are valid whenever the characteristic of the residue field does not divide the coefficient of the initial term. The output shows that we only have issues with the polynomial A since the coefficient is 6. To extend our findings to characteristic 3, we must refine the computation of the initial form of A, by writing A as an integer combination of polynomials with coefficients ±1:

A = 4 A4 + 6 A6 + 12 A12 + 120 A120.

The computation verifies that no matter what point we choose, we need only compare the initial terms of 4 A4 and Q = (6 A6 + 12 A12 + 120 A120)/3. This gives 3 cases to analyze, depending of the order between ω3 and (ω4 - 1). In case of equality there will be cancellations whenever val(-3 α4 + 2α3) > -ω3. If so, there will be no general formula for computing val(A), and we would have to make a substitution of the form

α3 = 3/2 α4 + y34     where val(y34)> -ω3

in case we want to compute the true valuation. Our genericity conditions will disallow this situation.

The following files are used to compute the decomposition of A, together with the Gröbner fans of these 4 polynomials and their intersection with the Dumbbell Cone, and the genericity conditions required to compute the tropical Igusa invariants.

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