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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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 α_{1},…,α_{6} in P^{1}. Furthermore, we may assume α_{1}= (0,1) and α_{6}=(0,1) (with -val(α_{1})="-Infinity", -val(α_{1})="+Infinity"), and the remaining α_{2},…,α_{5} lie in K^{*}. Their negative valuations
ω_{i} = -val(α_{i}) for i = 1,…,6, d_{34} = -val(α_{3} - α_{4})
and the initial forms in(α_{i}) satisfy the required conditions indicated in Table 1.Cell | Extended skeleta | Defining conditions | Lengths |
(I) | ω_{2} < ω_{3} < ω_{4} <ω_{5} |
L_{0} = (ω_{4} - ω_{3})/2 L_{1} = 2(ω_{5} - ω_{4}) L_{2} = (ω_{3} - ω_{2})/2 | |
(II) | ω_{2} < ω_{3} = ω_{4} <ω_{5} d_{34}<ω_{3} in(α_{3})=in(α_{4}) |
L_{0} = (ω_{4} - d_{34})/2 L_{1} = 2(ω_{5} - ω_{3}) L_{2} = (ω_{3} - ω_{2})/2 | |
(III) | ω_{2} < ω_{3} = ω_{4} <ω_{5} d_{34} = ω_{3} in(α_{3}) ≠ in(α_{4}) |
L_{0} = 0 L_{1} = 2(ω_{5} - ω_{3}) L_{2} = (ω_{3} - ω_{2})/2 | |
(IV) | ω_{2} < ω_{3} < ω_{4} ω_{4} = ω_{5} in(α_{4}) ≠ in(α_{5}) |
L_{0} = (ω_{4} - ω_{3})/2 L_{1} = 0 L_{2} = (ω_{3} - ω_{2})/2 | |
(V) | ω_{2} < ω_{4} ω_{2} = ω_{3 } ω_{4} = ω_{5} in(α_{2}) ≠ in(α_{3}) in(α_{4}) ≠ in(α_{5}) |
L_{0} = (ω_{4} - ω_{3})/2 L_{1} = 0 L_{2} = 0 | |
(VI) | ω_{2} < ω_{3} ω_{3} = ω_{4} = ω_{5} in(α_{3}) ≠ in(α_{4}) in(α_{3}) ≠ in(α_{5}) in(α_{4}) ≠ in(α_{5}) |
L_{0} = 0 L_{1} = 0 L_{2} = (ω_{3} - ω_{2})/2 | |
(VII) | ω_{2} = ω_{3} = ω_{4} = ω_{5} in(α_{i}) ≠ in(α_{j}) (for i≠ j) |
L_{0} = 0 L_{1} = 0 L_{2} = 0 |
The naive tropicalization induced by the hyperelliptic equation
g(x,y) = y^{2} - x (x- α_{2})(x- α_{3})(x- α_{4})(x- α_{5})
is never faithful. We use the tropical modification of R^{2} along the tropical polynomialF(X,Y) = max{Y, A + X, B + 2X} for A = (ω_{5} + ω_{4} + ω_{3}) / 2 and B = ω_{5} / 2.
A lifting of this tropical polynomial yields a faithful tropicalization on the minimal skeleta via the ideal I_{g,f} = ⟨ g(x,y), z-f(x,y) ⟩ ⊂ K[x^{±}, y^{±}, z^{±}], i.e. trop(f) = F wheref(x,y) = y - (- α_{5} α_{4} α_{3})^{1/2} x + ( - α_{5})^{1/2} x^{2}.
Since the lifting f(x,y) of the modification involves square roots of the branch points, we replace the 4 varying branch points α_{i} by squares of 4 new variables b_{i}, i.e.α_{2} = b_{2}^{2} , α_{3} = b_{3}^{2} , α_{4} = b_{4}^{2} and α_{5} = - b_{5}^{2}, so ω_{i} = - 2 val(b_{i}) for i = 2, 3, 4, 5 .
Therefore the hyperelliptic equation becomesg(x,y) = y^{2} - x (x - b_{2}^{2})(x - b_{3}^{2})(x - b_{4}^{2})(x + b_{5}^{2})
The combinatorial types of the tropical curves of Type II defined by the ideal I_{g,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 b_{5}, b_{4}, b_{3}, b_{2} in Singular (in a second step we replace b_{3} = b_{4} + b_{34}. In all cases, the (x,y)-projection is well understood: it looks like the Type III case except that the x^{3} 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 R^{3} 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 ω_{i} = ω_{j} for i ≠ j, the legs l_{i}, and l_{j} in the extended Berkovich skeleta of the curve X marked with the branch points α_{i} and α_{j} 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,ω_{i1}} with liftings z_{ij} = x - α_{ij} 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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Branch points | Singular Scripts | (x,y)-subdivision | (x,z)-subdivision | (y,z)-subdivision |
α_{2} = (t^{10}+t^{12})^{2} α_{3} = (t^{5}+t^{9})^{2} α_{4} = (3 t^{3}+t^{8})^{2} α_{5} = -(1+t^{5})^{2} | Standard Script (no refinement) | tex output ps output | tex output ps output | tex output ps output |
α_{2} = (t^{10}+t^{12})^{2} α_{3} = (t^{5}+t^{9})^{2} α_{4} = (3 t^{3}+t^{8})^{2} α_{5} = -(1+t^{5})^{2} | Refinement Script (t^{ε} and t^{ε'}) ε = 4 ε'= 2 | tex output ps output | tex output ps output | tex output ps output |
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This is the most challenging cell from the combinatorics perspective, since its the only case for which the chart σ_{4} 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 R^{3}, 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 α_{i} by squares of 4 new variables b_{i} for i = 2, 3, 4, 5. Furthermore, since the valuation of α_{3} - α_{4} is relevant in this cell, we will replace b_{3} by introducing a new variable b_{34} = b_{3}-b_{4}, i.e.
α_{2} = b_{2}^{2} , α_{3} = (b_{4} + b_{34})^{2} , α_{4} = b_{4}^{2} and α_{5} = - b_{5}^{2}, so ω_{i} = - 2 val(b_{i}) for i = 2, 4, 5 and d_{34} = - val(b_{34}) - val(b_{4}).
Therefore the hyperelliptic equation becomesg(x,y) = y^{2} - x (x - b_{2}^{2})(x - (b_{4}+b_{34})^{2})(x - b_{4}^{2})(x + b_{5}^{2})
To avoid working with non-real expressions, we sometimes replace α_{2} with - b_{2}^{2} (it will be indicated accordingly). In the new coordinates(v_{5}, v_{4}, v_{34}, v_{2}) := (-val(b_{5}), -val(b_{4}), -val(b_{34}), -val(b_{2})).
the Type II Cone is defined by the following inequalities:v_{5} > v_{4} , v_{4} > v_{34} , and v_{4} > v_{2}.
We record the 17 relevant coefficients for the (x,z)- and (y,z)-projections in terms of the parameters b_{5}, b_{4}, b_{34} and b_{2}: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.
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.
Monomials | Leading Terms | Cones |
x^{4} | (-2) * b4^2 | [0, 1, 2, 3, 4, 5, 6] |
x^{3} |
(-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] |
x^{2} | (2) * b2^2 * b4^2 * b5^2 | [0, 1, 2, 3, 4, 5, 6] |
Monomials | Leading Terms | Cones |
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] |
y^{3} z | (-2) * b4^2 * b5^3 | [0, 1, 2, 3, 4, 5, 6, 7, 8] |
y^{2} z^{2} | (4) * b5^5 | [0, 1, 2, 3, 4, 5, 6, 7, 8] |
y z^{3} | (-4) * b5^5 | [0, 1, 2, 3, 4, 5, 6, 7, 8] |
z^{4} | b5^5 | [0, 1, 2, 3, 4, 5, 6, 7, 8] |
y^{3} |
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] |
y^{2} 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 z^{2} | (-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] |
z^{3} | 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] |
y^{2} | (-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] |
z^{2} | (-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.
C_{0}: [1, 10, 34, 52, 38, 12],
C_{1}: [1, 14, 46, 64, 42, 12],
C_{2}: [1, 13, 45, 65, 43, 12],
C_{3}: [1, 19, 59, 74, 43, 11],
C_{4}: [1, 15, 55, 84, 58, 16],
C_{5}: [1, 19, 61, 79, 47, 12],
C_{6}: [1, 19, 66, 91, 56, 14],
C_{7}: [1, 14, 47, 67, 45, 13],
C_{8}: [1, 19, 68, 98, 63, 16].
In order to determine the domains of lineality of each combinatorial type, we further subdivide each piece C_{0},…, C_{8} by the projection of the Gröbner fan in R^{6} of the (y,z)-equation equation in the variables [b_{5}, b_{4}, b_{34}, b_{2}, 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.
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 I_{g,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 (b_{5},b_{4}, b_{34}, b_{5}) via the formulas:α_{2} = b_{2}^{2} , α_{3} = (b_{4} + b_{34})^{2} , α_{4} = b_{4}^{2} and α_{2} = - b_{5}^{2}
The only exception to this rule is provided by the non-generic cases in cells C_{4} and C_{6} (determined by Tables 1 and 2), where we use α_{2} = -b_{2}^{2}. This change turns the non-genericity conditions from in (b_{34})^{2} = -in (b_{2})^{2} to in (b_{34})^{2} = in (b_{2})^{2} (for C_{6}) and from in (b_{4})^{2} = -in (b_{2})^{2} to in (b_{4})^{2} = in (b_{2})^{2} (for C_{4}). In this way we can construct examples over Q[t] where these genericity conditions are not satisfied.Jump to:
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Parameters | Singular Scripts | (x,y)-subdivision | (x,z)-subdivision | (y,z)-subdivision |
b_{2} = 3 t^{5} b_{34} = 5 t^{7} b_{4} = 11 t^{2} b_{5} = 1+t^{2} | Standard Script (no refinement) | tex output ps output | tex output ps output | tex output ps output |
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Parameters | Singular Scripts | (x,y)-subdivision | (x,z)-subdivision | (y,z)-subdivision |
b_{2} = 3 t^{4} b_{34} = 5 t^{5} b_{4} = 11 t b_{5} = 1+t^{2} | Standard Script (no refinement) | tex output ps output | tex output ps output | tex output ps output |
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Parameters | Singular Scripts | (x,y)-subdivision | (x,z)-subdivision | (y,z)-subdivision |
b_{2} = 3 t^{3} b_{34} = 5 t^{6} b_{4} = 11 t b_{5} = 1+t^{2} | Standard Script (no refinement) | tex output ps output | tex output ps output | tex output ps output |
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Parameters | Singular Scripts | (x,y)-subdivision | (x,z)-subdivision | (y,z)-subdivision |
b_{2} = 3 t^{3} b_{34} = 5 t^{4} b_{4} = 11 t b_{5} = 1+t^{2} | Standard Script (no refinement) | tex output ps output | tex output ps output | tex output ps output |
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Parameters | Singular Scripts | (x,y)-subdivision | (x,z)-subdivision | (y,z)-subdivision |
b_{2} = 3 t^{3} b_{34} = 5 t^{5} b_{4} = 11 t b_{5} = 1+t^{2} | Standard Script (no refinement) | tex output ps output | tex output ps output | tex output ps output |
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Parameters | Singular Scripts | (x,y)-subdivision | (x,z)-subdivision | (y,z)-subdivision |
b_{2} = 3 t^{4} b_{34} = 5 t^{3} b_{4} = 11 t b_{5} = 1+t^{2} | Standard Script (no refinement) | tex output ps output | tex output ps output | tex output ps output |
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Parameters | Singular Scripts | (x,y)-subdivision | (x,z)-subdivision | (y,z)-subdivision |
b_{2} = 3 t^{5} b_{34} = 5 t^{6} b_{4} = 11 t^{3} b_{5} = 1+t^{2} | Standard Script (no refinement) | tex output ps output | tex output ps output | tex output ps output |
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Parameters | Singular Scripts | (x,y)-subdivision | (x,z)-subdivision | (y,z)-subdivision |
b_{2} = 3 t^{3} b_{34} = 5 t^{5} b_{4} = 11 t^{2} b_{5} = 1+t^{2} | Standard Script (no refinement) | tex output ps output | tex output ps output | tex output ps output |
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Parameters | Singular Scripts | (x,y)-subdivision | (x,z)-subdivision | (y,z)-subdivision |
b_{2} = 3 t^{3} b_{34} = 5 t^{4} b_{4} = 11 t^{2} b_{5} = 1+t^{2} | Standard Script (no refinement) | tex output ps output | tex output ps output | tex output ps output |
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Parameters | Singular Scripts | (x,y)-subdivision | (x,z)-subdivision | (y,z)-subdivision |
b_{2} = 5 t^{4} - 17 t^{5} b_{34} = 7 t^{3} + 11 t^{9} b_{4} = t^{2}+ t^{6} b_{5} = 1+t^{2} | Standard Script (no refinement) | tex output ps output | tex output ps output | tex output ps output |
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Parameters | Singular Scripts | (x,y)-subdivision | (x,z)-subdivision | (y,z)-subdivision |
b_{2} = 5 t^{3} b_{34} = 5 t^{3} + t^{9} b_{4} = 3 t + t^{7} b_{5} = 1+t^{2} | Standard Script (no refinement) | tex output ps output | tex output ps output | tex output ps output |
Parameters | Singular Scripts | (x,y)-subdivision | (x,z)-subdivision | (y,z)-subdivision |
b_{2} = 17 t^{3} + t^{6} b_{34} = 5 t^{3} + t^{9} b_{4} = 3 t + t^{7} b_{5} = 1+t^{2} | Standard Script (no refinement) | tex output ps output | tex output ps output | tex output ps output |
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Parameters | Singular Scripts | (x,y)-subdivision | (x,z)-subdivision | (y,z)-subdivision |
b_{2} = 17 t^{3} + 13 t^{4} b_{34} = 5 t^{2} - t^{5} b_{4} = 3 t + t^{7} b_{5} = 1+t^{2} | Standard Script (no refinement) | tex output ps output | tex output ps output | tex output ps output |
Parameters | Singular Scripts | (x,y)-subdivision | (x,z)-subdivision | (y,z)-subdivision |
b_{2} = 17 t^{3} + 13 t^{4} b_{34} = 4 t^{2} - t^{5} b_{4} = 2 t + t^{7} b_{5} = 1+t^{2} | Standard Script (no refinement) | tex output ps output | tex output ps output | tex output ps output |
b_{2} = 17 t^{9} + 13 t^{12} b_{34} = 4 t^{6} - t^{7} b_{4} = 2 t^{3} b_{5} = 1 | Standard Script (no refinement) | tex output ps output | tex output ps output | tex output ps output |
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Parameters | Singular Scripts | (x,y)-subdivision | (x,z)-subdivision | (y,z)-subdivision |
b_{2} = 17 t^{3} + t^{6} b_{34} = 5 t^{3} + t^{10} b_{4} = 4 t^{2} + t^{5} b_{5} = 1+t^{2} | Standard Script (no refinement) | tex output ps output | tex output ps output | tex output ps output |
Parameters | Singular Scripts | (x,y)-subdivision | (x,z)-subdivision | (y,z)-subdivision |
b_{2} = t^{3} + t^{6} b_{34} = t^{3} + t^{10} b_{4} = 4 t^{2} + t^{5} b_{5} = 1+t^{2} | Standard Script (no refinement) | tex output ps output | tex output ps output | tex output ps output |
b_{2} = t^{9} b_{34} = t^{9} + t^{10} b_{4} = t^{6} + t^{15} b_{5} = 1 + t^{6} | Standard Script (no refinement) | tex output ps output | tex output ps output | tex output ps output |
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Parameters | Singular Scripts | (x,y)-subdivision | (x,z)-subdivision | (y,z)-subdivision |
b_{2} = 3 t^{4} b_{34} = 5 t^{5} b_{4} = 11 t^{2} b_{5} = 1+t^{2} | Standard Script (no refinement) | tex output ps output | tex output ps output | tex output ps output |
Parameters | Singular Scripts | (x,y)-subdivision | (x,z)-subdivision | (y,z)-subdivision |
b_{2} = 4 t^{4} + t^{16} b_{34} = t^{5} + t^{10} b_{4} = 2 t^{2} + t^{5} b_{5} = 1 + t^{2} | Standard Script (no refinement) | tex output ps output | tex output ps output | tex output ps output |
b_{2} = 4 t^{12} + t^{13} b_{34} = t^{15} b_{4} = 2 t^{6} b_{5} = 1 | Standard Script (no refinement) | tex output ps output | tex output ps output | tex output ps output |
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Parameters | Singular Scripts | (x,y)-subdivision | (x,z)-subdivision | (y,z)-subdivision |
b_{2} = 3 t^{2} b_{34} = 5 t^{4} b_{4} = 11 t b_{5} = 1+t^{2} | Standard Script (no refinement) | tex output ps output | tex output ps output | tex output ps output |
Parameters | Singular Scripts | (x,y)-subdivision | (x,z)-subdivision | (y,z)-subdivision |
b_{2} = 4 t^{2} + t^{16} b_{34} = t^{4} + t^{10} b_{4} = 2 t + t^{5} b_{5} = 1 + t^{2} | Standard Script (no refinement) | tex output ps output | tex output ps output | tex output ps output |
b_{2} = 4 t^{2} + t^{3} b_{34} = t^{4} b_{4} = 2 t b_{5} = 1 | Standard Script (no refinement) | tex output ps output | tex output ps output | tex output ps output |
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Parameters | Singular Scripts | (x,y)-subdivision | (x,z)-subdivision | (y,z)-subdivision |
b_{2} = 3 t^{2} b_{34} = 5 t^{3} b_{4} = 11 t b_{5} = 1+t^{2} | Standard Script (no refinement) | tex output ps output | tex output ps output | tex output ps output |
Parameters | Singular Scripts | (x,y)-subdivision | (x,z)-subdivision | (y,z)-subdivision |
b_{2} = 4 t^{2} b_{34} = t^{3} b_{4} = 2 t b_{5} = 1 + t^{2} | Standard Script (no refinement) | tex output ps output | tex output ps output | tex output ps output |
b_{2} = 4 t^{6} + t^{7} b_{34} = t^{9} b_{4} = 2 t^{3} b_{5} = 1 + t^{2} | Standard Script (no refinement) | tex output ps output | tex output ps output | tex output ps output |
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Parameters | Singular Scripts | (x,y)-subdivision | (x,z)-subdivision | (y,z)-subdivision |
b_{2} = 17 t^{2} + t^{6} b_{34} = 5 t^{2} + t^{10} b_{4} = 4 t + t^{5} b_{5} = 1+t^{2} | Standard Script (no refinement) | tex output ps output | tex output ps output | tex output ps output |
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In addition to the modification along F and its lifting f, we compute a vertical modification along MAX{X,ω_{4}} with lifting u = x - α_{4}. This produces a faithful tropicalization at the level of the extended skeleta. We recover the resulting tropical curve in R^{4} by means of the planar (x,y), (x,z), (u,y) and (u,z)-projections.
Branch points | Singular Scripts | (x,y)-subdivision | (x,z)-subdivision | (u,y)-subdivision | (u,z)-subdivision | (y,z)-subdivision |
α_{2} = (t^{9}+t^{12})^{2} α_{3} = (t^{5}+t^{9})^{2} α_{4} = (3 t^{5}+t^{8})^{2} α_{5} = -(1+t^{5})^{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 |
α_{2} = (t^{9}+t^{12})^{2} α_{3} = (t^{5}+t^{9})^{2} α_{4} = (3 t^{5}+t^{8})^{2} α_{5} = -(1+t^{5})^{2} | Refinement Script (t^{ε} and t^{ε'}) 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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In addition to the modification along F and its lifting f, we compute a vertical modification along MAX{X,ω_{4}} with lifting u = x - α_{4}. This produces a faithful tropicalization at the level of the extended skeleta. We recover the resulting tropical curve in R^{4} 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.
Branch points | Singular Scripts | (x,y)-subdivision | (x,z)-subdivision | (u,y)-subdivision | (u,z)-subdivision | (y,z)-subdivision |
α_{2} = (t^{14}+t^{16})^{2} α_{3} = (t^{5}+t^{9})^{2} α_{4} = (3 +t^{8})^{2} α_{5} = - (1+t^{5})^{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 |
α_{2} = (t^{14}+t^{16})^{2} α_{3} = (t^{5}+t^{9})^{2} α_{4} = (3 +t^{8})^{2} α_{5} = - (1+t^{5})^{2} | Refinement Script (t^{ε} 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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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,ω_{4}} and MAX{X,ω_{2}} (in that order), with liftings z_{2} = x - α_{2}, and z_{4} = x - α_{4}, we produce a faithful tropicalization at the level of the extended skeleta. We can compute the resulting tropical curve in R^{4} by means of the planar (x,y), (z_{2},y) and (z_{4},y)-projections.
Branch points | Singular Scripts | (x,y)-subdivision | (z_{2},y)-subdivision | (z_{4},y)-subdivision |
α_{2} = (5 t^{14}+t^{16})^{2} α_{3} = (t^{14}+t^{19})^{2} α_{4} = (3 +t^{8})^{2} α_{5} = - (1+t^{5})^{2} | Standard Script (no refinement) | tex output ps output | tex output ps output | tex output ps output |
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In addition to the modification along F and its lifting f, we compute two successive vertical modifications along MAX{X,ω_{4}}, with liftings z_{3} = x - α_{3}, and z_{4} = x - α_{4}. This produces a faithful tropicalization at the level of the extended skeleta. We recover the resulting tropical curve in R^{5} by means of the planar (x,y), (z_{3},y) and (z_{4},y)-projections. Notice that the the last two agree with the (z_{4},y) and (z_{4},z)-projections so the latter ones need not be plotted.
Branch points | Singular Scripts | (x,y)-subdivision | (x,z)-subdivision | (z_{3},y)- and (z_{4},y)-subdivisions | (z_{3},z)- and (z_{4},z)-subdivision | (y,z)-subdivision |
α_{2}=(t^{14}+t^{16})^{2} α_{3}= (5+t^{9})^{2} α_{4}= (3 +t^{8})^{2} α_{5}= - (1+t^{5})^{2} | Standard Script (no ref.) | tex output ps output | tex output ps output |
tex output (z_{3}y) ps output (z_{3}y) tex output (z_{4}y) ps output (z_{4}y) |
tex output (z_{3}z) ps output (z_{3}y) tex output (z_{4}z) ps output (z_{4}y) |
tex output ps output |
α_{2}=(t^{14}+t^{16})^{2} α_{3}= (5+t^{9})^{2} α_{4}= (3 +t^{8})^{2} α_{5}= - (1+t^{5})^{2} | Refinement Script (t^{ε} only) ε = 8 | tex output ps output | tex output ps output |
tex output (z_{3}y) ps output (z_{3}y) tex output (z_{4}y) ps output (z_{4}y) |
tex output (z_{3}z) ps output (z_{3}y) tex output (z_{4}z) ps output (z_{4}y) |
tex output ps output |
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In addition to the modification along F and its lifting f, we compute two successive vertical modifications along MAX{X,ω_{4}}, with liftings z_{3} = x - α_{3}, and z_{4} = x - α_{4}. This produces a faithful tropicalization at the level of the extended skeleta. We recover the resulting tropical curve in R^{5} by means of the planar (x,y), (z_{3},y) and (z_{3},z)-projections. Notice that the the last two agree with the (z_{4},y) and (z_{4},z)-projections so the latter ones need not be plotted.
Branch points | Singular Scripts | (x,y)-subdivision | (x,z)-subdivision | (z_{3},y)- and (z_{4},y)-subdivisions | (z_{3},z)- and (z_{4},z)-subdivision | (y,z)-subdivision |
α_{2}=(t^{14}+t^{16})^{2} α_{3}= (2+t^{9})^{2} α_{4}= (1 +t^{8})^{2} α_{5}= - (2+t^{5})^{2} | Standard Script (no ref.) | tex output ps output | tex output ps output |
tex output (z_{3}y) ps output (z_{3}y) tex output (z_{4}y) ps output (z_{4}y) |
tex output (z_{3}z) ps output (z_{3}y) tex output (z_{4}z) ps output (z_{4}y) |
tex output ps output |
α_{2}=(t^{14}+t^{16})^{2} α_{3}= (2+t^{9})^{2} α_{4}= (1 +t^{8})^{2} α_{5}= - (2+t^{5})^{2} | Refinement Script (t^{ε} only) ε = 13 | tex output ps output | tex output ps output |
tex output (z_{3}y) ps output (z_{3}y) tex output (z_{4}y) ps output (z_{4}y) |
tex output (z_{3}z) ps output (z_{3}y) tex output (z_{4}z) ps output (z_{4}y) |
tex output ps output |
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In addition to the modification along F and its lifting f, we compute two successive vertical modifications along MAX{X,ω_{4}}, with liftings z_{3} = x - α_{3}, and z_{4} = x - α_{4}. This produces a faithful tropicalization at the level of the extended skeleta. We recover the resulting tropical curve in R^{5} by means of the planar (x,y), (z_{3},y) and (z_{3},z)-projections. Notice that the the last two agree with the (z_{4},y) and (z_{4},z)-projections so the latter ones need not be plotted.
Branch points | Singular Scripts | (x,y)-subdivision | (x,z)-subdivision | (z_{3},y)- and (z_{4},y)-subdivisions | (z_{3},z)- and (z_{4},z)-subdivision | (y,z)-subdivision |
α_{2}=(t^{14}+t^{16})^{2} α_{3}= (2)^{2} α_{4}= (1)^{2} α_{5}= - (2)^{2} | Standard Script (no ref.) | tex output ps output | tex output ps output |
tex output (z_{3}y) ps output (z_{3}y) tex output (z_{4}y) ps output (z_{4}y) |
tex output (z_{3}z) ps output (z_{3}y) tex output (z_{4}z) ps output (z_{4}y) |
tex output ps output |
α_{2}=(t^{14}+t^{16})^{2} α_{3}= (2)^{2} α_{4}= (1)^{2} α_{5}= - (2)^{2} | Refinement Script (t^{ε} only) ε = 13 | tex output ps output | tex output ps output |
tex output (z_{3}y) ps output (z_{3}y) tex output (z_{4}y) ps output (z_{4}y) |
tex output (z_{3}z) ps output (z_{3}y) tex output (z_{4}z) ps output (z_{4}y) |
tex output ps output |
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α_{2}=β_{2}^{2}, α_{3}= β_{3}^{2}, α_{4}= β_{4}^{2} and α_{5}= - β_{5}^{2}.
Violating The genericity conditions for both x^{3} and x^{4} requires the use of non-real numbers, since we must have(in (β_{3}))^{2} = in (α_{3}) = -in(α_{4}) = - ( in (β_{4}))^{2} and in (β_{4}) in (β_{4}) = in (β_{5}) (in (β_{3}) - in (β_{4})).
We take in (β_{5}) = 1, in (β_{3}) = (ι - 1), and in (β_{4}) = (1 + ι). We produce all the numerical examples by evaluating all the elimination ideals at the desired values of β_{2}, β_{3}, β_{4}, and β_{5} with Singular, by working with the field extension associated to the polynomial x^{2} + 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 ι^{1/2}.) As was mentioned, we only treat the refined modifications.Branch points | Singular Scripts | (x,y)-subdivision | (x,z)-subdivision | (z_{3},y)- and (z_{4},y)-subdivisions | (z_{3},z)- and (z_{4},z)-subdivision | (y,z)-subdivision |
α_{2}= (5t^{14}+t^{16})^{2} α_{3}= ((ι)^{1/2}-1+t^{18})^{2} α_{4}= ((1+(ι)^{1/2})^{2} α_{5}= - 1 |
Refinement Script with Symbolic ι^{1/2} Refinement Script with Numerical ι^{1/2} (t^{ε} and t^{ε'}) ε = 1 ε' = 2 | tex output ps output | tex output ps output |
tex output (z_{3}y) ps output (z_{3}y) tex output (z_{4}y) ps output (z_{4}y) |
tex output (z_{3}z) ps output (z_{3}y) tex output (z_{4}z) ps output (z_{4}y) |
tex output ps output |
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If we compute three successive vertical modifications along MAX{X,ω_{2}} with 3 different liftings
z_{2} = x - α_{2}, z_{3} = x - α_{3}, and z_{4} = x - α_{4},
we produce a faithful tropicalization at the level of the extended skeleta. We can compute the resulting tropical curve in R^{5} by means of the planar (x,y), (z_{2},y), (z_{3},y), and (z_{4},y)-projections. Notice that these 3 projections are indistinguishable. Their sole purpose is to isolate the leg marked by the corresponding branch point.Branch points | Singular Scripts | (x,y)-subdivision | (z_{2},y)-subdivision | (z_{3},y)-subdivision | (z_{4},y)-subdivision |
α_{2} = ( t^{10}+t^{12})^{2} α_{3} = (4 t^{10}+t^{19})^{2} α_{4} = (3 t^{10}+t^{18})^{2} α_{5} = - (12 t^{10})^{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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The following scripts allow to compute initial forms of the corresponding polynomials A, B, C, A', B', C', Q_{3} = AB - 3C, Q_{4} = AB - 4C, and Q_{4}' = A'B' - 4C'. We refer to the last two sections of the paper for the definition of these polynomials.
ω_{1} < ω_{2} < ω_{3} < ω_{4} < ω_{5} < ω_{6}
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.In all our scripts we input a weight vector ω=(ω_{6},ω_{5},d_{34},ω_{3},ω_{2},ω_{1}) in R^{6}, where -val(α_{i}) = ω_{i} for i=1,2,3, 5, 6, and -val(x_{34}) = d_{34}. Recall that x_{34} = α_{4} - α_{3}.
The vector ω lies in the Theta cone since it satisfies:ω_{1} < ω_{2} < ω_{3} <ω_{5} < ω_{6} and d_{34} < ω_{3}.
The Sage script at the begining of this section computes the Gröbner fan of Q_{4}' = A'B' - 4 C' by replacing Q_{4}' by the sum of its extremal monomials. Its f-vector equals [1, 32, 174, 396, 420, 168].Cell | Defining inequalities | Extremal rays | Lines | Sample point |
Θ_{0} | (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); |
Θ_{1} |
(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); |
Θ_{2} |
(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) |
A = 4 A_{4} + 6 A_{6} + 12 A_{12} + 120 A_{120}.
The computation verifies that no matter what point we choose, we need only compare the initial terms of 4 A_{4} and Q = (6 A_{6} + 12 A_{12} + 120 A_{120})/3. This gives 3 cases to analyze, depending of the order between ω_{3} and (ω_{4} - 1). In case of equality there will be cancellations whenever val(-3 α_{4} + 2α_{3}) > -ω_{3}. If so, there will be no general formula for computing val(A), and we would have to make a substitution of the formα_{3} = 3/2 α_{4} + y_{34} where val(y_{34})> -ω_{3}
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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