2.1. Invariant theory of binary forms

We recall some facts and results from invariant theory of binary forms. For detailed treatments of invariant theory, we refer the reader to [Reference Derksen and Kemper7, Reference Popoviciu Draisma19, Reference Sturmfels21]. Fix an algebraically closed field $K$ such that $\textrm{char}(K)=0$ .

2.1.1. A group action on binary forms

Let $n\geq{0}$ be a fixed integer. Let $A=K[a_{0},\ldots,a_{n}]$ be the polynomial ring in $n+1$ variables. We view $A$ as a graded ring with the standard grading $\textrm{deg}(a_{i})=1$ . Let $V_n$ be the $A$ -submodule of $A[x,z]$ consisting of homogeneous polynomials in $x$ and $z$ of total degree $n$ . The reductive group $ G \;:\!=\;{\textrm{SL}}_2(K)$ acts (as a right action) on the $A$ -module $V_n$ as follows:

(2) \begin{equation} g^{\sigma }(x,z) \;:\!=\; g(ax + bz, cx + dz), \quad \text{ for } g \in V_n \text{ and } \sigma = \begin{bmatrix} a\;\;\;\;\; & b \\[5pt] c\;\;\;\;\; & d \end{bmatrix} \in G. \end{equation}

Definition 5. (Universal binary form) We define the universal binary form $f$ of degree $n$ over $K$ as the binary form given by

\begin{equation*} f(x,z) = a_{0}x^{n} + a_{1}x^{n-1}z+ \cdots +a_{n}z^{n}\in V_n. \end{equation*}

A binary form over $K$ of degree $n$ is obtained by specializing the coefficients to $K$ .

We obtain an action of $G$ on $A$ by sending $a_{i}$ to the coefficient $c_i(f^{\sigma })$ of $x^{n-i} z^{i}$ in $f^{\sigma }$ for $0 \leq i \leq n$ . This means that $G$ acts on $A$ as follows:

(3) \begin{equation} F^{\sigma }(a_0, \ldots, a_n) \;:\!=\; F(c_0( f^{\sigma } ), \ldots, c_n(f^{\sigma })), \quad \text{ for } F \in A \text{ and } \sigma \in G. \end{equation}

Example 6. When $n = 2$ , from the definition

\begin{equation*} f^{\sigma }(x,z) = f(ax + bz, cx + dz), \quad \text { for } \sigma = \begin {bmatrix} a\;\;\;\;\; & b \\[5pt] c\;\;\;\;\; & d \end {bmatrix} \in G \end{equation*}

we get

\begin{equation*} \begin {bmatrix} c_0(f^{\sigma }) \\[5pt] c_1(f^{\sigma }) \\[5pt] c_2(f^{\sigma }) \end {bmatrix} = \begin {bmatrix} a^2\;\;\;\;\; & ac\;\;\;\;\; & c^2\\[5pt] 2 ab\;\;\;\;\; & ad + bc\;\;\;\;\; & 2cd \\[5pt] b^2\;\;\;\;\; & bd\;\;\;\;\; & d^2 \end {bmatrix} \begin {bmatrix} a_0 \\[5pt] a_1 \\[5pt] a_2 \end {bmatrix}. \end{equation*}

We can then see how $\sigma$ acts on the generators $a_0,a_1,a_2$ of $A = K[a_0, a_1, a_2]$ and we have

\begin{equation*} a_0^{\sigma } = a^2 a_0 + ac a_1 + c^2 a_2, \quad a_1^{\sigma } = 2ab a_0 + (ad + bc) a_1 + 2cd a_2 \quad \text { and } a_2^{\sigma } = b^2 a_0 + bd a_1 + d^2 a_2. \end{equation*}

Definition 7. (Separable binary forms) A binary form is said to be separable if its discriminant does not vanish.

Definition 8. (Invariants of binary forms) A homogeneous polynomial $F \in{A}$ is called $G$ -invariant if

\begin{equation*} F^{\sigma } = F \quad \text { for all } \sigma \in G = {\textrm {SL}}_2(K). \end{equation*}

We denote the graded ring generated by all homogeneous $G$ -invariant polynomials by $A^{G}$ .

Remark 9. Notice that there is also an action of ${\textrm{GL}}_2(K)$ on $A$ as in ( 4 ). Since $K$ is algebraically closed, a homogeneous polynomial $F \in A$ is ${\textrm{SL}}_2$ -invariant if and only if for any $\sigma \in{\textrm{GL}}_2$ we have

\begin{equation*} F^\sigma = \det (\sigma )^{\deg (F)} F. \end{equation*}

We say that a homogeneous polynomial $F\in{A}$ is ${\textrm{GL}}_{2}$ -invariant if it satisfies the identity above. This immediately implies that the ring of invariants for ${\textrm{GL}}_{2}$ and ${\textrm{SL}}_{2}$ is the same. We will use these interchangeably.

Since $G={\textrm{SL}}_{2}(K)$ is reductive, it is a well-known fact from invariant theory that the ring $A^G$ is finitely generated over $K$ ; see, for example, [Reference Derksen and Kemper7, Corollary 2.2.11]. To find generators of $A^G$ , we can use the notion of transvectants, or Überschiebung, which we explain briefly here. Let $g \in{V_{m}}$ and $h\in{V_{n}}$ with $m \geq n$ and let $r$ be an integer with $0\leq r \leq{n}$ . We define the bilinear map

\begin{equation*} \langle \cdot, \cdot \rangle _{r} \;:\; V_m \times V_n \xrightarrow []{} V_{m + n - 2r}, \quad (g,h)\mapsto \sum _{i=0}^{r} (\!-\!1)^i\binom {r}{i} \dfrac {\partial ^{r}{g}}{\partial ^{r-i}{x} \ \partial ^{i}{z}} \ \dfrac {\partial ^r{h}}{\partial ^{i}{x} \ \partial ^{r-i}{z}}. \end{equation*}

It turns out that this map is $G$ -invariant, i.e.,

\begin{equation*} \langle g^{\sigma }, h^{\sigma } \rangle _{r} = \langle g,h \rangle _{r}, \quad \text { for } (g,h) \in V_m \times V_n \text { and } \sigma \in G. \end{equation*}

The quantity $\langle g,h \rangle _{r}$ is called the $r$ th transvectant of $g$ and $h$ . Notice that when $n = m = r$ , the transvectant $\langle g,h \rangle _{r}$ has degree $0$ in $x,z$ so $\langle g,h \rangle _{r} \in A^G$ . A theorem by Gordan [Reference Gordan8] shows that one can obtain all invariants by taking (iterated if necessary) transvectants of $f$ , so this makes the set generators of $A^G$ explicit, at least in theory. In practice, the generators have only been found for binary forms of degrees up to $10$ ; see [Reference Popoviciu Draisma19, Chapter 1] or [Reference Derksen and Kemper7, Example 2.1.2].

Example 10.

1. 1. For binary quadrics, the invariant ring $A^G$ is generated by $\langle f,f \rangle _{2}$ , which is equal to the discriminant of $f$ .

2. 2. For binary cubics, the invariant ring is again generated by the discriminant

   \begin{equation*} \Delta = \langle \langle f, f \rangle _2,\langle f, f \rangle _2 \rangle _2. \end{equation*}

3. 3. For binary quartics $n=4$ , the invariant ring $A^G$ is generated by two algebraically independent invariants $I_{2}$ and $I_{3}$ of degrees $2$ and $3$ , respectively. In terms of transvectants, they are given by

   \begin{equation*} I_2 = \langle f,f \rangle _4 \quad \text {and} \quad I_3 = \langle f,\langle f,f \rangle _2 \rangle _4. \end{equation*}

4. 4. For binary quintics, the invariant ring $A^G$ is generated by four generators $I_{4}$ , $I_{8}$ , $I_{12}$ , and $I_{18}$ of degrees $4$ , $8$ , $12$ , and $18$ , respectively. These are not algebraically independent, as $I_{18}^2$ can be expressed in terms of the other three. We refer the reader to Section 2.1.2 for a more detailed exposition in this case.

2.1.2. Invariants of binary quintics

As discussed in Example 10, the ring of invariants for quintic binary forms is generated by four elements $I_{4}$ , $I_{8}$ , $I_{12}$ , and $I_{18}$ of degrees $4$ , $8$ , $12$ , and $18$ , respectively. Let $f$ be the universal quintic

\begin{equation*} f(x,z) = a_0 x^5 + a_1 x^4 z + a_2 x^3 z^2 + a_3 x^2 z^3 + a_4 x z^4 + a_5 z^5. \end{equation*}

The four invariants $I_4$ , $I_8$ , $I_{12}$ , and $I_{18}$ can be obtained by taking transvectants of $f$ in the following way:

(4) \begin{equation} \begin{aligned} I_{4}&=\langle f^2,f^2 \rangle _{10}, & I_{8}&=\langle f^4,f^4 \rangle _{20},\\[5pt] I_{12}&=\langle f^6,f^6 \rangle _{30}, & I_{18}&=\langle \langle f^5,f^6 \rangle _{10},f^7 \rangle _{35}. \end{aligned} \end{equation}

Note that these transvectants are different from the ones considered in [Reference Popoviciu Draisma19, Section 4.4].Footnote ¹ They still yield generators however, as one easily checks using the Poincaré series. The discriminant $\Delta$ of $f$ can be expressed in terms of the previous invariants as

(5) \begin{equation} \Delta = c_{0}I_{4}^2 + c_{1} I_{8} \end{equation}

where $c_0, c_1$ are the constants

\begin{equation*} c_{0} =-1/2746158938062848000000, \quad c_{1} = 1/46987474647852089270599680000000. \end{equation*}

Note that $\deg (\Delta ) = 8$ .

In Section 3, we will assume that $K$ is a complete non-archimedean field and associate a metric tree to a binary form over $K$ . This tree is invariant under the action of ${\textrm{GL}}_{2}$ . For quintics, there are three different types, see Figure 1. The invariants $I_4,I_8,I_{12},I_{18}$ and the discriminant $\Delta$ are however not enough to distinguish between the different types, as we will see in Example 32. In order to distinguish tree types, we need to introduce another invariant, which we call the $H$ -invariant.

Definition 11. ( $H$ -invariant) The $H$ -invariant of the quintic $f = a_0 x^5 + a_1 x^4 z + \ldots + a_5 z^5$ is defined as follows:

(6) \begin{equation} H = \beta I_{12} - 396 \alpha ^3 I_{4}^{3}, \end{equation}

where $\alpha$ and $\beta$ are given by

\begin{align*} \alpha &= 2^{-17} \cdot 3^{-7} \cdot 5^{-3} \cdot 7^{-1},\\[5pt] \beta &= 2^{-50} \cdot 3^{-27} \cdot 5^{-14} \cdot 7^{-7} \cdot 11^{-4} \cdot 13^{-3} \cdot 17^{-1} \cdot 19^{-1} \cdot 23^{-1} \cdot 29^{-1}. \end{align*}

This invariant is of degree $12$ and constructed to satisfy the following property:

\begin{equation*} H(0,0,1,0,a_4,a_5) \quad \text { is equal to the discriminant of the cubic } \quad x^{3} + a_4x + a_5. \end{equation*}

The motivation behind creating this invariant is as follows. For binary quintics, there are three unmarked tree types, see Figure 1. For the two non-trivial ones, we choose a non-trivial vertex of valency two. By applying a projective linear transformation, we can ensure that the roots are grouped as $\{\infty,\lambda _{1}\}$ and $\{0,1,\lambda _{2}\}$ , where $v(\lambda _{1})\lt v(\lambda _{2})=0$ . After scaling the binary form so that the smallest valuation of its coefficients is zero, the coefficients of $x^5$ and $x^4z$ have positive valuation. This binary form hence reduces to a cubic and the discriminant of this cubic distinguishes between the two remaining types.

2.1.3. Invariants of $(4,1)$ forms

Let $B = K[b_0, \ldots, b_4, c_0, c_1]$ be a polynomial ring. For an integer $n \geq 0$ , we denote by $W_n$ the space of homogeneous polynomials in $B[x,z]$ of degree $n$ . Here, we are interested in $(4,1)$ -binary forms, which are the elements $(q, \ell )$ of $W_4 \oplus W_1$ . There is a natural group action of $G ={\textrm{SL}}_2(K)$ on $(4,1)$ -forms as follows:

\begin{equation*} (q(x,z), \ell (x,z))^{\sigma } = ( q^{\sigma }(x,z), \ell ^{\sigma } (x,z)) = ( q(\sigma (x,z)), \ell (\sigma (x,z))). \end{equation*}

Definition 12. (Universal $(4,1)$ -form) We define the universal $(4,1)$ -form $(q,\ell )$ as

\begin{equation*} q(x,z) = b_0x^4 + b_1 x^3 z + \ldots + b_4 z^4 \quad \text {and} \quad \ell (x,z) = c_0 x + c_1 z. \end{equation*}

The action of $G$ on $(q,\ell )$ defines an action of $G$ on $B$ where $\sigma \in G$ acts on $B$ by sending $ b_0, \ldots, b_4, c_0, c_1$ to the coefficients of $(q,\ell )^{\sigma }$ . We denote by $B^G$ the ring of $G$ -invariant polynomials in $B$ . This ring is again finitely generated since $G$ is reductive. It has $5$ generators $j_2, j_3, j_5,j_6$ , and $j_9$ of respective degrees $2,3,5,6$ , and $9$ , and they can be obtained using transvectants as follows:

(7) \begin{equation} \begin{aligned} j_{2}&=\langle q,q \rangle _{4}, & j_{3}&=\langle \langle q,q \rangle _{2},q \rangle _{4}, \\[5pt] j_{5}&=\langle q,\ell ^4 \rangle _{4}, & j_{6}&=\langle \langle q,q \rangle _{2},\ell ^{4} \rangle _{4},\\[5pt] j_{9}&=\langle \langle q,\langle q,q \rangle _{2} \rangle _1, \ell ^6 \rangle _6. \end{aligned} \end{equation}

We refer the reader to [Reference Popoviciu Draisma19, Section 5.4.1] for more details.

The invariants $I_4, I_8, I_{12}, I_{18}, \Delta, H$ for binary quintics and the invariants $j_2, j_3, j_5, j_6, j_9$ for $(4,1)$ -forms are polynomials with rational coefficients. Scaling these invariants, we may assume that their coefficients are integer and are coprime; recall that $\textrm{char}(K) = 0$ . The reason behind this is that we are working with a valued field $K$ in which integer scalars might have a positive valuation. This justifies the following assumption:

Assumption 13. We scale all the invariants so that the greatest common divisor of their coefficients is $1$ , and we do not change the notation. These scaled invariants are computed in [ 24 ] and are the ones used in our results.

Write $B_{4,1}=B$ for the coefficient ring corresponding to $(4,1)$ -forms and $B_{5}$ for the coefficient ring corresponding to quintics. The universal $(4,1)$ -form $(q,\ell )$ gives a canonical quintic $f=q\cdot \ell$ . From this, we obtain an injective ring homomorphism

\begin{equation*} B_{5}\to {B_{4,1}}. \end{equation*}

This ring homomorphism is $G ={\textrm{SL}}_{2}(K)$ -equivariant, so we obtain an induced injective ring homomorphism

\begin{equation*} B_{5}^{G}\to {B^{G}_{4,1}} \end{equation*}

of invariants. We identify $B^{G}_{5}$ with its image, so that we have an inclusion $B_{5}^{G}\subset{}{B^{G}_{4,1}}$ . The generators from Section 2.1.2 can then be expressed in terms of the $j_{i}$ explicitly as follows:

(8) \begin{equation} \begin{aligned} I_4 &= \frac{2}{3}\,j_2\, j_6 - \frac{1}{2}\, j_3\, j_5, \\[5pt] I_8 &= \frac{14}{9}\,j_2^2\, j_6^2 + \frac{22}{27}\, j_2^3\, j_5^2 + \frac{5}{27}\, j_3^2\, j_5^2 - \frac{14}{9}\, j_2\, j_3\, j_5\, j_6, \\[5pt] I_{12} &= \frac{4400}{243}\, j_2^3\, j_6^3 - \frac{11}{243}\, j_3^2\, j_6^3 - \frac{242}{9}\, j_2^2\, j_3\, j_5\, j_6^2 + \frac{2479}{81}\, j_2^4\, j_5^2\, j_6 + \frac{692}{81}\, j_2\, j_3^2\, j_5^2\, j_6\\[5pt] & \ \ \ + \frac{7156}{243}\, j_2^3\, j_3\, j_5^3 - \frac{92}{243}\, j_3^3\, j_5^3, \\[5pt] I_{18} &= - \frac{625}{729}\, j_2^6\, j_5^3\, j_9 - \frac{512}{729}\, j_2^3\, j_3^2\, j_5^3\, j_9 - \frac{4}{729}\, j_2^3\, j_3\, j_6^3\, j_9 + \frac{1}{729}\, j_3^3\, j_6^3\, j_9 + \frac{1}{3}\, j_2^4\, j_3\, j_5^2\, j_6\, j_9 \\[5pt] & \ \ \ + \frac{4}{243}\, j_2^5\, j_5\, j_6^2\, j_9 - \frac{1}{243}\, j_2^2\, j_3^2\, j_5\, j_6^2\, j_9; \end{aligned} \end{equation}

see the computations in [24]. For $\Delta$ and $H$ , we use (6) and (7).

Finally, we summarize the invariants used in Theorems 1, 2, and 3, and their respective degrees as follows:

Table 1.

Invariants of binary quintics (on the left) and (4,1)-forms (on the right) together with their degrees.

2.2. Picard curves

Let $K$ denote a field of characteristic $0$ . A Picard curve $X$ over $K$ is a smooth projective curve of genus $3$ in $\mathbb{P}^{2}$ given by an equation of the form

\begin{equation*} y^3\ell (x,z) = q(x,z), \end{equation*}

where $q$ and $\ell$ are homogeneous polynomials of degrees $4$ and $1$ , respectively, in $K[x,z]$ . Here, the smoothness of $X$ is equivalent to the separability of the quintic $q(x,z)\cdot \ell (x,z)$ . For a Picard curve, the five distinct roots of this quintic are the branch points of the $\mathbb{Z}/3\mathbb{Z}$ -covering of $\mathbb{P}^{1}$ induced by the rational map

\begin{equation*} \pi \colon X \dashrightarrow \mathbb {P}^1, \quad [x\;:\;y\;:\;z] \mapsto [x\;:\;z]. \end{equation*}

Either using the fact that $X$ is smooth or an explicit computation, we find that it gives a morphism, so that it is defined everywhere. The Galois group of $\pi$ is $G=\mathbb{Z}/3\mathbb{Z}$ and it acts as follows on $X$ . Let $\zeta \in{K}$ be a primitive third root of unity. There is then an automorphism

\begin{equation*} \alpha \;:\;[x\;:\;y\;:\;z]\mapsto [x\;:\;\zeta {y}\;:\;z] \end{equation*}

of order three and the quotient map $X\to X/\langle \alpha \rangle =\mathbb{P}^{1}$ is $\pi$ . For each of the five points $P_{i}$ in the zero set of the quintic $q(x,z)\cdot \ell (x,z)$ , there is a unique point $Q_{i}$ lying over $P_{i}$ . The four points $Q_{1}$ ,…, $Q_{4}$ corresponding to the quartic $q(x,z)$ differ from the remaining point $Q_{5}$ in the sense that $\alpha$ acts by $\zeta$ on the tangent spaces of $Q_{1}$ ,…, $Q_{4}$ and as $\zeta ^{2}$ on the tangent space of $Q_{5}$ . This gives a second, and equivalent, definition of a Picard curve as a $\mathbb{Z}/ 3\mathbb{Z}$ -covering of $\mathbb{P}^{1}$ branched over five points with a specified action of $\mathbb{Z}/3\mathbb{Z}$ on the tangent spaces of the ramification points, see [Reference Achter and Pries2]. This rigidification can also be given in terms of differential forms, see [Reference Cléry and van der Geer6, Section 5].

3.1. Compactifications and tropicalizations

Let $U$ be an irreducible variety over $K$ with Berkovich analytification $U^{\textrm{an}}$ . We can define tropical separating sets for partitions of subsets of $U^{\textrm{an}}$ as follows.

The partitions of $U^{\textrm{an}}$ we are interested in are given using compactifications of $U$ as follows. Let $\mathcal{X}\to{\textrm{Spec}(R)}$ be a proper flat $R$ -scheme and let $U\to{\mathcal{X}}$ be an open immersion. Consider the reduction map

(9) \begin{equation} \textrm{red}\;:\;U^{\textrm{an}}\to \mathcal{X}_{s} \end{equation}

as in [Reference Gubler9, Section 4], where $\mathcal{X}_{s}$ is the special fiber of $\mathcal{X}$ . We now obtain a partition of $U^{\textrm{an}}$ by taking the inverse image under $\textrm{red}(\cdot{})$ of a partition of $\mathcal{X}_{s}$ .

3.2. Tropical invariants

We start with the definition of tropical weighted projective space.

Definition 16. (Tropical weighted projective space) Let $(\mathbb{T}\mathbb{A}^n)^* = \overline{\mathbb{R}}^n\setminus \{(\infty,\ldots,\infty )\}$ be the punctured tropical affine plane. For a fixed $n$ -tuple of natural numbers $(a_1,\ldots,a_n)$ , we define an action of $\mathbb{R}$ on $(\mathbb{T}\mathbb{A}^n)^*$ as follows:

\begin{equation*} \lambda \odot (x_1,\ldots,x_n)\;:\!=\; (x_1,\ldots,x_n)+\lambda \cdot (a_{1},\ldots,a_{n}). \end{equation*}

The (set-theoretic) quotient of $(\mathbb{T}\mathbb{A}^n)^*$ by this action is called the tropical weighted projective space and is denoted by $\mathbb{T}\mathbb{P}(a_1,\ldots,a_n)$ .

Now, let $U/K$ be an irreducible variety and let $G/K$ be a group scheme acting on $U$ . Suppose that there exists a geometric quotient $V=U/G$ for $U$ and $G$ . The notion of a tropical separating from Section 3.1 now allows us to define a set of tropical invariants.

Since points of $V^{\textrm{an}}$ correspond to orbits in $U^{\textrm{an}}$ , we find that a set of tropical invariants separates a partition of a subset of $U^{\textrm{an}}$ that is stable under the action of $G$ .

Example 18. Consider the ring $K[x,y]$ . The group $S_{2}\simeq{}\mathbb{Z}/2\mathbb{Z}$ acts on $K[x,y]$ by $x\mapsto{y}$ and $y\to{x}$ . The invariant ring is $K[xy,x+y]$ and, on the level of schemes, the map $\textrm{Spec}(K[x,y])\to \textrm{Spec}(K[xy,x+y])$ gives the geometric quotient. Consider the standard embedding of $\mathbb{A}^{2}=\textrm{Spec}(K[x,y])$ into $\mathbb{P}^{2}$ . The group action then extends to $\mathbb{P}^{2}$ . The boundary is isomorphic to $\mathbb{P}^{1}$ and we decompose $\mathbb{P}^{2}$ into

\begin{equation*} \overline {S}_{0} = \{[x\;:\;y\;:\;z] \mid z\neq {0}\},\ \overline {S}_{1}=\{[1\;:\;1\;:\;0]\} \text { and } \overline {S}_{2}=\{[x\;:\;y\;:\;0] \mid x\neq {y}\}. \end{equation*}

This induces a partition

\begin{equation*} S_{0}\sqcup {}S_{1}\sqcup {S_{2}}={\textrm {Spec}}(K[xy,x+y])^{\textrm {an}} \end{equation*}

since the group action preserves the $\overline{S_{i}}$ . We now note that the standard embedding

\begin{equation*}\textrm {Spec}(K[xy,x+y])=\mathbb {A}^{2}\to \mathbb {P}^{2}\end{equation*}

is not tropically separating since $\textrm{char}(K) \neq 2$ . If we however consider the embedding into $\mathbb{P}^{3}$ given by $\{xy,x+y,(x-y)^2,1\}$ , then this does form a tropical set of invariants.

We will see many more examples of tropical invariants in Sections 3.3 and 3.4.

3.3. Moduli of tropical curves

We now review moduli of stable curves of genus zero. The functor $\mathcal{M}_{0,n}\;:\; \textbf{Sch} \to \textbf{Sets}$ from the category of schemes to the category of sets, defined by

(10) \begin{equation} \mathcal{M}_{0,n}(S)=\{\text{smooth curves } C\to{S} \text{ of genus zero with }n \text{ marked points}\}/\sim{} \end{equation}

for $n\geq{3}$ is representable by a scheme $M_{0,n}$ . Note that there are indeed no non-trivial elements of $\textrm{Aut}(\mathbb{P}^{1})=\textrm{PGL}_{2}$ that fix three or more elements of $\mathbb{P}^{1}$ . We refer to $M_{0,n}$ as the moduli space of smooth $n$ -marked curves of genus zero. There is a natural compactification of this space, given by replacing smooth curves by stable curves in (11). We denote the corresponding scheme by $\overline{M}_{0,n}$ . It is proper and flat over $\textrm{Spec}(\mathbb{Z})$ . The boundary locus $\Delta =\overline{{M}}_{0,n}\backslash{M_{0,n}}$ corresponds to non-smooth stable curves.

The group $S_{n}$ acts on the above moduli spaces by permuting the marked points. Since $S_{n}$ is finite and $M_{0,n}$ and $\overline{M}_{0,n}$ are quasi-projective, there exist geometric quotients $M_{0,n}/S_{n}$ and $\overline{M}_{0,n}/S_{n}$ which fit into a commutative diagram

The scheme $\overline{M}_{0,n}/S_{n}$ is again proper over $\mathbb{Z}$ . The geometric points of $M_{0,n}/S_{n}$ (resp. $\overline{M}_{0,n}/S_{n}$ ) can be identified with smooth (resp. stable) curves of genus $0$ with $n$ unordered points.

We now describe their abstract tropicalizations as in [Reference Abramovich, Caporaso and Payne1, Section 4]. We start with the original setup without quotients. Consider phylogenetic trees with $n$ marked leaves, together with a length function $\ell$ on the non-leaves. These are the points of the tropical moduli space $M_{0,n}^{\textrm{trop}}$ . It can be given the structure of a generalized cone complex as in [Reference Abramovich, Caporaso and Payne1, Section 2]. It can also be given as the tropicalization of the Plücker map as in [Reference Maclagan and Sturmfels15, Theorem 6.4.12]. We obtain an abstract tropicalization map

\begin{equation*} \textrm {trop}\;:\;M^{\textrm {an}}_{0,n}\to {M^{\textrm {trop}}_{0,n}} \end{equation*}

by the following procedure. A point $P\in{M^{\textrm{an}}_{0,n}}$ can be represented by an $L$ -valued point of $M_{0,n}$ , and we use the natural map

(11) \begin{equation} M_{0,n}(L)\to{\overline{M}_{0,n}(L)}=\overline{M}_{0,n}(R_{L}) \end{equation}

to obtain a stable model $\mathcal{X}\to \textrm{Spec}(R_{L})$ with special fiber $\mathcal{X}_{s}\to \textrm{Spec}(k_{L})$ . Here, $L$ is a complete valued field extension of $K$ with valuation ring $R_{L}$ and residue field $k_{L}$ . Let $T$ be the marked dual intersection graph of $\mathcal{X}_{s}$ , together with the natural edge length function $\ell$ induced from $\mathcal{X}$ . Note that there is a unique leaf for every marked point. We set $\textrm{trop}(P)=(T,\ell )$ .

We can also express the tropicalization of a point in $M_{0,n}^{{\textrm{an}}}$ in terms of Berkovich skeleta as in [Reference Baker, Payne and Rabinoff4]. Namely, a point $P\in M_{0,n}^{{\textrm{an}}}$ corresponds to a marked curve $(\mathbb{P}^{1},(P_{1},\ldots,P_{n}))$ , where $P_{i}\in \mathbb{P}^{1}(L)$ are distinct $L$ -valued points. The minimal skeleton $\Sigma$ of this marked curve as in [Reference Baker, Payne and Rabinoff4, Section 2.6]Footnote ² is then a marked metric tree whose leaves correspond to the points $P_{i}$ . The underlying graph of this marked metric tree is the same as $T$ and it gives rise to these same edge length function $\ell$ . By abuse of notation, we will write $\Sigma =(T,\ell )$ , where we either view $(T,\ell )$ as a marked finite graph with a length function or a marked metric tree.

Let $\mathcal{P}([n])$ be the power set of $[n]=\{1,\ldots,n\}$ and let $\Xi \subset \mathcal{P}([n])$ be a partition. We write $S_{\Xi }\subset S_{n}$ for the subgroup of permutations that preserve $\Xi$ . Note that $S_{\Xi }$ is isomorphic to a product of symmetry groups $S_{k}$ for $k\leq{n}$ . We now define an action of $S_{\Xi }$ on $M^{\textrm{trop}}_{0,n}$ . We view an element of $M^{\textrm{trop}}_{0,n}$ as a metric tree $\Sigma =(T,\ell )$ , together with an injection $i\;:\;\{1,\ldots,n\}\to{L(T)}$ , where $L(T)$ is the set of (infinite) leaves of $T$ . By permuting the leaves indexed by the partition $\Xi$ , we obtain an action of $S_{\Xi }$ on $M^{\textrm{trop}}_{0,n}$ , where the latter is viewed as an object in the category of generalized cone complexes. We now note that categorical quotients in the category of generalized cone complex exist since arbitrary finite colimits exist, see [Reference Abramovich, Caporaso and Payne1, Remark 2.6.1]. From this, we obtain the following definition of $M^{\textrm{trop}}_{0,n}/S_{\Xi }$ .

Example 24. Let $n=5$ . Then, there are exactly three unmarked types: $\textrm{I}$ , $\textrm{II}$ , and $\textrm{III}$ ; they are depicted in Figure 1. Type $\textrm{III}$ corresponds to a folded positive orthant of dimension $2$ , since the automorphism group of the underlying graph is $\mathbb{Z}/2\mathbb{Z}$ . Figure 4 represents the space $M^{\textrm{trop}}_{0,5}/S_{5}$ .

Figure 4.

The space $M^{\textrm{trop}}_{0,5}/S_{5}$ .

Example 25. Suppose that $\Xi =\{\{1,2,\ldots,n-1\},\{n\}\}$ . In this case, we will write $S_{\Xi }=S_{n-1}$ for brevity. The corresponding tropical moduli space $M^{\textrm{trop}}_{0,n}/S_{n-1}$ parametrizes tropical trees with one marked point.

For instance, consider the case where $n=5$ and $\Xi =\{\{1,2,\ldots,4\},\{5\}\}$ , giving rise to an action of $S_{4}$ on $M^{\textrm{trop}}_{0,5}$ . The quotient $M^{\textrm{trop}}_{0,5}/S_{4}$ is the moduli space of phylogenetic trees with $4$ unmarked leaves and $1$ marked leaf, see Figure 5. There are five different corresponding phylogenetic tree types: $\textrm{I}$ , $\textrm{II}.1$ , $\textrm{II}.2$ , $\textrm{III}.1$ , and $\textrm{III}.2$ , see Figure 2.

Figure 5.

The space $M^{\textrm{trop}}_{0,5}/S_{4}$ .

In Figures 4 and 5 , the (possibly folded) positive orthants are glued with respect to degeneration of the corresponding tree types, which is shown in Figure 6. We refer to [Reference Markwig16 , Section 2] and [ Reference Chan5, Section 4] for more details.

Figure 6.

Degenerations of trees.

For the symmetrized moduli space $M_{0,n}/S_{\Xi }$ , we can again introduce an abstract tropicalization map as follows. Let $\Xi \subset \mathcal{P}[n]$ be a partition and let $P$ be an $L$ -valued point of $M^{\textrm{an}}_{0,n}/S_{\Xi }$ . This again gives a phylogenetic tree $(T,\ell )$ with a marking $\{1,\ldots,n\}\to L(T)$ , but we identify two markings $\phi _{1},\phi _{2}\;:\;\{1,\ldots,n\}\to L(T)$ if they are related by an element of $S_{\Xi }$ . This gives a map

(12) \begin{equation} \textrm{trop}\;:\;M^{\textrm{an}}_{0,n}/S_{\Xi }\to{M^{\textrm{trop}}_{0,n}/S_{\Xi }}. \end{equation}

3.4. Trees and tropical binary forms

Let $V=V_{k_{1}}\oplus \cdots \oplus{V_{k_{r}}}$ be the standard ${\textrm{SL}}_{2}$ -module and write $A^{G}$ for the ring of invariants. We consider $\textrm{Proj}(A^{G})$ , where the grading is induced by the degree of an invariant. We are interested in the affine open $U=D_{+}(\Delta )$ for $\Delta$ the discriminant. The set of $L$ -valued points $U(L)$ of $U$ can be identified with tuples of binary forms $(f_{1},\ldots,f_{r})$ over $L$ without common zeros and up to ${\textrm{GL}}_{2}$ -equivalence. Here $\textrm{deg}(f_{i})=k_{i}$ .

We now consider an $L$ -valued point of $U$ . This gives a set of binary forms $(f_{1},\ldots,f_{r})$ over $L$ . We write $f=\prod _{i=1}^{r}f_{i}$ for the resulting $(k_{1} + \cdots + k_{r})$ -binary form. Let $Z(f)$ and $Z(f_i)$ denote the zero sets of $f$ and $f_i$ , respectively, for $1 \leq i \leq r$ . Note that $Z(f) = \sqcup _{i=1}^{r} \ Z(f_i)$ gives a well-defined $L$ -valued point of the moduli space ${M}_{0,n}/S_{\Xi }$ , where $n = k_1 + \cdots + k_r$ and $\Xi = \{ \{1, \ldots, k_1\}, \{k_1 +1, \ldots, k_1 + k_2\}, \ldots, \{ k_1 + \cdots + k_{r-1} + 1, \ldots, n\} \}$ . Indeed, taking a different equivalent binary form changes the zero sets by the action of ${\textrm{GL}}_{2}$ , so that the induced map

\begin{equation*} \left (\mathbb {P}^{1}, \ \bigsqcup _{i=1}^{r} Z(f_i)\right ) \to \left (\mathbb {P}^{1}, \ \bigsqcup _{i=1}^{r} Z(f_i^\sigma )\right ) \end{equation*}

is an isomorphism, which means that we obtain the same point in ${M}_{0,n}/S_{\Xi }$ .

The partition on the Berkovich analytification of $M_{0,n}/S_{\Xi }$ given in Remark 26 induces a partition of $U^{\textrm{an}}$ . We use this as our definition of a tropical invariant of a binary form.

For a given set of invariants $h_{i}$ , we obtain a rational map $U\to \mathbb{P}(a_{0},\ldots,a_{n})$ , where $\textrm{deg}(h_{i})=a_{i}$ . If the $h_{i}$ include a set of generators of the ring of invariants, then this is automatically a morphism since the set of generators generate the nullcone. If the set of generators is a projective tropical separating set, then we call this a set of projective tropical invariants.

As promised in Section 2.1.2, we now show that the invariants $I_{4}$ , $I_{8}$ , $I_{12}$ , $I_{18}$ , and $\Delta$ are not enough to distinguish between the unmarked trees of a quintic.

Example 32. Let $K=\mathbb{C}\{\{t\}\}$ be the field of Puiseux series over the complex numbers with $v(t)=1$ . Consider the two quintics given by

\begin{align*} f_{2}& = xz(x-z)(x-t^2z)(x-2z),\\[5pt] f_{3}& = xz(x-z)(x-tz)(x-(1+2t)z). \end{align*}

The tree of $f_{2}$ is of Type $\textrm{II}$ and the tree of $f_{3}$ is of Type $\textrm{III}$ . However, in both cases, the tropical invariants are the same:

\begin{equation*} [v(I_{4}) \;:\; v(I_{8}) \;:\; v(I_{12}) \;:\; v(I_{18}) \;:\; v(\Delta )] = [0\;:\;0\;:\;0\;:\;2\;:\;4]. \end{equation*}

We thus see that we cannot distinguish between the two tree types using these invariants. This also implies that we cannot distinguish the various reduction types of Picard curves using these invariants. They are however enough to decide whether a tree is of Type $\textrm{I}$ or not.

The results in this paper show that there exists a set of projective tropical invariants for quintics and $(4,1)$ -forms. In general, we conjecture that there exists a finite set of projective tropical invariants for any set of binary forms. Moreover, there should be a practical algorithm that can calculate these tropical invariants.

4.1. Universal families

In this section, we explain the general idea of the proofs. We begin by recalling the setup in Sections 3.3 and 3.4 in the case of binary quintics and $(4,1)$ -forms.

Let $f$ be a separable binary quintic or $(4,1)$ -form over $K$ . The zero set $Z(f)$ of $f$ consists of five points in $\mathbb{P}^{1}$ , which we also consider as points of the corresponding Berkovich analytification $\mathbb{P}^{1,\textrm{an}}$ . These are often called points of type $1$ in the literature. These five points define a natural metric tree $\Sigma _{f}\subset \mathbb{P}^{1,{\textrm{an}}}$ that contains $Z(f)$ . If we choose a marking $\{1,\ldots,5\}\to Z(f)$ , then this is the minimal Berkovich skeleton of the marked curve $(\mathbb{P}^{1},P_{1},\ldots,P_{5})$ as defined in [Reference Baker, Payne and Rabinoff4, Definition 4.20] in terms of semistable vertex sets. If $f=(f_{4},f_{1})$ is a $(4,1)$ -form, then we assume that the marking is chosen so that $P_{1},\ldots,P_{4}\in Z(f_{4})$ .

The group ${\textrm{GL}}_{2}(K)$ acts on $\mathbb{P}^{1,{\textrm{an}}}$ through Möbius transformations. The individual $\Sigma _{f}$ is not invariant under this action, since $Z(f)$ is not invariant. The abstract marked metric tree $\Sigma _{f}$ however is invariant, up to suitable permutations of the markings. More precisely, if $f$ is a binary quintic, then we consider $\Sigma _{f}$ as an element of the moduli space $M^{\textrm{trop}}_{0,5}/S_{5}$ of unmarked metric trees on five leaves, and if $f=(f_{4},f_{1})$ is a $(4,1)$ -form, then we consider $\Sigma _{f}$ as an element of the moduli space $M^{\textrm{trop}}_{0,5}/S_{4}$ of metric trees with five leaves and one marked point. For every separable binary form in the same $\textrm{GL}_{2}$ -orbit, this gives the same metric tree, so that we can associate an element of $M^{\textrm{trop}}_{0,5}/S_{5}$ or $M^{\textrm{trop}}_{0,5}/S_{4}$ with any $\textrm{GL}_{2}$ -equivalence class of separable binary quintics or $(4,1)$ -forms. Note that if two separable binary forms lie in the same $\textrm{GL}_{2}$ -orbit, then they also have the same projective invariants since a transformation $\sigma$ scales an invariant $I$ by $\textrm{det}(\sigma )^{\textrm{deg}(I)}$ , see Remark 9. To summarize, applying a transformation in $\textrm{GL}_{2}$ to a binary quintic or $(4,1)$ -form does not change

1. 1. the metric tree of the binary form,

2. 2. the projective invariants of the binary form.

For a given binary quintic or $(4,1)$ -form defined over a valued field, we can now apply a projective transformation so that the resulting binary form is

(13) \begin{equation} f(x,z) = xz(x-z)(x-\lambda _{1}z)(x-\lambda _{2}z). \end{equation}

Here, we send the marked point to $\infty$ . By the discussion above, this does not change the associated metric tree, nor the projective invariants. We now view the binary form in (14) as being defined over the ring $A_{0}=K[\lambda _{1},\lambda _{2}]$ . The latter is the coordinate ring of a two-dimensional affine space $Y_{0}=\mathbb{A}^{2}_{K}$ over $K$ . For specializations of the $\lambda _{i}$ outside the vanishing locus $Z(\Delta _{\lambda })$ of $ \Delta _{\lambda } \;:\!=\; \lambda _{1}\lambda _{2}(\lambda _{1}-1)(\lambda _{2}-1)(\lambda _{1}-\lambda _{2})$ , we again obtain a separable binary form. We write $Y$ for this open subspace of $Y_{0}$ and $U$ for the space of separable binary forms as in Section 3.4. We then have a natural map

\begin{equation*}Y\to U\end{equation*}

sending $(\lambda _{1},\lambda _{2})$ to the separable binary quintic in (14). This map is surjective by construction.

We will write down parametrizations of the strata of $Y^{{\textrm{an}}}$ that correspond to the different tree types in $U^{{\textrm{an}}}$ . This will be done in terms of universal algebras. For each tree type, we will give a free algebra $A=R[t_{i},\mu _{i}]$ with an injective map $R[\lambda _{1},\lambda _{2}] \to A$ . We will view this as an inclusion, so that $\lambda _{i}\in{A}$ . Note that the binary form in (14) automatically becomes a binary form over $A$ through this map. Let $R_{L}$ be the valuation ring of a valued field extension $K\subset{L}$ . We then consider $R_{L}$ -valued points $\psi :A\to{R_{L}}$ with $v(\psi (t_{i}))\gt 0$ . Such an $R_{L}$ -valued point corresponds to a specific binary form of the given type. We will also write $v(t_{i})\gt 0$ if $\psi$ is understood. The universal algebras are constructed in such a way that every separable binary form of the given type occurs as an $R_{L}$ -valued point with $v(t_{i})\gt 0$ .

Table 2.

The chosen universal families. The $t_{i}$ parametrize the different edge lengths and the $\mu _{i}$ the different algebraic residue classes.

To explicitly construct these universal families, we fix a tree type and write down the implied conditions on the valuations of the $\lambda _{i}$ . We then parametrize these conditions using additional parameters $t_{i}$ and $\mu _{i}$ . Our choices for these universal families can be found in Table 2, see Figure 7 for the corresponding trees.

Figure 7.

Trees corresponding to the universal families in Table 2.

Example 34. To give an example, let us explain the universal family for Type III.2 in more detail. By assumption, we have the conditions

\begin{equation*} v(\lambda _{1}) \gt 0 \quad \text { and } \quad v(\lambda _{2}-1) \gt 0. \end{equation*}

Note that we can assume by permuting the $\lambda _{i}$ that $v(\lambda _{1})\leq v(\lambda _{2}-1)$ . To parametrize the different edge lengths, we then write

\begin{equation*} \lambda _{1}=t_{1}\mu _{1} \quad \text { and } \quad \lambda _{2}=1+t_{2}\mu _{2}. \end{equation*}

Here, we specialize the $\mu _{i}$ to elements with valuation zero and the $t_{i}$ to elements in $R$ , so that $v(t_{1})=v(\lambda _{1})$ and $v(t_{2})=v(\lambda _{2}-1)$ . In particular, note that the two non-trivial edge lengths are given exactly by $v(t_{i})$ . By our assumption on the edge lengths, we then have $v(t_{1})\leq v(t_{2})$ .

In these universal algebras, we have various natural reduction maps. We will be particularly interested in reducing elements modulo $I=\mathfrak{m}+(t_{1},t_{2})$ . Here, we think of the $t_{i}$ as elements of $\mathfrak{m}$ whose valuations we can freely alter. In this vein, we will also refer to working modulo $I$ as working modulo $\mathfrak{m}$ . For a given tree type with universal algebra $A$ , we can consider the universal binary form $f$ from (14) as being defined over $A$ . The invariants of this binary form are thus elements of $A$ .

In particular, we now see that for any specialization $\psi :A\to{R_{L}}$ , we have that $\psi (I)\in{R_{L}}$ . We write $I\in{A^{\times }}$ if for every specialization $\psi$ as above, we have that $v(\psi (I))=0$ .

4.2. Proof of Theorem 1

In this subsection, we prove Theorem 1. In the proofs, we are free to choose the universal family of either $\textrm{II}.1$ or $\textrm{II}.2$ , and similarly for $\textrm{III}.1$ and $\textrm{III}.2$ . We do the necessity of the statement first and finish with the sufficiency in Section 4.2.4.

4.2.1. Type I

Computing the reduction modulo $\mathfrak{m}$ of the discriminant $\Delta$ , we obtain

\begin{equation*} \overline {\Delta } = \overline {\lambda }_1^2 \ \overline {\lambda }_2^2 \ (\overline {\lambda }_1 - 1)^2 \ (\overline {\lambda }_2 - 1)^2 \ ( \overline {\lambda }_1 -\overline {\lambda }_2 )^2. \end{equation*}

So we deduce that $v(\Delta ) = 0$ , hence the condition $8 v(I) - \deg (I)v(\Delta ) \geq 0$ is satisfied for any $I \in S$ .

4.2.2. Type II

Consider the universal family for Type $\textrm{II}.1$ in Table 2. We calculate the invariants $I \in S$ and find that they are divisible by $t_{1}^{\textrm{deg}{I}/2}$ . The reduction of $H/t_{1}^{\textrm{deg}(H)/2}$ modulo $\mathfrak{m}$ is

\begin{equation*} \overline {H/t_{1}^{\textrm {deg}(H)/2}}=-22\overline {\mu }_{1}^2\overline {\mu }_{2}^2(\overline {\mu }_{1}-\overline {\mu }_{2})^2. \end{equation*}

Since $p\neq{2,11}$ , we find that the latter is non-zero. We thus obtain

\begin{equation*} v(H) = \deg (H) v(t_1)/2 = 6 v(t_1), \quad \text {and} \quad v(I) \geq \deg (I)v(t_1)/ 2 \quad \text {for } I \in S\setminus \{ H \}. \end{equation*}

Therefore,

\begin{equation*} 12v(I)-\textrm {deg}(I)v(H)\geq {0} \text { for all } I\in {S}. \end{equation*}

On the other hand, computing the reduction modulo $\mathfrak{m}$ of $I_{4}/t_{1}^2$ and $I_{18}/t_{1}^{9}$ , we find that

\begin{align*} \overline{I_{4}/t_{1}^{2}}&=-2(\overline{\mu }_{1}^2-\overline{\mu }_{1}\overline{\mu }_{2}+\overline{\mu }_{2}^{2}),\\[5pt] \overline{I_{18}/t_{1}^{9}}&=-\overline{\mu }_{1}^2\overline{\mu }_{2}^2(\overline{\mu }_{1}-2\overline{\mu }_{2})(\overline{\mu }_{1}+\overline{\mu }_{2})(2\overline{\mu }_{1}-\overline{\mu }_{2})(\overline{\mu }_{1}-\overline{\mu }_{2})^2. \end{align*}

It is not so hard to check that, since $p\neq{3}$ , the quantities $\overline{I_{4}/t_{1}^{2}}$ and $\overline{I_{18}/t_{1}^{9}}$ cannot be simultaneously zero. We thus find that

\begin{equation*} v(I_{4}) = 2 v(t_1) \quad \text {or} \quad v(I_{18}) = 9 v(t_1). \end{equation*}

Our computations give $v(\Delta/ t_1^{6}) \geq 0$ , so we deduce that $v(\Delta ) \gt 4 v(t_1)$ and thus

\begin{equation*} v(\Delta ) - 2v(I_{4}) \gt 0 \quad \text {or} \quad 9v(\Delta ) - 4v(I_{18}) \gt 0. \end{equation*}

Combining these, we obtain the statement of the theorem.

4.2.3. Type III

Consider the universal family for Type $\textrm{III}.2$ in Table 2. We compute the reduction of $I_{4}$ modulo $\mathfrak{m}$ to obtain

\begin{equation*} \overline {I}_4 = -2. \end{equation*}

So, since $p \neq 2$ , we deduce that $v(I_4) = 0$ . Computing $\Delta$ and $H$ , we obtain that $\Delta$ is divisible by $t_{1}^{2}t_{2}^{2}$ and $\overline{H}=0$ . So we deduce that

\begin{equation*} v(\Delta ) - 2 v(I_4) \gt 0 \quad \text {and} \quad v(H) - 3 v(I_4) \gt 0. \end{equation*}

4.2.4. Finishing the proof

Finally, we check sufficiency. Suppose that the condition in (I) is satisfied. In particular,

\begin{equation*} 8v(I_4)-4v(\Delta )\geq {0} \text { and } 8v(I_{18})-18v(\Delta )\geq {0}. \end{equation*}

But these imply that

\begin{equation*} v(\Delta ) - 2v(I_4)\leq 0 \text { and } 9v(\Delta ) - 4v(I_{18})\leq 0, \end{equation*}

so the conditions in (II) and (III) cannot be satisfied. Now suppose that the conditions in (II) are satisfied. The condition

\begin{equation*} v(\Delta )- 2v(I_4) \gt 0 \text { or } 9v(\Delta ) - 4v(I_{18}) \gt 0 \end{equation*}

gives

\begin{equation*} 8v(I_4)-\deg (I_4)v(\Delta ) \lt 0 \text { or } 8v(I_{18})-\deg (I_{18})v(\Delta ) \lt 0, \end{equation*}

and therefore, the condition in (I) cannot be satisfied. On the other hand, the inequality $12v(I_4) - 4v(H) \geq 0$ implies $v(H) - 3v(I_4)\leq 0$ , which means that the conditions in (III) cannot be satisfied either. Finally, suppose that the conditions in (III) are satisfied. Hence,

\begin{equation*} 8v(I_4)-\deg (I_4)v(\Delta ) \lt 0 \text { and } 12v(I_4)-\deg (I_4)v(H) \lt 0, \end{equation*}

so that the conditions in (I) and (II) cannot be satisfied. This finishes the proof.

4.3. Proof of Theorem 2

In this subsection, we prove Theorem 2. The strategy is the same as in Section 4.2: we first calculate the invariants for each universal family and show that the given inequalities hold. This gives the necessity of the conditions. The sufficiency in this case is trivial, so this concludes the proof. We start with trees of Type $\textrm{II}.1$ since the marking does not matter for trees of Type $\textrm{I}$ .

4.4. Proof of Theorem 3

For trees of Type $\textrm{I}$ , there is nothing to prove.

4.4.1. Type II

We assume that the quintic $f$ has tree Type $\textrm{II}$ and use the universal family for Type $\textrm{II}.1$ in Table 2. Recall that the length $L(e_1)$ of the unique edge in the tree in this case is simply the valuation of $t_1$ .

We compute the invariants $\Delta, I_4$ , and $I_{18}$ and find that $\Delta \in t_1^6{A}$ , $I_4 \in t_1^2{A}$ , and $I_{18} \in t_1^{9}A$ , where $A$ is the universal algebra in Section 4.1. Computing the reductions, we see

\begin{align*} \overline{\Delta/ t_1^6} &= \overline{\mu }_1^2 \ \overline{\mu }_2^2 \ (\overline{\mu }_1 - \overline{\mu }_2)^2, \\[5pt] \overline{I_4/t_1^2} &= -2 \ (\overline{\mu }_{1}^{2} - \ \overline{\mu }_{1} \overline{\mu }_{2} + \ \overline{\mu }_{2}^{2}), \\[5pt] \overline{I_{18}/ t_1^9} &= - \overline{\mu }_1^2 \ \overline{\mu }_2^2 \ (\overline{\mu }_1 + \overline{\mu }_2) \ (\overline{\mu }_1 - 2 \overline{\mu }_2) \ (2 \overline{\mu }_1 - \overline{\mu }_2) \ (\overline{\mu }_1 - \overline{\mu }_2)^2. \end{align*}

Notice that, since $p \neq 3$ , the two quantities $\overline{I_4/t_1^2}$ and $\overline{I_{18}/ t_1^9}$ cannot vanish simultaneously. So we obtain

\begin{align*} v(\Delta ) = 6 v(t_1) \quad \text{and} \quad \left (v(I_4) = 2 v(t_1) \quad \text{or} \quad v(I_{18}) = 9 v(t_1) \ \right ). \end{align*}

From this, we deduce that

\begin{equation*} v(t_1) = \frac {1}{2} \left (v(\Delta ) - 2v(I_4)\right ) \quad \text {or} \quad v(t_1) = \frac {1}{3} \left (2 v(\Delta ) - v(I_{18})\right ), \end{equation*}

and the valuation of $t_1$ , the length of the unique non-trivial edge, is then the maximum of these two quantities, i.e.,

\begin{equation*} L(e_1) = \max \left (\frac {1}{2} \left (v(\Delta ) - 2v(I_4)\right ), \ \ \frac {1}{3} \left (2 v(\Delta ) -v(I_{18})\right ) \right ). \end{equation*}

4.4.2. Type III

Before we give the proof of Theorem 3 for trees of Type $\textrm{III}$ , we shortly discuss the various formulas occurring in that theorem. For a tree of Type $\textrm{III}$ , we can only recover the edge lengths from the quintic invariants up to a permutation of the edges, see Example 24. A set of representatives of the edges here is given by $(e_{1},e_{2})$ with $L(e_{1})\leq{L(e_{2})}$ . For trees of Type $\textrm{III}.2$ , this symmetry continues to hold, so the formulas do not change. For trees of Type $\textrm{III}.1$ , there is no such symmetry, meaning that we can single out the edge next to the marked point. See Section 4.4.3 for the formulas in this case.

Now suppose that $f$ has tree Type $\textrm{III}$ . Recall that the lengths $L(e_1)$ and $L(e_2)$ of the edges $e_1$ and $e_2$ are, respectively, $v(t_1)$ and $v(t_2)$ . Computing the invariants $\Delta, I_4$ and $I_{18}$ , we obtainFootnote ³

\begin{align*} \Delta = \ &{\boldsymbol{t}_{\textbf{1}}^{\textbf{2}} \ \boldsymbol{t}_{\textbf{2}}^{\textbf{2}}} \ \mu _1^2 \ \mu _2^2 \ (t_2 \mu _2 + 1)^2 \ (\!-\!t_1 \mu _1 + t_2 \mu _2 + 1)^2 \ (t_1 \mu _1 - 1)^2 \end{align*}

\begin{align*} I_{18} = \ &{ \boldsymbol{(\mu _1 t_1 - \mu _2 t_2) (\mu _1 t_1 + \mu _2 t_2)}} (\!-\!\mu _1 t_1 + \mu _2 t_2 + 2) (\mu _2^2 t_2^2 - \mu _1 t_1 + 2 \mu _2 t_2 + 1)\\[5pt] \ &( \mu _2^2 t_2^2 + \mu _1 t_1 - 1) (\!-\!2 \mu _1 \mu _2 t_1 t_2 + \mu _2^2 t_2^2 - \mu _1 t_1 + 2 \mu _2 t_2 + 1) (\mu _1 \mu _2 t_1 t_2 - \mu _2 t_2 - 1)\\[5pt] \ &( \mu _1 \mu _2 t_1 t_2 + \mu _2 t_2 + 1) (\mu _1 \mu _2 t_1 t_2 - \mu _1 t_1 + 1) (\mu _1 \mu _2 t_1 t_2 + \mu _1 t_1 - 1) \\[5pt] \ &( \mu _1 \mu _2 t_1 t_2 + \mu _1 t_1 - 2 \mu _2 t_2 - 1) (\mu _1 \mu _2 t_1 t_2 + 2 \mu _1 t_1 - \mu _2 t_2 - 1) (\!-\!\mu _1^2 t_1^2 + \mu _2 t_2 + 1)\\[5pt] \ &(\!-\!\mu _1^2 t_1^2 + 2 \mu _1 \mu _2 t_1 t_2 + 2 \mu _1 t_1 - \mu _2 t_2 - 1) (\mu _1^2 t_1^2 - 2 \mu _1 t_1 + \mu _2 t_2 t_2 + 1) \end{align*}

and $I_4 \in A$ with $\overline{I}_4 = -2$ . So we deduce that

\begin{equation*} v(\Delta ) = 2 v(t_1) + 2 v(t_2) \quad \text {and} \quad v(I_4) = 0. \end{equation*}

If $v(t_1) \lt v(t_2)$ , then we find $v(I_{18}) = 2 v(t_1)$ and this gives

\begin{equation*} L(e_1) = v(t_1) = \frac {1}{2}\left (v(I_{18}) - \frac {9}{2} v(I_{4})\right ). \end{equation*}

If, on the other hand, $v(t_1) = v(t_2)$ , then we have

\begin{equation*} L(e_1) = v(t_1) = \frac {1}{4} \left ( v(\Delta ) - 2 v(I_4) \right ). \end{equation*}

Therefore, in both cases, we obtain

\begin{equation*} L(e_1) = v(t_1) = \min \left (\frac {1}{2}\left (v(I_{18}) - \frac {9}{2} v(I_{4})\right ), \frac {1}{4} \left ( v(\Delta ) - 2 v(I_4) \right ) \right ). \end{equation*}

The length of the second edge is $v(t_2)$ and can be computed using $\Delta$ as

\begin{equation*} L(e_2) = v(t_2) = \frac {1}{2} (v(\Delta ) - 2v(I_4)) - v(t_1). \end{equation*}

Hence, we deduce that

\begin{align*} L(e_1) &= \min \left (\frac{1}{2}\left (v(I_{18}) - \frac{9}{2} v(I_{4})\right ), \frac{1}{4} \left ( v(\Delta ) - 2 v(I_4) \right ) \right ),\\[5pt] L(e_2) &= \frac{1}{2} (v(\Delta ) - 2v(I_4)) - L(e_1). \end{align*}

4.4.3. Type III.1

Now let $(q,\ell )$ be a $(4,1)$ -form with tree Type $\textrm{III}.1$ . Let $e_{1}$ be the edge adjacent to $\infty$ and $e_2$ the second edge in the tree Type $\textrm{III}.1$ , see Figure 2. Using the universal family for Type $\textrm{III}.1$ and computing the invariants we find $j_2 \in t_1^2 A^{\times }$ , $j_5 = 1$ , $\Delta \in t_1^6 t_2^2 A$ and $I_4 \in t_1^2 A^{\times }$ and

\begin{equation*} \overline {j_2/ t_1^2} = \overline {\mu }_1^2, \quad \overline {\Delta/ (t_1^6 t_2^2)} = \overline {\mu }_1^4 \overline {\mu }_2^2, \quad \text {and} \quad \overline {I_4/ t_1^2} = - 2 \overline {\mu }_1^2. \end{equation*}

So we deduce that

\begin{equation*} L(e_1) = v(t_1) = \frac {1}{10}(5 v(j_2) - 2 v(j_5)), \end{equation*}

and

\begin{equation*} L(e_2) = v(t_2) = \frac {1}{2} \left ( v(\Delta ) - 2 v(I_{4}) \right ) - L(e_{1}). \end{equation*}

4.5. Proof of Corollary 4

We now recall from [Reference Helminck11] how the reduction type of a Picard curve $y^3\ell (x,z)=q(x,z)$ can be recovered from the $(4,1)$ -marked tree of $(q,\ell )$ . We refer the reader to [Reference Helminck11, Section 1.2] for the definition of the reduction type of a curve. We note here that our assumption that $K$ is algebraically closed is not restrictive. Namely, if we are interested in the reduction type of a curve over a complete discretely valued field $K$ , then its reduction type is completely determined by the reduction type of the base change over $\overline{K}$ , see [Reference Helminck11, Remark 3.6]. Finally, we note that the notion of an edge length used here is the same as the notion of thickness of the nodal point corresponding to the edge in question, which is used in other sources.

Let $X$ be a Picard curve over $K$ . The branch locus $B$ of the covering

\begin{equation*} X\to \mathbb {P}^{1} \end{equation*}

given by $[x\;:\;y\;:\;z] \mapsto [x:z]$ is the zero locus of $q\cdot \ell$ . The minimal skeleton of the marked curve $(\mathbb{P}^{1,{\textrm{an}}},B)$ is then the $(4,1)$ -marked tree of $(q,\ell )$ . By applying a projective transformation, we can assume that the zero of $\ell$ is $\infty$ .

For tame coverings, we have that the inverse image of a skeleton is a skeleton (see [Reference Helminck11, Theorem 3.1] or [Reference Helminck12, Theorem 1.1]), so we obtain a map

\begin{equation*} \Sigma ' \to \Sigma, \end{equation*}

where $\Sigma$ is the $(4,1)$ -marked tree and $\Sigma '$ is its inverse image under the morphism of Berkovich analytifications $X^{\textrm{an}}\to \mathbb{P}^{1,\textrm{an}}$ . Consider the dehomogenized polynomial $q=q(x,1)$ . The criteria in [Reference Helminck11, Section 3.1] allow us to reconstruct $\Sigma '$ explicitly in terms of the piecewise-linear function $ - \textrm{log}|q|$ on $\mathbb{P}^{1, \textrm{an}} \backslash{B}$ . This function can in turn be obtained from potential theory. We then find that over an edge in $\Sigma$ , there are three edges if and only if the slope of $ - \textrm{log}|q|$ is divisible by three. If it is not divisible by three, then the length of an edge has to be divided by three. That is, the expansion factor $d_{e'/e}$ in this case is three. These data are enough to determine the skeleton for Picard curves, as the weights of the vertices are determined by the Riemann–Hurwitz conditions. The resulting graphs can be found in Figure 3.

Proof of Corollary 4. By Theorem 2, the $(4,1)$ -marked tree type of $(q,\ell )$ is determined by the tropical invariants. The edge lengths of the $(4,1)$ -marked tree type are then given by Theorem 3. To obtain the edge lengths for the curve $X$ , we use the formula

\begin{equation*} d_{e'/e}L(e')=L(e), \end{equation*}

where $d_{e'/e}$ is the expansion factor, see [Reference Amini, Baker, Brugallé and Rabinoff3, Definition 2.4, Theorem 4.23].