# Rational unicuspidal plane curves of low degree

Kristin DeVleming  Department of Mathematics and Statistics, University of
Massachusetts, Amherst, MA 01003, USA<EMAIL_ADDRESS>and  Nikita Singh
Department of Mathematics and Statistics, University of Massachusetts,
Amherst, MA 01003, USA<EMAIL_ADDRESS>
###### Abstract.

We completely classify all plane curves of degree at most $30$ with a unique
cuspidal (locally unibranch) singular point and rational normalization in
terms of the Newton pairs parameterizing the cusp. We distinguish between
prime and composite degree in the classification and study the relationship
between prime or composite degree and the number of distinct topological types
of cuspidal singularities. Motivated by wall-crossing in moduli of curves, we
also survey several geometric properties of rational unicuspidal plane curves.

##  1\. Introduction

It is a difficult open question to effectively classify all possible
singularities that can appear on a degree $d$ plane curve
$C\subset\mathbb{P}^{2}_{\mathbb{C}}$. In this paper, we explore the
classification question for rational unicuspidal curves and provide a complete
classification of low degree curves.

######  Definition 1.1.

A plane curve $C\subset\mathbb{P}^{2}_{\mathbb{C}}$ is cuspidal if it is
locally irreducible at all singular points. It is unicuspidal if is has a
unique singular point and this point is cuspidal.

######  Definition 1.2.

A projective curve $C$ is rational if it has rational normalization
$C^{\nu}\cong\mathbb{P}^{1}$.

Classifying the allowable cusps on plane curves is related to many difficult
conjectures and results in algebraic geometry, such as the Coolidge-Nagata
problem, determining the maximal number of cusps on a plane curve, or rigidity
questions related to the curve and its complement. More recently, the
classification of such curves has proved necessary for problems in moduli of
curves and surfaces. For instance, rational unicuspidal limits of smooth
quintic curves give rise, under semistable reduction, to hyperelliptic genus 6
curves (see, e.g. [ADL19, §7.1] or [DS22, Theorem D, Example 1.11]), and
similar rational unicuspidal plane curves of higher degree can give rise to
lower gonality (non-planar) limits of degree $d$ planar curves.

Geometrically, the rationality of the plane curve corresponds to its complete
replacement in semistable reduction. Given a generically smooth family of
genus $g\geq 2$ curves $\mathcal{C}$ over a pointed curve $0\in T$ with
central fiber $\mathcal{C}_{0}$, we may compute the stable replacement of
$\mathcal{C}_{0}$–the associated limit of the family in
$\overline{\mathcal{M}}_{g}$, the moduli space of stable genus $g$ curves–by
resolving the surface $\mathcal{D}\to\mathcal{C}$ and running a relative
minimal model program on $\mathcal{D}$ over $T$ (suppressing any base change
in the notation). The minimal model program contracts rational curves in the
fiber over $0$. If $\mathcal{C}_{0}$ is a rational unicuspidal plane curve,
the central fiber of the resolution $\mathcal{D}_{0}$ is a configuration of
curves with a rational tail that is the strict transform of $\mathcal{C}_{0}$.
This rational tail is contracted as the first step of the minimal model
program and $\mathcal{C}_{0}$ completely disappears via semistable reduction.
Furthermore, if no rational tails exist in $\mathcal{D}_{0}$, the surface
$\mathcal{D}$ is already minimal so is its own minimal model. In this sense,
rational unicuspidal curves are geometrically significant for moduli of curves
and the “starting point” for semistable reduction.

Their geometric interest is further illustrated in moduli of pairs, which is
the motivation for considering rational unicuspidal curves in
$\mathbb{P}^{2}$. For each $\alpha\in(0,1)\cap\mathbb{Q}$, there exists a
projective moduli space $\mathcal{M}_{d}^{\alpha}$ compactifying the space of
pairs $(\mathbb{P}^{2},\alpha C_{d})$ where $C_{d}$ is a smooth degree $d\geq
4$ plane curve. This space is constructed using K-stability for
$\alpha\in(0,\frac{3}{d})$ ([ADL19, Xu23]), the theory of KSB and KSBA
stability for $\alpha\in(\frac{3}{d},1)$ ([Kol23]), or using [ABB+23] for
$\alpha=\frac{3}{d}$. For each $\alpha$, this space parameterizes some notion
of stable pairs, but the stability condition changes as $\alpha$ varies, so
the objects parameterized by the boundary of the moduli space may change. By
[ADL19, ABIP23], there is a finite collection of values of $\alpha$ where the
stability condition changes, meaning that there exist rational numbers
$0<\alpha_{1}<\alpha_{2}<\dots<\alpha_{n}<1$ such that, for
$\alpha,\alpha^{\prime}\in(\alpha_{i},\alpha_{i+1})$,
$\mathcal{M}_{d}^{\alpha}\cong\mathcal{M}_{d}^{\alpha^{\prime}}$ and for each
$\alpha_{i}$ we have a diagram (up to seminormalization):

${\mathcal{M}_{d}^{\alpha_{i}-\epsilon}}$${\mathcal{M}_{d}^{\alpha_{i}+\epsilon}}$${\mathcal{M}_{d}^{\alpha_{i}}}$

The singularities of curves $C_{d}$ correspond to certain walls $\alpha_{i}$.
For example, if ${\mathrm{lct}}(C_{d})=\beta$ for some $\beta\geq\frac{3}{d}$,
then $\beta$ is a value corresponding to a wall-crossing, i.e.
$\beta=\alpha_{i}$ for some $i$. In general, in the wall-crossing for moduli
of pairs $(\mathbb{P}^{2},\alpha C_{d})$, the rational unicuspidal curves and
their intrinsic geometry control behavior of certain walls and surfaces that
may arise as degenerations of the ambient space, as the following examples
illustrate.

######  Example 1.3.

There exists a rational unicuspidal curve $C_{0}$ of degree $5$ with a cusp
locally analytically of the form $x^{2}+y^{13}=0$ (c.f. Theorem 2.9.4). This
curve appears in $\mathcal{M}_{5}^{\alpha}$ for $\alpha\ll 1$, but is unstable
for $\alpha>\frac{8}{15}$ [ADL19, §7]. In the associated moduli spaces of
quintic plane curves, there is a wall crossing at $\alpha_{i}=\frac{8}{15}$
where the pair $(\mathbb{P}^{2},(\alpha_{i}-\epsilon)C_{0})$ appears in the
moduli space $\mathcal{M}_{5}^{\alpha_{i}-\epsilon}$ but is replaced by
$(X_{26},(\alpha_{i}+\epsilon)C_{h})$ in
$\mathcal{M}_{5}^{\alpha_{i}+\epsilon}$. Here, $X_{26}$ is a partial smoothing
of the $\frac{1}{4}(1,1)$ singularity on $\mathbb{P}(1,4,25)$ and $C_{h}$ is a
hyperelliptic genus 6 curve. The birational geometry of the replacement is
shown in the following diagram, with details left to [DeV22, §5].

smooth quinticexceptional $E\cong\mathbb{P}(1,2,13)$singular points
fromweighted blow-
up$\tilde{C_{0}}$$(\tilde{C_{0}})^{2}=-1$$D$$t$$0$$\mathbb{A}^{1}_{t}$$Y_{t}=\mathbb{P}^{2}_{t}$$\tilde{X_{0}}$$\rightsquigarrow$
---
smooth quintic$E^{+}$$\tilde{C}_{0}^{+}$flip of $\tilde{C}_{0}$$D^{+}$strict
transform of
$D$$t$$0$$\mathbb{A}^{1}_{t}$$Y^{+}_{t}=\mathbb{P}^{2}_{t}$$\tilde{X_{0}}^{+}$$\rightsquigarrow$
---
smooth quintic$C_{0}$: quintic curve with$A_{12}$ singularity
$x^{2}+y^{13}$$t$$0$$\mathbb{A}^{1}_{t}$$X_{t}=\mathbb{P}^{2}_{t}$$X_{0}=\mathbb{P}^{2}_{0}$$\rightsquigarrow$
---
smooth quintic$\frac{1}{4}(1,25)$ singularity on $X_{26}$$C_{h}$ = image of
$D^{+}$smooth $C_{h}$ is
hyperelliptic$t$$0$$\mathbb{A}^{1}_{t}$$Z_{t}=\mathbb{P}^{2}_{t}$$\rightsquigarrow$
---
weighted blow upAtiyah flop$\scriptstyle{\text{contract
}\widetilde{X}_{0}^{+}}$ Figure 1. Replacement of the $A_{12}$ quintic curve.

The birational operations to produce the pair $(X_{26},C_{h})$ are explicitly
tied to geometric properties of the unicuspidal plane curve: first, $C_{0}$
must have been rational with minimal resolution $\widetilde{C}_{0}$ such that
$\widetilde{C}_{0}$ has negative self-intersection in the surface
$\widetilde{X}_{0}$ in order to be flopped from the top left diagram to the
top right diagram. Additionally, the complement $\mathbb{P}^{2}-C_{0}$ must
have had negative Kodaira dimension to be contracted to a point in the minimal
model, producing the normal surface $X_{26}$. These properties will be
explored in general in Section 3.

Furthermore, starting with a sufficiently general family of smooth quintics
degenerating to $C_{0}$, we see that the curve $D$ in the exceptional divisor
of the first weighted blow-up is smooth and itself is isomorphic to the stable
replacement of $C_{0}$. So, the geometry of $C_{0}$ in this example gives rise
to interesting smooth degenerations of quintic plane curves and a normal
degeneration of $\mathbb{P}^{2}$. In fact, this can be generalized for any
unicuspidal rational curve as in Theorem 2.9.4, with $2,5,13$ replaced by the
appropriate Fibonacci numbers.

######  Example 1.4.

There exists a rational octic curve $C_{0}$ with a unique cusp of the form
$y^{3}=x^{22}$ (c.f. Theorem 2.9.5). Consider a general family of smooth
octics $C_{t}$ degenerating to $C_{0}$ over $\mathbb{A}^{1}_{t}$. The log
canonical threshold of the curve $C_{0}$ is
$\frac{1}{3}+\frac{1}{22}<\frac{1}{2}$, so $(\mathbb{P}^{2},\frac{1}{2}C_{0})$
is not log canonical and is not a stable pair in $\mathcal{M}_{8}^{1/2}$. In
this case, computing the stable replacement of the limit
$(\mathbb{P}^{2},\frac{1}{2}C_{0})$ (the associated limit in the KSB moduli
space) gives rise to an slc surface of general type. Generically for $t\neq
0$, the fiber $(\mathbb{P}^{2},\frac{1}{2}C_{t})$ is log smooth, and the
associated double cover of $\mathbb{P}^{2}$ branched over $C_{t}$ is a smooth
Horikawa surface of general type. However, the pair
$(\mathbb{P}^{2},\frac{1}{2}C_{0})$ is not slc, so the associated branched
cover is not slc. To compute the limit of the family of Horikawa surfaces in
the KSB moduli space, we must find the stable replacement of
$(\mathbb{P}^{2},\frac{1}{2}C_{0})$. The birational geometry of computing the
replacement is indicated in the following diagram.

smooth octicexceptional $E\cong\mathbb{P}(1,3,22)$singular points fromweighted
blow-
up$\tilde{C_{0}}$$(\tilde{C_{0}})^{2}=-2$$D$$t$$0$$\mathbb{A}^{1}_{t}$$Y_{t}=\mathbb{P}^{2}_{t}$$\tilde{X_{0}}$$\rightsquigarrow$
---
smooth octic$C_{0}$: octic curve withcusp singularity
$y^{3}+x^{22}$$t$$0$$\mathbb{A}^{1}_{t}$$X_{t}=\mathbb{P}^{2}_{t}$$X_{0}=\mathbb{P}^{2}_{0}$$\rightsquigarrow$
---
smooth octicsingular pointimage of
$\tilde{C}_{0}$$D$$t$$0$$\mathbb{A}^{1}_{t}$$Z_{t}=\mathbb{P}^{2}_{t}$$\tilde{X_{0}}^{+}$$\rightsquigarrow$
---
weighted blow upsmall contraction Figure 2. Replacement of the rational octic
curve with cusp $y^{3}=x^{22}$.

If $f:Y\to Z$ is the small contraction of the threefolds illustrated above and
$\Delta$ is the divisor giving the family of curves in $Y$ (with image
$\Delta_{Z}$ in $Z$), note that
$(K_{Y}+\frac{1}{2}\Delta)\cdot\widetilde{C}_{0}=1+\frac{1}{2}(-1)=0$, so
$K_{Z}+\frac{1}{2}\Delta_{Z}$ is Cartier and relatively ample over
$\mathbb{A}^{1}_{t}$ by computation. Therefore, the replacement of
$(\mathbb{P}^{2},\frac{1}{2}C_{0})$ in $\mathcal{M}_{8}^{1/2}$ is the non-
normal surface $Z_{0}=\widetilde{X}_{0}^{+}\cup E$. The pair
$(Z_{0},\frac{1}{2}D)$ is log canonical and the double cover of this central
fiber is the KSB stable limit of the associated family of Horikawa surfaces.
One could similarly compute the replacement of this pair in
$\mathcal{M}_{8}^{1/2+\epsilon}$ after the
$K_{Y}+(\frac{1}{2}+\epsilon)\Delta$ flip of $\widetilde{C}_{0}$.

In both of these cases, the smoothness of the replacement curve is tied to the
rationality of the unicuspidal curve. The rationality allows the curve to be
the fiber of a small contraction, flip, or flop of the total space in the
first step of the minimal model problem. As the curve only intersects the
exceptional divisor transversally, this does not affect the smoothness of the
curve $D$ in the exceptional divisor.

Motivated by these examples, we expect that there is a relationship between
non-planar limits of degree $d$ plane curves and unicuspidal rational degree
$d$ plane curves where a wall crossing as above replaces the rational
unicuspidal curve with a smooth non-planar curve. This problem is explored in
more detail in [DS22], focusing on the case that $d$ is prime. More broadly,
however, to effectively enumerate the walls $\alpha_{i}$ in the moduli
problem, one needs an understanding of the possible plane curve singularities.
Thus, for many applications, the following question is essential:

######  Question 1.5.

What are the rational unicuspidal plane curves of degree $d$?

For any degree $d$, there always exists the rational unicuspidal plane curve
$x^{d-1}z=y^{d}$ (Theorem 2.9) as the generalization of the usual cuspidal
cubic. We are interested in other cuspidal singularities. Furthermore, the
work in [DS22] suggests that there exist very few examples of smooth non-
planar limits of smooth plane curves of prime degree, related to the existence
of prime degree rational unicuspidal plane curves. This relationship is not
sharp, but leads us to investigate the following refinement of the previous
question:

######  Question 1.6.

If $d$ is prime, when do there exist rational unicuspidal plane curves of
degree $d$ other than $x^{d-1}z=y^{d}$ (up to isomorphism)?

It turns out that the answer to this question is closely intertwined with many
longstanding open questions in number theory, notably the infinitude of
Fibonacci primes and one of Landau’s problems on primes of the form $n^{2}+1$.
We explore this in more detail in Section 4.

In what follows, we introduce two perspectives that have been historically
fruitful to yield partial answers to the problem. In Section 2, we classify
curve singularities by combinatorial properties associated to the Newton pairs
parameterizing the cusp, and in Section 3, we connect this perspective with
classification of curve singularities by the logarithmic Kodaira dimension of
the curve. The main results of this article are the following.

######  Theorem 1.7.

For degree $d\leq 30$, there is explicit list of all unicuspidal rational
plane curves of degree $d$, enumerated in Table LABEL:t:bigtable.

Additionally, we make two statements about the possible number of non-
isomorphic curves for composite versus prime degree.

######  Theorem 1.8.

For any integer $d>1$, there are at least as many non-isomorphic rational
unicuspidal plane curves of degree $d$ as there are ordered factorizations of
$d$.

######  Theorem 1.9.

There exist infinitely many prime degrees for which there exist at least two
non-isomorphic rational unicuspidal plane curves.

Conjecturally, there exist at least three different infinite families of prime
degrees for which there exists more than one rational unicuspidal plane curve.
However, we expect for a general prime number $p$, there only exists (up to
isomorphism) the rational unicuspidal curve $x^{p-1}z=y^{p}$. These
conjectures and expectations will be discussed in Section 4.

Acknowledgements. We would like to thank David Stapleton for helpful
conversations. Research of KD was supported in part by NSF grant DMS-2302163.

##  2\. Classification of cuspidal curves by their Newton pairs and associated
semigroup

We recall the classical invariants parametrizing a cusp. For additional
background references see [BK86, Ch. 8], [Liu14, Ch. 2], and [Moe08, Ch. 2].

Let $p\in C\subset X$ be a cuspidal point $p$ on the curve $C$ on a smooth
surface $X$. In an analytic neighborhood of $p$, the curve $C$ can be written
parametrically as

$(x,y)=(t^{a},c_{1}t^{b_{1}}+c_{2}t^{b_{2}}+\dots)$

with $1<a<b_{1}<b_{2}<\dots\in\mathbb{Z}$ such that $a$ does not divide
$b_{1}$, $\gcd(a,b_{1},b_{2},\dots)=1$, and $c_{i}\neq 0$ for each $i$. This
parametrization determines the geometry of the curve. We begin by recalling
several preliminary definitions.

######  Definition 2.1.

Let $C$ be a reduced curve with normalization $\nu:C^{\nu}\rightarrow C$ and
let $p\in C$ be a point. The $\delta$-invariant of $C$ at $p$ is the number

$\delta_{p}=\mathrm{length}_{p}\left(\nu_{*}(\mathcal{O}_{C^{\nu}})/\mathcal{O}_{C}\right).$

The sum of the $\delta$-invariants controls the difference between the
arithmetic and geometric genera of $C$:

$\chi(\mathcal{O}_{C^{\nu}})-\chi(\mathcal{O}_{C})=\sum_{p\in C}\delta_{p}.$

######  Remark 2.2.

If $C$ is a rational unicuspidal plane curve of degree $d$ with cusp at $p\in
C$, then we must have the $\delta_{p}$ is actually equal to the arithmetic
genus because $g(\mathcal{O}_{C^{\nu}})=0$. Therefore, we have the equality

$\delta_{p}=\frac{(d-1)(d-2)}{2}.$

######  Definition 2.3.

The multiplicity sequence of a cusp is the sequence encoding the multiplicity
of the exceptional divisors in the minimal resolution of the cusp. Let
$(X_{0},C_{0}):=(X,C)$ and let

$\pi_{i}:(X_{i},C_{i})\to(X_{i-1},C_{i-1})$

be the blow up of the singular point of $C_{i-1}$ with exceptional divisor
$E_{i}$. Set $C_{i}=(\pi_{i})^{-1}_{*}C_{i-1}$. Let
$\pi=\pi_{n}\circ\pi_{n-1}\dots\circ\pi_{1}$ be the minimal embedded
resolution of the cusp $p$. The multiplicity sequence of the cusp $p\in C$ is
the sequence $\overline{m}_{p}:=(m_{1},m_{2},\dots,m_{n})$, where $m_{i}$ is
the multiplicity of the exceptional divisor $E_{i}$ in $\pi_{i}^{*}(C_{i-1})$.
This satisfies the inequalities $m_{1}\geq m_{2}\geq\dots\geq m_{n}=1$. For
simplicity, as $C_{i-1}$ is smooth if and only if $m_{i}=1$ (and hence $C_{j}$
smooth for all $i\leq j\leq n$), we omit all the multiplicities $m_{j}=1$ for
$j\geq i$.

To further simplify the notation, we will write multiplicity sequences as

$\overline{m}_{p}=(m_{1},m_{2},\dots,m_{n})=((k_{1})_{i_{1}},(k_{2})_{i_{2}},\dots,(k_{n})_{i_{n}})$

where we collect repeated entries and the subscript indicates the number of
times each entry is repeated. For instance, instead of writing $(4,2,2,2,2)$,
we will write $(4,2_{4})$.

The $\delta$-invariant (Definition 2.1) of a cusp singularity can be read off
from its multiplicity sequence:

(2.1) $\delta_{p}=\sum_{i=0}^{n}\frac{m_{i}(m_{i}-1)}{2}$

######  Definition 2.4.

Another invariant of the cusp is the collection of Newton pairs that
parameterize the cusp. Define $g_{i}:=\gcd(a,b_{1},b_{2},\dots,b_{i})$. Then,
there is a finite sequence $i_{1}<i_{2}<\dots<i_{k}$ at which $g_{i}$
decreases, i.e. $i_{1}=1$,

$g_{i_{1}}=\dots=g_{i_{2}-1}>g_{i_{2}}=\dots=g_{i_{3}-1}>g_{i_{3}}=\dots>g_{i_{k}}=1.$

The monomials $t^{b_{i_{j}}}$ appearing in the parameterization are called the
characteristic terms.

Define $i_{0}=0$, $b_{0}=0$, and $g_{i_{0}}=a$. For $1\leq j\leq k+1$, let
$P_{j}=g_{i_{j-1}}$ and for $1\leq j\leq k$, let
$Q_{j}=b_{i_{j}}-b_{i_{j-1}}$. The $k$ Newton pairs of the cusp are the $k$
pairs

$(p_{j},q_{j})=\left(\frac{P_{j}}{P_{j+1}},\frac{Q_{j}}{P_{j+1}}\right)\text{
for }1\leq j\leq k.$

We can write $P_{j}=p_{j}p_{j+1}\dots p_{k}$ and $Q_{j}=q_{j}p_{j+1}\dots
p_{k}$. The $\delta$-invariant of a cusp singularity can also be expressed in
terms of the $P_{i}$ [Liu14, 2.1.1]:

(2.2)
$\delta_{p}=\frac{1}{2}\left((P_{1}-1)(Q_{1}-1)+\sum_{j=2}^{k}(P_{j}-1)Q_{j}\right).$

######  Remark 2.5.

The pairs $(P_{1},Q_{1}),(P_{2},Q_{2}),\dots,(P_{k},Q_{k})$ are called the
Puiseux pairs of the cusp. These can be read off from the Newton pairs, and
vice versa. The multiplicity sequence can also be determined from the Newton
pairs using an extended Euclidean algorithm as in [Moe08, Theorem 2.2.6], and
vice versa.

######  Definition 2.6.

The semigroup associated to a cuspidal singularity $p\in C$, denoted
$W_{p}\subset\mathbb{N}$, is the set of all possible local intersection
multiplicities of $C$ with other curves at $p$:

$W_{p}:=\\{\dim_{\mathbb{C}}\mathcal{O}_{C,p}/(f)\mid
f\in\mathcal{O}_{C,p},f\neq 0\\}\subset\mathbb{N}.$

The semigroup $W_{p}$ has a set of minimal generators
$\\{0,w_{1},w_{2},\dots,w_{k+1}\\}$, and each $w_{i}$ can be expressed in
terms of the Newton and Puiseux pairs of the cusp:

$w_{1}=P_{1},\quad w_{2}=Q_{1},\quad w_{j}=p_{j-2}w_{j-1}+Q_{j-1}\quad 3\leq
j\leq k+1.$

######  Example 2.7.

In the case of a cusp $p\in C$ with one Newton pair $(a,b)$, the curve can be
analytically locally parametrized by $(x,y)=(t^{a},t^{b})$ with $\gcd(a,b)=1$.
In this case, $C$ has analytic local equation $y^{a}=x^{b}$,
$\delta$-invariant $(a-1)(b-1)/2$, and semigroup $W_{p}=\langle 0,a,b\rangle.$

Finally, if $p\in C$ is a cusp with semigroup $W_{p}$, define

$R_{p}(k):=\\#\\{W_{p}\cap[0,k)\\}$

to be the counting function of elements in $W_{p}$ between $0$ and $k-1$. It
has been shown that if a curve $C$ is a cuspidal degree $d$ plane curve, these
counting functions must satisfy particular constraints related to the degree
$d$ of the curve. The multiplicity, Newton pairs, and intervals in the
semigroup have been widely used to study plane curves, e.g. [MS89, Ore02,
FdBLMHN07, BL14].

In [BL14, Thm. 6.5, Rem. 6.6], Borodzik and Livingston prove the following
strong result on existence of rational cuspidal curves:

######  Theorem 2.8.

[BL14] For a rational cuspidal curve of degree $d$ with $n$ cusps
$p_{1},\dots,p_{n}$ and counting functions $R_{p_{1}},\dots,R_{p_{n}}$, then
for any $j\in\\{-1,\dots,d-2\\}$,

(2.3) $\min_{\begin{subarray}{c}k_{1},\dots,k_{n}\in\mathbb{Z};\\\
k_{1}+\dots+k_{n}=jd+1\end{subarray}}(R_{p_{1}}(k_{1})+\dots+R_{p_{n}}(k_{n}))=\frac{(j+1)(j+2)}{2}.$

With this background, we will classify low degree curves by the number of
Newton pairs parameterizing the cusp using Equation 2.3. Currently, there
exists the following classification of unicuspidal curves with cusp
parameterized by one or two Newton pairs. We denote by $\phi_{j}$ the
Fibonacci numbers, defined by $\phi_{0}=0$, $\phi_{1}=1$, and
$\phi_{j+1}=\phi_{j}+\phi_{j-1}$.

######  Theorem 2.9.

[FdBLVMHN06, FdBLMHN07] A rational unicuspidal plane curve of degree $d$ with
singularity parametrized by one Newton pair $(p_{1},q_{1})$ exists if and only
if $(d,p_{1},q_{1})$ is in the following list.

  1. (1)

$d$ is arbitrary and $(p_{1},q_{1})=(d-1,d)$

  2. (2)

$d$ is even and $(p_{1},q_{1})=(d/2,2d-1)$

  3. (3)

$d=\phi_{j-1}^{2}+1=\phi_{j-2}\phi_{j}$ where $j\geq 5$ is odd and
$(p_{1},q_{1})=(\phi_{j-2}^{2},\phi_{j}^{2})$

  4. (4)

$d=\phi_{j}$ where $j\geq 5$ is odd and
$(p_{1},q_{1})=(\phi_{j-2},\phi_{j+2})$

  5. (5)

$d=8$ and $(p_{1},q_{1})=(3,22)$

  6. (6)

$d=16$ and $(p_{1},q_{1})=(6,43)$.

######  Theorem 2.10.

[Liu14, Bod16] A rational unicuspidal plane curve of degree $d$ with
singularity parametrized by two Newton pairs $(p_{1},q_{1}),(p_{2},q_{2})$,
exists if and only if $(d,p_{i},q_{i})$ appear in the following list.

  1. (1)

$d=\phi_{2k-1}\phi_{2k+1}(l\phi_{2k-1}^{2}+\phi_{2k-3}^{2})$ for some $k\geq
2$ and $l\geq 0$ but $(k,l)\neq(2,0)$, and

$(p_{1},q_{1}),(p_{2},q_{2})=(l\phi_{2k-1}^{2}+\phi_{2k-3}^{2},l\phi_{2k+1}^{2}+\phi_{2k-1}^{2}+2),(\phi_{2k-1}^{2},l\phi_{2k-1}^{2}+\phi_{2k-3}^{2})$

  2. (2)

$d=\phi_{2k+1}(l\phi_{2k-1}^{2}+\phi_{2k-3}^{2})$ for some $k\geq 3$ and
$l\geq 0$, and

$(p_{1},q_{1}),(p_{2},q_{2})=(l\phi_{2k-1}^{2}+\phi_{2k-3}^{2},l\phi_{2k+1}^{2}+\phi_{2k-1}^{2}+2),(\phi_{2k-1},l\phi_{2k-1}+\phi_{2k-5})$

  3. (3)

$d=nm$ for some $n\geq 3,m\geq 2$, and
$(p_{1},q_{1}),(p_{2},q_{2})=(n-1,n),(m,nm-1)$

  4. (4)

$d=2nm$ for some $n\geq 2,m\geq 2$, and
$(p_{1},q_{1}),(p_{2},q_{2})=(n,4n-1),(m,nm-1)$

  5. (5)

$d=n^{2}+1$ for some $n\geq 3$, and
$(p_{1},q_{1}),(p_{2},q_{2})=(n-1,n),(n,(n+1)^{2})$

  6. (6)

$d=8n^{2}+4n+1$ for some $n\geq 2$, and
$(p_{1},q_{1}),(p_{2},q_{2})=(n,4n+1),(4n+1,(2n+1)^{2})$

  7. (7)

$d=\phi_{4k+2}$ for some $k\geq 2$,
$(p_{1},q_{1}),(p_{2},q_{2})=(\phi_{4k}/3,\phi_{4k+4}/3),(3,1)$

  8. (8)

$d=2\phi_{4k+2}$ for some $k\geq 2$,
$(p_{1},q_{1}),(p_{2},q_{2})=(\phi_{4k}/3,\phi_{4k+4}/3),(6,1)$

Beyond the case of one or two Newton pairs, very little is known. The first
main result of this article is the following.

######  Theorem 2.11.

A unicuspidal rational curve of degree $d\leq 30$, with cusp parametrization
given by 3 Newton pairs, is one of the curves in Table 1.

Table 1. Rational unicuspidal curves with cusp parameterized by three Newton pairs. degree | Newton pairs |  sample parametrization111This parametrization only includes the characteristic terms in the local equation of the cusp, which are the data contained in the Newton pairs. A cusp with parametrization $(x,y)=(t^{a},t^{b_{i_{1}}}+t^{b_{i_{2}}}+\dots)$ where $g_{i_{j}}>g_{i_{j+1}}$ (notation as in Definition 2.4), has the same multiplicity sequence and collection of Newton pairs as $(x,y)=(t^{a},u_{i_{1}}t^{b_{i_{1}}}+u_{i_{2}}t^{b_{i_{2}}}+\dots)$ where each $u_{i_{j}}$ is a unit in the power series ring such that $\deg u_{i_{j}}t^{b_{i_{j}}}<b_{i_{j+1}}$ and for any monomial $t^{b}$ appearing in $u_{i_{j}}t^{b_{i_{j}}}$, $\gcd(a,b_{i_{j}},b)=\gcd(a,b_{i_{j}})$. | multiplicity sequence
---|---|---|---
12 | $(2,3),(2,5),(2,3)$ | $(x,y)=(t^{8},t^{12}+t^{22}+t^{25})$ | $(8,4_{4},2_{3})$
16 | $(2,7),(2,3),(2,3)$ | $(x,y)=(t^{8},t^{28}+t^{34}+t^{37})$ | $(8_{3},4_{3},2_{3})$
16 | $(3,4),(2,7),(2,3)$ | $(x,y)=(t^{12},t^{16}+t^{30}+t^{33})$ | $(12,4_{6},2_{3})$
18 | $(2,3),(2,5),(3,5)$ | $(x,y)=(t^{12},t^{18}+t^{33}+t^{38})$ | $(12,6_{4},3_{3},2)$
18 | $(2,3),(3,8),(2,5)$ | $(x,y)=(t^{12},t^{18}+t^{34}+t^{39})$ | $(12,6_{4},4,2_{4})$
19 | $(2,3),(2,7),(3,7)$ | $(x,y)=(t^{12},t^{18}+t^{39}+t^{46})$ | $(12,6_{5},3_{4})$
20 | $(4,5),(2,9),(2,3)$ | $(x,y)=(t^{16},t^{20}+t^{38}+t^{41})$ | $(16,4_{8},2_{3})$
24 | $(2,7),(2,3),(3,5)$ | $(x,y)=(t^{12},t^{42}+t^{51}+t^{56})$ | $(12_{3},6_{3},3_{3},2)$
24 | $(2,7),(3,5),(2,5)$ | $(x,y)=(t^{12},t^{42}+t^{52}+t^{57})$ | $(12_{3},6_{3},4,2_{4})$
24 | $(3,11),(2,5),(2,3)$ | $(x,y)=(t^{12},t^{44}+t^{54}+t^{57})$ | $(12_{3},8,4_{4},2_{3})$
24 | $(2,3),(2,5),(4,7)$ | $(x,y)=(t^{16},t^{24}+t^{44}+t^{51})$ | $(16,8_{4},4_{3},3)$
24 | $(2,3),(4,11),(2,7)$ | $(x,y)=(t^{16},t^{24}+t^{46}+t^{53})$ | $(16,8_{4},6,2_{6})$
24 | $(3,4),(2,7),(3,5)$ | $(x,y)=(t^{18},t^{24}+t^{45}+t^{50})$ | $(18,6_{6},3_{3},2)$
24 | $(3,4),(3,11),(2,5)$ | $(x,y)=(t^{18},t^{24}+t^{46}+t^{51})$ | $(18,6_{6},4,2_{4})$
24 | $(5,6),(2,11),(2,3)$ | $(x,y)=(t^{20},t^{24}+t^{46}+t^{49})$ | $(20,4_{10},2_{3})$
27 | $(2,3),(3,8),(3,8)$ | $(x,y)=(t^{18},t^{27}+t^{51}+t^{59})$ | $(18,9_{4},6,3_{4},2)$
28 | $(2,3),(3,10),(3,10)$ | $(x,y)=(t^{18},t^{27}+t^{57}+t^{67})$ | $(18,9_{5},3_{6})$
28 | $(6,7),(2,13),(2,3)$ | $(x,y)=(t^{24},t^{28}+t^{54}+t^{57})$ | $(24,4_{12},2_{3})$
30 | $(2,3),(2,5),(5,9)$ | $(x,y)=(t^{20},t^{30}+t^{55}+t^{64})$ | $(20,10_{4},5_{3},4)$
30 | $(2,3),(5,14),(2,9)$ | $(x,y)=(t^{20},t^{30}+t^{58}+t^{67})$ | $(20,10_{4},8,2_{8})$
30 | $(4,5),(2,9),(3,5)$ | $(x,y)=(t^{24},t^{30}+t^{57}+t^{62})$ | $(24,6_{8},3_{3},2)$
30 | $(4,5),(3,14),(2,5)$ | $(x,y)=(t^{24},t^{30}+t^{58}+t^{63})$ | $(24,6_{8},4,2_{4})$

For low degree curves, we also show that there is only one curve with more
than three Newton pairs.

######  Theorem 2.12.

For degree $d\leq 30$, the only unicuspidal rational curve of degree $d$ with
cusp given by $\geq 4$ Newton pairs is the curve in Table 2.

Table 2. Rational unicuspidal curves with cusp parameterized by at least four Newton pairs. degree | Newton pairs |  sample parametrization222See footnote 1. | multiplicity sequence
---|---|---|---
24 | $(2,3),(2,5),(2,3),(2,3)$ | $(x,y)=(t^{16},t^{24}+t^{44}+t^{50}+t^{53})$ | $(16,8_{4},4_{3},2_{3})$

Combining these results, we have:

######  Corollary 2.13.

(= Theorem 1.7) Any unicuspidal rational curve of degree $d$ for $1\leq d\leq
30$ appears in Theorems 2.9, 2.10, 2.11, or 2.12. The exhaustive list of all
possible unicuspidal rational curves of degree $d\leq 30$ appears in Table
LABEL:t:bigtable.

The key tool to prove Theorems 2.11 and 2.12 comes from low dimensional
topology and the study of the distribution of the semigroup associated to the
cusp as proven in [BL14] using Equation 2.3. This allows one to produce a list
of candidates for unicuspidal rational curves. From the list of potential
candidates, we use blow ups and blow downs to exhibit each curve as a
birational modification of a known unicuspidal rational curve of lower degree,
thus proving its existence.

###### Proof.

(Of Theorems 2.11 and 2.12.) First, note that that a bound on the number of
Newton pairs of the cusp in terms of the degree of the curve is derived in
[DS22, Remark 5.5]. This bound is far from sharp, but for a curve of degree
$d$ with cusp parameterized by $k$ Newton pairs, we have

$(d-1)(d-2)\geq(2^{k}-1)2^{k}.$

If $k=5$, this implies that $d>30$ so beyond the scope of Theorems 2.11 and
2.12. Thus, we only need to consider curves with cusps parameterized by three
or four Newton pairs.

We use a computer program to generate a list of possible cusp candidates
defined by three and four Newton pairs. The program is included in Appendix A.
We describe the program for three pairs (as the program for four is similar).

For three pairs, given a degree $d$, the program first generates a finite list
of possible characteristic terms of the parameterization $a,b_{1},b_{2},b_{3}$
and uses them to generate the related Puiseux pairs
$(P_{1},Q_{1}),(P_{2},Q_{2}),(P_{3},Q_{3})$ and Newton pairs
$(p_{1},q_{1}),(p_{2},q_{2}),(p_{3},q_{3})$ such that:

$\displaystyle P_{1}$ $\displaystyle=a=p_{1}p_{2}p_{3}$ $\displaystyle P_{2}$
$\displaystyle=\gcd(P_{1},b_{1})=p_{2}p_{3}$ $\displaystyle P_{3}$
$\displaystyle=\gcd(P_{2},b_{2})=p_{3}$ $\displaystyle Q_{1}$
$\displaystyle=b_{1}=q_{1}p_{2}p_{3}$ $\displaystyle Q_{2}$
$\displaystyle=b_{2}-b_{1}=q_{2}p_{3}$ $\displaystyle Q_{3}$
$\displaystyle=b_{3}-b_{2}=q_{3}$

where the maximum of $a,b_{i}$ is bounded by the genus of the curve.

The program then checks a series of properties:

  1. (1)

Validity for each Newton pair: $\gcd(p_{i},q_{i})=1$, and that the sequence
$\\{(P_{i},Q_{i})\\}$ satisfies the conditions in Definition 2.4.

  2. (2)

The $\delta$-invariant is equal to the genus of a smooth degree $d$ curve (so
the normalization is rational):
$\frac{(d-1)(d-2)}{2}=\frac{1}{2}\left((P_{1}-1)(Q_{1}-1)+(P_{2}-1)(Q_{2})+(P_{3}-1)(Q_{3})\right)$.

  3. (3)

The semigroup counting property 2.3 is satisfied: the generators for relevant
semigroup $W=\langle 0,w_{1},w_{2},w_{3},w_{4}\rangle$ are computed by

$\displaystyle w_{1}$ $\displaystyle=P_{1}$ $\displaystyle w_{2}$
$\displaystyle=Q_{1}$ $\displaystyle w_{3}$ $\displaystyle=p_{1}w_{2}+Q_{2}$
$\displaystyle w_{4}$ $\displaystyle=p_{2}w_{3}+Q_{3}$

Since there is only one cusp, the semigroup counting property says for any $l$
up to $d-2$, the size of $W\cap[0,ld)$ should be equal to
$\frac{(l+1)(l+2)}{2}$.

After these checks, we are left with a list of possible curves of degree $d$
defined by three Newton pairs. The program in fact yields the list in Theorem
2.11 and Theorem 2.12. However, the checks and semigroup counting property 2.3
do not guarantee the existence of such a curve. So, to complete the proof, we
must verify the existence of each cusp. We do this by explicit construction
with the following two lemmas.

######  Lemma 2.14.

Let $k$ and $n$ be positive integers, $n\geq 2$. Suppose $C$ is a unicuspidal
plane curve of degree $d=(k+1)n$ with multiplicity sequence of the cusp
$\overline{m}=(kn,n_{2k},\overline{m}^{\prime})$, where
$\overline{m^{\prime}}$ indicates the rest of the sequence. Then, $C$ can be
transformed by blow ups and blow downs to a unicuspidal curve $C^{\prime}$ of
degree $n$ with multiplicity sequence $\overline{m}^{\prime}$. In particular,
as these monoidal transformations are invertible, $C$ exists if and only if
$C^{\prime}$ exists.

###### Proof.

Suppose $C$ is a plane curve of degree $d=(k+1)n$ with a cusp $p\in C$ with
mulitplicity sequence $\overline{m}=(kn,n_{2k},\overline{m}^{\prime})$. For
$T$ the tangent line to $C$ at $p$, we have $m_{1}+m_{2}\leq(T\cdot C)_{p}\leq
d$ by [Moe08, (2.3)], so $(k+1)n\leq(T\cdot C)_{p}\leq(k+1)n$, so $(T\cdot
C)_{p}=T\cdot C=(k+1)n=d$.

Repeatedly blow up the intersection point of $C$ and $T$, and let $E_{i}$ be
the exceptional divisor of the $i$th blow up. After $k+1$ blow ups, denote by
$\widehat{C}$ and $\widehat{T}$ the strict transform of $C$ and $T$. The dual
graph of the configuration of curves is given below, where we adopt the
convention that we use straight edges to indicate transverse intersections (as
one would in a typical dual graph) but curved edges to indicate higher
multiplicity intersections:

$\widehat{C}$$\begin{array}[]{c}E_{1}\\\ -(k+1)\end{array}$
$E_{k+1}$$E_{k}$$-2$$E_{2}$$-2$$\widehat{T}$$-1$

Here, $\widehat{C}$ has multiplicity sequence $(n_{k},\overline{m}^{\prime})$
and $\widehat{C}\cdot E_{k+1}=n$. Here, the intersections of $E_{i}$ and $T$
are transverse, but the intersection of $\widehat{C}$ and $E_{k+1}$ is the
singular point of $\widehat{C}$. Contracting the chain $E_{k}\cup\dots\cup
E_{2}\cup\widehat{T}$ yields the Hirzebruch surface $\mathbb{F}_{k+1}$ with
negative section $E_{1}$ and fiber $E_{k+1}$. The curve $\widehat{C}$ does not
intersect the negative section.

On the Hirzebruch surface, blow up the cusp of $\widehat{C}$ (the intersection
of $\widehat{C}$ and $E_{k+1}$) $k$ times. Denote the exceptional divisor of
the $i$th blow up by $F_{i}$ and the strict transform of $\widehat{C}$,
$E_{k+1}$, and $E_{1}$ by $C^{\prime}$, $E_{k+1}^{\prime}$, and
$E_{1}^{\prime}$, respectively. From the multiplicity sequence of
$\widehat{C}$, the configuration of curves has dual graph:

$\begin{array}[]{c}E_{1}^{\prime}\\\ -(k+1)\end{array}$
$E_{k+1}^{\prime}$$-1$$F_{1}$$-2$$F_{k}$$-1$$C^{\prime}$

Here, $C^{\prime}\cdot F_{k}=n$ (meeting at the cusp of $C^{\prime}$) and
$C^{\prime}$ has multiplicity sequence $\overline{m}^{\prime}$. We contract
$E_{1}^{\prime}\cup E_{k+1}^{\prime}\cup F_{1}\cup\dots\cup F_{k-1}$ to a
smooth point in $\mathbb{P}^{2}$. The image of $F_{k}$ is a line in
$\mathbb{P}^{2}$ and as $C^{\prime}\cdot F_{k}=n$, $C^{\prime}$ is a degree
$n$ unicuspidal plane curve with multiplicity sequence
$\overline{m}^{\prime}$. ∎

Because $C^{\prime}$ has lower degree than $C$, this provides an inductive
approach to proving the existence of $C$. From this lemma, we can verify the
existence of all but two curves in Table 1. In Table 3, we use Lemma 2.14 to
reduce the existence of each curve $C$ to a lower degree curve $C^{\prime}$,
and in some cases reduce $C^{\prime}$ to a lower degree curve
$C^{\prime\prime}$.

Table 3. Proving existence of curves by reducing their degree. degree |  multiplicity sequence of $C$ |  existence of $C$ is equivalent to existence of $C^{\prime}$ with $d^{\prime}$, $\overline{m}^{\prime}$ |  existence of $C^{\prime}$ is equivalent to existence of $C^{\prime\prime}$ with $d^{\prime\prime}$, $\overline{m}^{\prime\prime}$
---|---|---|---
12 |  $(8,4_{4},2_{3})$ |  $d^{\prime}=4,\overline{m}^{\prime}=(2_{3})$ |
16 |  $(8_{3},4_{3},2_{3})$ |  $d^{\prime}=8,\overline{m}^{\prime}=(4_{3},2_{3})$ |  $d^{\prime\prime}=4,\overline{m}^{\prime\prime}=(2_{3})$
|  $(12,4_{6},2_{3})$ |  $d^{\prime}=4,\overline{m}^{\prime}=(2_{3})$ |
18 |  $(12,6_{4},3_{3},2)$ |  $d^{\prime}=6,\overline{m}^{\prime}=(3_{3},2)$ |
|  $(12,6_{4},4,2_{4})$ |  $d^{\prime}=6,\overline{m}^{\prime}=(4,2_{4})$ |
20 |  $(16,4_{8},2_{3})$ |  $d^{\prime}=4,\overline{m}^{\prime}=(2_{3})$ |
24 |  $(12_{3},6_{3},3_{3},2)$ |  $d^{\prime}=12,\overline{m}^{\prime}=(6_{3},3_{3},2)$ |  $d^{\prime\prime}=6,\overline{m}^{\prime\prime}=(3_{3},2)$
24 |  $(12_{3},6_{3},4,2_{4})$ |  $d^{\prime}=12,\overline{m}^{\prime}=(6_{3},4,2_{4})$ |  $d^{\prime\prime}=6,\overline{m}^{\prime\prime}=(4,2_{4})$
24 |  $(12_{3},8,4_{4},2_{3})$ |  $d^{\prime}=12,\overline{m}^{\prime}=(8,4_{4},2_{3})$ |  $d^{\prime\prime}=4,\overline{m}^{\prime\prime}=(2_{3})$
24 |  $(16,8_{4},4_{3},3)$ |  $d^{\prime}=8,\overline{m}^{\prime}=(4_{3},3)$ |  $d^{\prime\prime}=4,\overline{m}^{\prime\prime}=(3)$
24 |  $(16,8_{4},6,2_{6})$ |  $d^{\prime}=8,\overline{m}^{\prime}=(6,2_{6})$ |  $d^{\prime\prime}=2,\overline{m}^{\prime\prime}=(1)$ ($C^{\prime\prime}$ is smooth)
24 |  $(18,6_{6},3_{3},2)$ |  $d^{\prime}=6,\overline{m}^{\prime}=(3_{3},2)$ |
24 |  $(18,6_{6},4,2_{4})$ |  $d^{\prime}=6,\overline{m}^{\prime}=(4,2_{4})$ |
24 |  $(20,4_{10},2_{3})$ |  $d^{\prime}=4,\overline{m}^{\prime}=(2_{3})$ |
27 |  $(18,9_{4},6,3_{4},2)$ |  $d^{\prime}=9,\overline{m}^{\prime}=(6,3_{4},2)$ |  $d^{\prime\prime}=3,\overline{m}^{\prime\prime}=(2)$
28 |  $(24,4_{12},2_{3})$ |  $d^{\prime}=4,\overline{m}^{\prime}=(2_{3})$ |
30 |  $(20,10_{4},5_{3},4)$ |  $d^{\prime}=10,\overline{m}^{\prime}=(5_{3},4)$ |  $d^{\prime\prime}=5,\overline{m}^{\prime\prime}=(4)$
30 |  $(20,10_{4},8,2_{8})$ |  $d^{\prime}=10,\overline{m}^{\prime}=(8,2_{8})$ |  $d^{\prime\prime}=2,\overline{m}^{\prime\prime}=(1)$ ($C^{\prime\prime}$ is smooth)
30 |  $(24,6_{8},3_{3},2)$ |  $d^{\prime}=6,\overline{m}^{\prime}=(3_{3},2)$ |
30 |  $(24,6_{8},4,2_{4})$ |  $d^{\prime}=6,\overline{m}^{\prime}=(4,2_{4})$ |

For the only curve in Table 2, we can apply Lemma 2.14 twice to show that the
existence of $C$ is equivalent to the existence of a degree $8$ curve
$C^{\prime}$ with multiplicity sequence of the cusp
$\overline{m}^{\prime}=(4_{3},2_{3})$, which is equivalent to the existence of
a degree $4$ curve $C^{\prime\prime}$ with
$\overline{m}^{\prime\prime}=(2_{3})$.

In summary, we can reduce the existence of every curve to either a smooth
conic or a degree 4, 5, or 6 unicuspidal rational curve, and then appeal to
[DS22, Table 1] to verify their existence. Indeed, there exist curves of
degree 4 with multiplicity sequences $(2_{3})$ and $(3)$; there exists a curve
of degree 5 with multiplicity sequence $(4)$; and there exist curves of degree
6 with multiplicity sequences $(3_{3},2)$ and $(4,2_{4})$ which are explicitly
given in [DS22, Table 1].

The only curves in Table 1 in Theorem 2.11 that are not present in Table 3 are
the degree 19 curve and the first degree 28 curve. We use the following lemma
to verify the existence of these curves. In the notation below, if $a=3$ and
$s=2$ (respectively, $3$) this recovers the degree 19 curve (respectively, the
first degree 28 curve).

######  Lemma 2.15.

Let $C$ be the degree $a+1$ plane curve $zx^{a}=y^{a+1}$, which has a cuspidal
singularity as in Theorem 2.9.1 at $[0:0:1]$. By a prescribed sequence of blow
ups and blow downs, $C$ can be transformed into a rational unicuspidal plane
curve $C^{\prime}$ of degree $d=a^{2}s+1$ with multiplicity sequence
$((a-1)as,as_{2a-1},a_{2s})$. In particular, because $C$ exists, $C^{\prime}$
also exists.

###### Proof.

$C$ has multiplicity sequence $(a)$ at the point $p=[0:0:1]$. Let $T$ be the
inflectional line at the point $q=[1:0:0]$ given by $z=0$. This meets the
curve to order $a+1$ at $q$. Let $L$ be the line $y=0$ connecting $p$ and $q$,
so $L$ meets $C$ to order $a$ at $p$ and transversely at $q$.

Blow up $q$ and denote the exceptional divisor by $\Sigma$, and blow up the
intersection of $\Sigma$ and the strict transform of $L$ $s-1$ times, denoting
the exceptional divisor of the $i$th blow up by $E_{i}$ to obtain the dual
graph:

$\widehat{T}$$0$$\widehat{\Sigma}$$-s$${E}_{s-1}$$-1$$E_{s-2}$$-2$$E_{1}$$-2$$\widehat{L}$$-1$$\widehat{C}$

Here, $\widehat{C}$, $\widehat{T}$, and $\widehat{L}$ denote the strict
transforms, and the intersection of every pair of curves not including
$\widehat{C}$ is transverse. The intersection of $\widehat{C}$ and
$\widehat{T}$ occurs at a smooth point of $\widehat{C}$ and $\widehat{T}$ and
they meet to order $a$. This is also the (transverse) intersection point of
$\widehat{C}$ and $\widehat{\Sigma}$. The intersection of $\widehat{C}$ and
$\widehat{L}$ is the cusp with multiplicity sequence $(a)$. Now, contract the
chain $\widehat{L}\cup E_{1}\cup\dots\cup E_{s-2}$ to yield the Hirzebruch
surface $\mathbb{F}_{s}$. Let $\overline{C}$ be the image of $\widehat{C}$ and
$\overline{E}_{s-1}$ the image of $E_{s-1}$. The curve $\overline{C}$ has a
cusp with multiplicity sequence $(a_{s})$ where it meets $\overline{E}_{s-1}$.

Now, blow up a point on $\overline{E}_{s-1}$ away from the cusp of
$\overline{C}$ and away from the intersection with the negative section of
$\mathbb{F}_{s}$, denoting the exceptional divisor by $F_{1}$. Blow up a point
on $F_{1}$ away from the intersection with $\overline{E}_{s-1}$ and denote the
exceptional divisor by $F_{2}$. Repeat until we have a collection of $s$
exceptional divisors with dual graph:

$\widehat{T}$$0$$\widehat{\Sigma}$$-s$$\overline{E}_{s-1}$$-1$$F_{1}$$-2$$F_{s-1}$$-2$$F_{s}$$-1$$\overline{C}$

and now contract $\overline{E}_{s-1}\cup F_{1}\cup\dots\cup F_{s}$ to
$\mathbb{P}^{1}\times\mathbb{P}^{1}$ where the images of $\widehat{T}$ and
$\widehat{\Sigma}$ are the vertical and horizontal sections, respectively, and
the image of $\overline{C}$ meets $\widehat{T}$ to order $a$ at a smooth point
of both curves, and meets $\widehat{\Sigma}$ at two points: transversely at
the intersection of $\widehat{T}$ and $\widehat{\Sigma}$, and has a cusp with
multiplicity sequence $(a_{2s})$ at the other intersection point.

Next, blow up the point $t\in\widehat{T}$ that is intersection of
$\widehat{T}$ and $\widehat{\Sigma}$ $a$ times to separate $\widehat{T}$ and
$\overline{C}$, and contract the union of all but the last exceptional divisor
of this blow up and the strict transform of $\widehat{\Sigma}$. This produces
the Hirzebruch surface $\mathbb{F}_{a}$, with negative section the image of
$\widehat{T}$, and the image of the last exceptional divisor is a fiber
$G_{0}$ of this surface. The image of $C$, $\widetilde{C}$, is a unicuspidal
curve with multiplicity sequence $(as_{a},a_{2s})$ and meets the fiber $G_{0}$
in one smooth point $g_{0}$ at the cusp of $\widetilde{C}$.

Finally, blow up $g_{0}$ on $G_{0}$, denoting the exceptional divisor by
$H_{1}$. Let $h_{1}$ be the intersection of the strict transform of
$\widetilde{C}$ and $H_{1}$. Now, blow up $h_{1}$ on $H_{1}$, with exceptional
divisor $H_{2}$, and iterate until there exist $a-1$ exceptional divisors:

$\overline{T}$$-a$$\overline{G}_{0}$$-1$$H_{1}$$-2$$H_{a-2}$$-2$$H_{a-1}$$-1$$\widetilde{C}$

Contracting the union $\overline{T}\cup\overline{G}_{0}\cup H_{1}\cup\dots\cup
H_{a-2}$ yields the surface $\mathbb{P}^{2}$ with a unicuspidal curve
$C^{\prime}$ with multiplicity sequence $((a-1)as,as_{2a-1},a_{2s})$, as
claimed. Furthermore, the image of $H_{a-1}$ is a line in $\mathbb{P}^{2}$,
and computing the degree of $C^{\prime}$ via the intersection of
$\widetilde{C}$ and the pullback of this line, we find that $C^{\prime}$ has
degree $a^{2}s+1$. ∎

By Lemma 2.14 and Lemma 2.15, we have verified the existence of every curve
generated by the program in Appendix A, so this concludes the proof of
Theorems 2.11 and 2.12. ∎

##  3\. Classification of cuspidal curves by logarithmic Kodaira dimension

While the previous section yielded a classification of curves based on the
$\delta$-invariant of the singular point and its Newton pairs, we will now
connect that perspective with the logarithmic Kodaira dimension of the
complement of the curve.

######  Definition 3.1.

If $C\subset\mathbb{P}^{2}$, the Kodaira dimension of open surface
$\mathbb{P}^{2}-C$ is defined as $\kappa(\mathbb{P}^{2}-C):=\kappa(X,D)$ where
$\pi:X\to\mathbb{P}^{2}$ is a minimal log resolution of $C$ and
$D=\textrm{Supp}\pi^{-1}C$.

We summarize the current knowledge of the classification of unicuspidal
rational plane curves by Kodaira dimension. Just as in the case of
classification by Newton pairs, there is no complete classification in
general. However, it is complete if we assume $\kappa(\mathbb{P}^{2}-C)<2$.
For a more detailed detailed survey, see [FdBLVMHN06].

###  3.1. Kodaira dimension $-\infty$

If $\kappa(\mathbb{P}^{2}-C)=\infty$, then $C$ is either an AMS curve or a
curve classified by Kashiwara in [Kas87]. AMS curves, short for Abhyankar-Moh-
Suzuki, are described up to projective equivalence in [Ton00] and refer to
those unicuspidal curves $C$ with cusp $p$ such that the tangent line to $C$
at $p$ meets $C$ only at $p$. All curves in Lemma 2.14 are AMS curves.

####  3.1.1. AMS curves

The AMS curves are one of the following two types. Let $d=n_{1}\dots n_{r}$
where $r\geq 1$, $n_{i}$ are positive integers, where $n_{i}\geq 2$ for each
$i$.

  1. (1)

If $n_{1}>2$, the AMS curves are degree $d$ rational unicuspidal plane curves
with cusp parameterized by the $r$ Newton pairs

$(n_{1}-1,n_{1}),(n_{2},n_{1}n_{2}-1),(n_{3},n_{2}n_{3}-1),\dots,(n_{k},n_{k-1}n_{k}-1),\dots,(n_{r},n_{r-1}n_{r}-1).$

If $r=1$ and $d=n_{1}$, then the cusp just has one Newton pair

$(d-1,d).$

These are precisely the curves in Theorem 2.9.1.

For $r=2$, these are the curves given in Theorem 2.10.3.

  2. (2)

If $n_{1}=2$ (and necessarily $r\geq 2$), so $d=2n_{2}\dots n_{r}$, the AMS
curves are degree $d$ rational unicuspidal plane curves with cusp parametrized
by the $r-1$ Newton pairs

$(n_{2},4n_{2}-1),(n_{3},n_{2}n_{3}-1),\dots,(n_{k},n_{k-1}n_{k}-1),\dots,(n_{r},n_{r-1}n_{r}-1).$

If $r=2$ and $d=2n_{2}$, then the cusp just has one Newton pair
$(n_{2},4n_{2}-1)=(d/2,2d-1)$. These are precisely the curves in Theorem
2.9.2.

For $r=3$, these are the curves given in Theorem 2.10.4.

######  Remark 3.2.

By direct computation, the multiplicity sequence of the cusp of the first type
of AMS curve is $\overline{m}=((n_{1}-1)n_{2}\dots n_{r},(n_{2}\dots
n_{r})_{2(n_{1}-1)},\overline{m}^{\prime})$ and that of the cusp of the second
type is $\overline{m}=((n_{2}\dots n_{r})_{3},\overline{m}^{\prime})$. In
particular, Lemma 2.14 applies for all AMS curves with $k=n_{1}-1$, so they
can be explicitly constructed from lower degree curves via an inductive
process.

Note that in the construction of AMS curves, the order of the factorization
matters: for $d$ even, $d\geq 6$, the factorization $d=2n$ yields the cusp
with a single Newton pair $(d/2,2d-1)$, but the factorization $d=n\cdot 2$
yields a cusp with two Newton pairs $(n-1,n),(2,d-1)$. In other words, each
ordered factorization of $d$ gives rise to a topologically distinct
unicupsidal plane curve. We state this observation as a result.

######  Theorem 3.3.

(= Theorem 1.8) For any degree $d$, there are at least as many non-isomorphic
rational unicuspidal plane curves as there are ordered factorizations of $d$.

######  Remark 3.4.

Counting the number of ordered factorizations of an integer $n$ is known as
Kalmár’s problem [Kal30, Hwa00]. The associated sequence $m(n)$ of ordered
factorizations of $n$ can be found in [SI20, A074206] and has surprising
connections to the Riemann zeta function.

This is evidence of a relationship between non-primeness of degree with many
different cuspidal curves of that degree. See Section 4 for more on prime
degree curves.

####  3.1.2. Kashiwara curves

The Kashiwara curves are unicuspidal rational curves with Kodaira dimension
$\kappa(\mathbb{P}^{2}-C)=-\infty$ that are not AMS curves. They come in six
families. Following the notation in [FdBLVMHN06], we enumerate them below. In
what follows, $N\geq 1$ is a positive integer, $l\geq 0$ is a nonnegative
integer, and $\lambda_{i}\geq 0$ is a nonnegative integer. Furthermore, if
$l=0$, then $\lambda_{i}\geq 1$.

  1. (1)

Type $II_{ge}$. The curves of this type have degree $d=\phi_{2l+3}\phi_{2l+5}$
and cusp parameterized by a single Newton pair
$(\phi_{2l+3}^{2},\phi_{2l+5}^{2})$. These are the curves in Theorem 2.9.3.

  2. (2)

Type $II_{sp}$. The curves of this type has degree $d=\phi_{2l+3}$ and cusp
parameterized by a single Newton pair $(\phi_{2l+1},\phi_{2l+5})$. These are
the curves in Theorem 2.9.4.

  3. (3)

Type $II^{+}_{ge}$. The curves of type
$II^{+}(l,N;\lambda_{1},\dots,\lambda_{N})_{ge}$ have degree
$d=\phi_{2l+3}\phi_{2l+5}n_{1}\dots n_{N}$ where

$n_{i}=\left\\{\begin{array}[]{c}\lambda_{i}\phi_{2l+3}^{2}+\phi_{2l+3}\phi_{2l-1}-1\text{
for }i\text{ odd }\\\
\lambda_{i}\phi_{2l+3}^{2}+\phi_{2l+3}(\phi_{2l+3}-\phi_{2l-1})-1\text{ for
}i\text{ even }\end{array}\right.$

The cusp is parameterized by $N+1$ Newton pairs:

$\left(n_{1},\tfrac{\phi_{2l+5}^{2}n_{1}-1}{\phi_{2l+3}^{2}}\right),\left(n_{2},\tfrac{n_{1}n_{2}-1}{\phi_{2l+3}^{2}}\right),\dots,\left(n_{k},\tfrac{n_{k-1}n_{k}-1}{\phi_{2l+3}^{2}}\right),\dots,\left(n_{N},\tfrac{n_{N-1}n_{N}-1}{\phi_{2l+3}^{2}}\right),\left(\phi_{2l+3}^{2},n_{N}\right).$

  4. (4)

Type $II^{+}_{sp}$. The curves of type
$II^{+}(l,N;\lambda_{1},\dots,\lambda_{N})_{sp}$ have degree
$d=\phi_{2l+5}n_{1}\dots n_{N}$ where

$n_{i}=\left\\{\begin{array}[]{c}\lambda_{i}\phi_{2l+3}^{2}+\phi_{2l+3}\phi_{2l-1}-1\text{
for }i\text{ odd }\\\
\lambda_{i}\phi_{2l+3}^{2}+\phi_{2l+3}(\phi_{2l+3}-\phi_{2l-1})-1\text{ for
}i\text{ even }\end{array}\right.$

The cusp is parameterized by $N+1$ Newton pairs:

$\left(n_{1},\tfrac{\phi_{2l+5}^{2}n_{1}-1}{\phi_{2l+3}^{2}}\right),\left(n_{2},\tfrac{n_{1}n_{2}-1}{\phi_{2l+3}^{2}}\right),\dots,\left(n_{k},\tfrac{n_{k-1}n_{2}k-1}{\phi_{2l+3}^{2}}\right),\dots,\left(n_{N},\tfrac{n_{N-1}n_{N}-1}{\phi_{2l+3}^{2}}\right),\left(\phi_{2l+3},\tfrac{n_{N}+1}{\phi_{2l+3}}\right).$

  5. (5)

Type $II^{-}_{ge}$. The curves of type
$II^{-}(l,N;\lambda_{1},\dots,\lambda_{N})_{ge}$ have degree
$d=\phi_{2l+3}\phi_{2l+1}n_{1}\dots n_{N}$ where

$n_{i}=\left\\{\begin{array}[]{c}\lambda_{i}\phi_{2l+3}^{2}+\phi_{2l+3}\phi_{2l-1}-1\text{
for }i\text{ even }\\\
\lambda_{i}\phi_{2l+3}^{2}+\phi_{2l+3}(\phi_{2l+3}-\phi_{2l-1})-1\text{ for
}i\text{ odd }\end{array}\right.$

The cusp is parameterized by $N+1$ Newton pairs:

$\left(n_{1},\tfrac{\phi_{2l+1}^{2}n_{1}-1}{\phi_{2l+3}^{2}}\right),\left(n_{2},\tfrac{n_{1}n_{2}-1}{\phi_{2l+3}^{2}}\right),\dots,\left(n_{k},\tfrac{n_{k-1}n_{k}-1}{\phi_{2l+3}^{2}}\right),\dots,\left(n_{N},\tfrac{n_{N-1}n_{N}-1}{\phi_{2l+3}^{2}}\right),\left(\phi_{2l+3}^{2},n_{N}\right).$

  6. (6)

Type $II^{-}_{sp}$. The curves of type
$II^{-}(l,N;\lambda_{1},\dots,\lambda_{N})_{sp}$ have degree
$d=\phi_{2l+1}n_{1}\dots n_{N}$ where

$n_{i}=\left\\{\begin{array}[]{c}\lambda_{i}\phi_{2l+3}^{2}+\phi_{2l+3}\phi_{2l-1}-1\text{
for }i\text{ even }\\\
\lambda_{i}\phi_{2l+3}^{2}+\phi_{2l+3}(\phi_{2l+3}-\phi_{2l-1})-1\text{ for
}i\text{ odd }\end{array}\right.$

The cusp is parameterized by $N+1$ Newton pairs:

$\left(n_{1},\tfrac{\phi_{2l+1}^{2}n_{1}-1}{\phi_{2l+3}^{2}}\right),\left(n_{2},\tfrac{n_{1}n_{2}-1}{\phi_{2l+3}^{2}}\right),\dots,\left(n_{k},\tfrac{n_{k-1}n_{k}-1}{\phi_{2l+3}^{2}}\right),\dots,\left(n_{N},\tfrac{n_{N-1}n_{N}-1}{\phi_{2l+3}^{2}}\right),\left(\phi_{2l+3},\tfrac{n_{N}+1}{\phi_{2l+3}}\right).$

######  Remark 3.5.

In [FdBLVMHN06], the types are enumerated by a different set of invariants,
the decorations of the Eisenbud-Neumann splice diagrams, but these can be
converted to the Newton pairs. We illustrate this below for Type $II^{+}_{sp}$
and the process is similar for all other types. From the invariants $p$ and
$a$ given in [FdBLVMHN06, §6.3] to describe the Kashiwara curves, we compute
the relevant Newton pairs $(p_{1},q_{1}),...,(p_{N+1},q_{N+1})$ using the
formulas $q_{1}=a_{1}$, $q_{k}=a_{k}-p_{k}p_{k-1}a_{k-1}$ for $k>1$ (see
[FdBLVMHN06, §2.1]).

From [FdBLVMHN06, §6.3], the Kashiwara curves of type $II^{+}_{sp}$ are given
by $(p_{i},a_{i})$ where:

$\displaystyle p_{k}$ $\displaystyle=n_{k}\text{ for }1\leq k\leq N\text{, and
}p_{N+1}=\phi_{2l+3}$ $\displaystyle a_{k}$
$\displaystyle=\frac{\phi_{2l+5}^{2}n_{1}^{2}...n_{k-1}^{2}n_{k}-1}{\phi_{2l+3}^{2}}\text{
for }1\leq k\leq N\text{, and
}a_{N+1}=\frac{\phi_{2l+5}n_{1}^{2}...n_{N}^{2}+1}{\phi_{2l+3}}$

Therefore,

$\displaystyle\text{for }k=1:\quad q_{1}$ $\displaystyle=a_{1}$
$\displaystyle=\frac{\phi^{2}_{2l+5}n_{1}-1}{\phi_{2l+3}^{2}}$
$\displaystyle\text{for }2\leq k\leq N:\quad q_{k}$
$\displaystyle=a_{k}-p_{k}p_{k-1}a_{k-1}$
$\displaystyle=\frac{\phi_{2l+5}^{2}n_{1}^{2}...n_{k-1}^{2}n_{k}-1}{\phi_{2l+3}^{2}}-n_{k}n_{k-1}\left(\frac{\phi_{2l+5}^{2}n_{1}^{2}...n_{k-2}^{2}n_{k-1}-1}{\phi_{2l+3}^{2}}\right)$
$\displaystyle=\frac{\phi_{2l+5}^{2}n_{1}^{2}...n_{k-1}^{2}n_{k}-1-\phi_{2l+5}^{2}n_{1}^{2}...n_{k-1}^{2}n_{k}-n_{k}n_{k-1}}{\phi_{2l+3}^{2}}$
$\displaystyle=\frac{n_{k}n_{k-1}-1}{\phi_{2l+3}^{2}}$ $\displaystyle\text{for
}k=N+1:\quad q_{N+1}$ $\displaystyle=a_{N+1}-p_{N+1}p_{N}a_{N}$
$\displaystyle=\frac{\phi_{2l+5}^{2}n_{1}^{2}...n_{N}^{2}+1}{\phi_{2l+3}}-\phi_{2l+3}n_{N}\left(\frac{\phi_{2l+5}^{2}n_{1}^{2}...n_{N-1}^{2}n_{N}-1}{\phi^{2}_{2l+3}}\right)$
$\displaystyle=\frac{\phi_{2l+5}n_{1}^{2}...n_{N}^{2}+1-\phi_{2l+5}n_{1}^{2}...n_{N}^{2}+n_{N}}{\phi_{2l+3}}$
$\displaystyle=\frac{n_{N}+1}{\phi_{2l+3}}$

and we obtain the Newton pairs as claimed.

###  3.2. Kodaira dimension $0$

As proved by Tsunoda in [Tsu81], there exist no unicuspidal rational curves
with $\kappa(\mathbb{P}^{2}-C)=0$.

###  3.3. Kodaira dimension $1$

In [Ton01], Tono has classified all unicuspidal rational curves with
$\kappa(\mathbb{P}^{2}-C)=1$. There are four families. We follow the notation
in [Ton01].

  1. (1)

Type I(a). These curves have degree $d=a^{2}+1$ where $a\geq 3$ and the
parameterization of the cusp is given by the two Newton pairs:

$(a-1,a),(a,(a+1)^{2}).$

These curves are precisely the curves in Theorem 2.10.5.

  2. (2)

Type I(b). These curves have degree $d=a^{2}s+1$ where $a\geq 3$ and $s\geq 2$
and the parameterization of the cusp is given by three Newton pairs:

$(a-1,a),(s,as+1),(a,as+1).$

The only curves of degree at most 30 of Type I(b) are the degree 19 curve and
the first degree 28 curve in Theorem 2.11.

  3. (3)

Type II(a). These curves have degree $d=8n^{2}+4n+1$ where $n\geq 2$. The
parameterization of the cusp is given by the two Newton pairs:

$(n,4n+1),(4n+1,(2n+1)^{2}).$

These curves are precisely the curves in Theorem 2.10.6.

  4. (4)

Type II(b). These curves have degree $d=2(4n+1)^{2}s-4n(2n+1)$ where $n\geq 2$
and $s\geq 1$. The parameterization of the cusp is given by the three Newton
pairs:

$(n(4s-1),(s-1)(4n+1)),(4s-1,(4n+1)s-n),(4n+1,(4n+1)s-n).$

Because all curves of Type II(b) have degree greater than 30, they do not
appear in the classification in Theorem 2.11.

######  Remark 3.6.

Both Type I(a) and I(b) curves can be constructed by blow ups and blow downs
from the plane curve $zx^{a-1}=y^{a}$ using Lemma 2.15.

###  3.4. Kodaira dimension $2$

There exists no complete classification of curves with
$\kappa(\mathbb{P}^{2}-C)=2$. In fact, there are only two families of known
examples, due to Orevkov in [Ore02].

  1. (1)

Family $C_{4k}$. For $k=1$, the curve $C_{4}$ is a degree $8$ curve with cusp
parameterized by a single Newton pair $(3,22)$. For $k>1$, the curve $C_{4k}$
has degree $\phi_{4k+2}$ and has cusp given by the two Newton pairs
$(\frac{\phi_{4k}}{3},\frac{\phi_{4k+4}}{3}),(3,1)$. These curves correspond
to the curves in Theorem 2.9.5 and Theorem 2.10.7, for $k=1$ and $k>1$
respectively.

  2. (2)

Family $C_{4k}^{*}$. For $k=1$, the curve $C_{4}^{*}$ is a degree $16$ curve
with cusp parameterized by a single Newton pair $(6,43)$. For $k>1$, the curve
$C_{4k}^{*}$ has degree $2\phi_{4k+2}$ and has cusp given by the two Newton
pairs $(\frac{\phi_{4k}}{3},\frac{\phi_{4k+4}}{3}),(6,1)$. These curves
correspond to the curves in Theorem 2.9.6 and Theorem 2.10.8, for $k=1$ and
$k>1$ respectively.

####  3.4.1. A question on many Newton pairs

For each family in the classification by Kodaira dimension parameterized by at
most two Newton pairs or of degree at most 30, we have connected it with
Section 2. Observe that all known curves with $\kappa(\mathbb{P}^{2}-C)\geq 0$
are parameterized by at most three Newton pairs. This motivates the following
question.

######  Question 3.7.

If $C$ is a rational unicuspidal plane curve parameterized by at least four
Newton pairs, then is $\kappa(\mathbb{P}^{2}-C)=-\infty$? In particular, is
$C$ necessarily an AMS curve or a Kashiwara curve?

Running a computer program suggests that the answer is yes.

###  3.5. Self-intersections and log canonical thresholds

Motivated by the moduli perspective in the introduction, we conclude this
section with several results on the self-intersection of $C$ in its minimal
resolution, denoted by $\widetilde{C}^{2}=n$, and its log canonical threshold
${\mathrm{lct}}(C)$. These numbers have geometric significance and are
especially important for moduli problems. For example, if
$(\mathbb{P}^{2},\alpha C_{d})$ is a pair in $\mathcal{M}_{d}^{\alpha}$, we
must have ${\mathrm{lct}}(C_{d})\geq\alpha$ (if $\alpha\geq\frac{3}{d}$, this
is by definition, and if $\alpha<\frac{3}{d}$, this is a consequence of
Odaka’s result that K-semistable pairs are klt [Oda13]). Furthermore, if
$\widetilde{C}$ is to be contractible (as in Examples 1.3 and 1.4), then we
must have $\widetilde{C}^{2}<0$.

######  Definition 3.8.

Let $C\subset X$ be a curve in a smooth surface $X$. The log canonical
threshold of the curve, denoted ${\mathrm{lct}}(C)$, is

${\mathrm{lct}}(C)=\sup_{r\in\mathbb{R}}\\{(X,rC)\text{ is log canonical
}\\}.$

For the definition of log canonical singularity, see [KM98]. This is a measure
of how ‘bad’ a curve singularity is, with smaller log canonical threshold
corresponding to a ‘worse’ singularity. If $C$ is smooth, then
${\mathrm{lct}}(C)=1$. For a unicuspidal curve with cusp $p\in C$ with Puiseux
pairs $\\{(P_{i},Q_{i})\\}_{i=1}^{k}$, the log canonical threshold is given by

(3.1) ${\mathrm{lct}}(C)=\frac{1}{P_{1}}+\frac{1}{Q_{1}}.$

######  Definition 3.9.

Let $C\subset X$ be a projective curve in a smooth projective surface. Let
$\pi:\widetilde{X}\to X$ be the minimal log resolution of the pair $(X,C)$,
and let $\widetilde{C}=\pi^{-1}_{*}C$ be the strict transform of $C$. The
self-intersection in the minimal resolution is the integer
$\widetilde{C}^{2}$. We will refer to this number just as the self-
intersection.

This self-intersection number is related to the embedding of $C$ in $X$ and
the Kodaira dimension of the open surface $X-C$. For a rational unicuspidal
curve with Puiseux pairs $\\{(P_{i},Q_{i})\\}_{i=1}^{k}$, the self-
intersection in the minimal resolution can be computed directly (see, e.g.
[Liu14]):

(3.2) $\widetilde{C}^{2}=3d-1-P_{1}-\sum_{i=1}^{k}Q_{i}.$

For each of the curves in the previous section, we compute the log canonical
threshold and the self intersection.

####  3.5.1. Kodaira dimension $-\infty$

######  Proposition 3.10.

For all AMS curves of degree $d=n_{1}n_{2}\dots n_{r}$, the log canonical
threshold is $\frac{1}{(n_{1}-1)n_{2}\dots n_{r}}+\frac{1}{d}$ and the self-
intersection is $n_{r}>0$.

###### Proof.

We directly apply the formulas 3.1 and 3.2. The Puiseux pairs
$\\{(P_{i},Q_{i})\\}_{i=1}^{k}$ are obtained from the Newton pairs
$\\{(p_{i},q_{i})\\}_{i=1}^{k}$ by the formulas $P_{i}=p_{i}p_{i+1}\dots
p_{k}$ and $Q_{i}=q_{i}p_{i+1}\dots p_{k}$. ∎

######  Proposition 3.11.

For the Kashiwara curves, the log canonical thresholds and self-intersections
are given in Table 4.

Table 4. Log canonical thresholds and self-intersection of Kashiwara curves. type | degree | log canonical threshold | self intersection
---|---|---|---
Type $II_{ge}$ | $\phi_{2l+3}\phi_{2l+5}$ | $\frac{1}{\phi_{2l+3}^{2}}+\frac{1}{\phi_{2l+5}^{2}}$ | $0$
Type $II_{sp}$ | $\phi_{2l+3}$ | $\frac{1}{\phi_{2l+1}}+\frac{1}{\phi_{2l+5}}$ | $-1$
Type $II^{+}_{ge}$ | $\phi_{2l+3}\phi_{2l+5}n_{1}\dots n_{N}$ | $\frac{1}{(n_{1}...n_{N})\phi_{2l+3}^{2}}+\frac{1}{(\phi_{2l+5}^{2}n_{1}-1)(n_{2}...n_{N})}$ | $0$
Type $II^{+}_{sp}$ | $\phi_{2l+5}n_{1}\dots n_{N}$ | $\frac{1}{(n_{1}...n_{N})\phi_{2l+3}}+\frac{\phi_{2l+3}}{(\phi_{2l+5}^{2}n_{1}-1)(n_{2}...n_{N})}$ | $-1$
Type $II^{-}_{ge}$ | $\phi_{2l+3}\phi_{2l+1}n_{1}\dots n_{N}$ | $\frac{1}{(n_{1}...n_{N})\phi_{2l+3}^{2}}+\frac{1}{(\phi_{2l+1}^{2}n_{1}-1)(n_{2}...n_{N})}$ | $0$
Type $II^{-}_{sp}$ | $\phi_{2l+1}n_{1}\dots n_{N}$ | $\frac{1}{(n_{1}...n_{N})\phi_{2l+3}}+\frac{\phi_{2l+3}}{(\phi_{2l+1}^{2}n_{1}-1)(n_{2}...n_{N})}$ | $-1$

###### Proof.

The log canonical threshold is a direct consequence of formula 3.1. We give
the computation for the self-intersection of Kashiwara curves of type
$II^{+}_{sp}$ as the computations for the other cases are similar. This is an
application of the formula 3.2 and the Fibonacci identities

(3.3) $\phi_{n-2}+\phi_{n+2}=3\phi_{n}$

(3.4) $\phi_{n}^{2}-\phi_{n+r}\phi_{n-r}=(-1)^{n-r}\phi_{r}^{2}.$

By formula 3.2, $\widetilde{C}^{2}=3d-1-P_{1}-\sum_{i=1}^{N+1}Q_{i}$. For
curves of type $II^{+}_{sp}$, we compute

$\displaystyle-Q^{\prime}$ $\displaystyle=-Q_{1}-Q_{2}+...-Q_{N+1}$
$\displaystyle=-\frac{\phi_{2l+5}^{2}(n_{1}...n_{N})+(n_{2}...n_{N})-(n_{1}...n_{N})+(n_{3}...n_{N})-\dots-(n_{N-1}n_{N})+1-n_{N}-1}{\phi_{2l+3}}$
$\displaystyle=-\frac{(\phi_{2l+5}^{2}+1)(n_{1}...n_{N})}{\phi_{2l+3}},$
$\displaystyle\text{so }\widetilde{C}^{2}$
$\displaystyle=3d-1-P_{1}-Q^{\prime}$
$\displaystyle=3\phi_{2l+5}(n_{1}...n_{N})-1-\phi_{2l+3}(n_{1}...n_{N})-\frac{(\phi_{2l+5}^{2}+1)(n_{1}...n_{N})}{\phi_{2l+3}}$
$\displaystyle=\frac{3\phi_{2l+3}\phi_{2l+5}(n_{1}...n_{N})-\phi_{2l+3}^{2}(n_{1}...n_{N})-\phi_{2l+5}^{2}(n_{1}...n_{N})-(n_{1}...n_{N})}{\phi_{2l+3}}-1$
$\displaystyle=\frac{(n_{1}...n_{N})}{\phi_{2l+3}}(3\phi_{2l+3}\phi_{2l+5}-\phi_{2l+3}^{2}-\phi_{2l+5}^{2}-1)-1$
$\displaystyle=\frac{(n_{1}...n_{N})}{\phi_{2l+3}}(\phi_{2l+3}\phi_{2l+5}-(\phi_{2l+5}-\phi_{2l+3})^{2}-1)-1$
$\displaystyle=\frac{(n_{1}...n_{N})}{\phi_{2l+3}}(\phi_{2l+3}\phi_{2l+5}-\phi_{2l+4}^{2}-1)-1$
$\displaystyle=\frac{(n_{1}...n_{N})}{\phi_{2l+3}}(-(-1)^{2l+3}\phi_{2}-1)-1\text{
(by \ref{fibonacci})}$
$\displaystyle=\frac{(n_{1}...n_{N})}{\phi_{2l+3}}(0)-1$ $\displaystyle=-1.$

∎

Following [DS22], rational unicuspidal plane curves with log canonical
threshold at least $\frac{3}{d}$ are geometrically interesting as they give
rise to smooth non-planar limits of plane curves contained non-normal surfaces
(as in Example 1.4). The following question was posed in [DS22, Question,
p.39]:

######  Question 3.12.

For what degree $d$ does there exist a unicuspidal rational plane curve of
degree $d$ with log canonical threshold at least $\frac{3}{d}$?

We expect only few curves to have large log canonical threshold at least
$\frac{3}{d}$. No curves with $\kappa=-\infty$ have large ${\mathrm{lct}}$:

######  Lemma 3.13.

For all curves with Kodaira dimension $-\infty$, the log canonical threshold
is less than $\frac{3}{d}$.

###### Proof.

The computation for AMS curves is straightforward. The computation for
Kashiwara curves again uses the Fibonacci identities 3.3, 3.4, and we
illustrate it below for curves of type $II^{+}_{sp}$ (the remaining cases are
similar).

By 3.1, the log canonical threshold is $\frac{1}{P_{1}}+\frac{1}{Q_{1}}$. We
aim to show that this is less than $\frac{3}{d}$. For curves of type
$II^{+}_{sp}$, the desired inequality is:

$\displaystyle\frac{1}{P_{1}}+\frac{1}{Q_{1}}<\frac{3}{d}$
$\displaystyle\iff\frac{1}{(n_{1}...n_{N})\phi_{2l+3}}+\frac{\phi_{2l+3}}{(\phi_{2l+5}^{2}n_{1}-1)(n_{2}...n_{N})}<\frac{3}{\phi_{2l+5}(n_{1}...n_{N})}$
$\displaystyle\iff\frac{1}{\phi_{2l+3}n_{1}}+\frac{\phi_{2l+3}}{\phi_{2l+5}^{2}n_{1}-1}<\frac{3}{\phi_{2l+5}n_{1}}$
$\displaystyle\iff\frac{\phi_{2l+5}^{2}n_{1}-1+\phi_{2l+3}^{2}n_{1}}{\phi_{2l+3}n_{1}(\phi_{2l+5}^{2}n_{1}-1)}<\frac{3}{\phi_{2l+5}n_{1}}$
$\displaystyle\iff\frac{n_{1}(\phi_{2l+5}^{2}+\phi_{2l+3}^{2})-1}{\phi_{2l+3}n_{1}(\phi_{2l+5}^{2}n_{1}-1)}<\frac{3}{\phi_{2l+5}n_{1}}$
$\displaystyle\iff\phi_{2l+5}(n_{1}(\phi_{2l+5}^{2}+\phi_{2l+3}^{2})-1)<3\phi_{2l+3}(\phi_{2l+5}^{2}n_{1}-1)$
$\displaystyle\iff\phi_{2l+5}\phi_{2l+3}^{2}n_{1}+\phi_{2l+5}(\phi_{2l+5}^{2}n_{1}-1)<3\phi_{2l+3}(\phi_{2l+5}^{2}n_{1}-1)$
$\displaystyle\iff\phi_{2l+5}\phi_{2l+3}^{2}n_{1}<(3\phi_{2l+3}-\phi_{2l+5})(\phi_{2l+5}^{2}n_{1}-1)$
$\displaystyle\iff\phi_{2l+5}\phi_{2l+3}^{2}n_{1}<\phi_{2l+1}(\phi_{2l+5}^{2}n_{1}-1)\text{
(by \ref{fibonacci1})}$
$\displaystyle\iff\phi_{2l+1}<\phi_{2l+1}\phi_{2l+5}^{2}n_{1}-\phi_{2l+5}\phi_{2l+3}^{2}n_{1}$
$\displaystyle\iff\phi_{2l+1}<\phi_{2l+5}n_{1}(\phi_{2l+1}\phi_{2l+5}-\phi_{2l+3})$
$\displaystyle\iff\phi_{2l+1}<\phi_{2l+5}n_{1}(-1)(-1)^{2l+1}\text{ (by
\ref{fibonacci})}$ $\displaystyle\iff\phi_{2l+1}<\phi_{2l+5}n_{1},$

and as $n_{1}\geq 1$ and $l\geq 0$, this last inequality holds. Therefore, the
log canonical threshold is less than $\frac{3}{d}$. ∎

####  3.5.2. Kodaira dimension $1$

######  Proposition 3.14.

For the unicuspidal rational curves with Kodaira dimension $1$, the log
canonical thresholds and self intersections are given in the Table 5.

Table 5. Log canonical thresholds and self-intersection of Tono curves. type | degree | log canonical threshold | self intersection
---|---|---|---
Type I(a) | $a^{2}+1,a\geq 3$ | $\frac{1}{a(a-1)}+\frac{1}{a^{2}}$ | $1-a\leq-2$
Type I(b) | $a^{2}s+1,a\geq 3$ | $\frac{1}{as(a-1)}+\frac{1}{a^{2}s}$ | $1-a\leq-2$
Type II(a) | $8n^{2}+4n+1,n\geq 2$ | $\frac{1}{n(4n+1)}+\frac{1}{(4n+1)^{2}}$ | $-n\leq-2$
Type II(b) | $2(4n+1)^{2}s-4n(2n+1),n\geq 2$ | $\frac{1}{n(4n+1)(4s-1)}+\frac{1}{(4n+1)^{2}(s-1)}$ | $-n\leq-2$

###### Proof.

The proof is again a direct application of 3.1 and 3.2. ∎

By a straightforward calculation, we observe the following, related to
Question 3.12.

######  Lemma 3.15.

All unicuspidal rational curves of Kodaira dimension $1$ have log canonical
threshold less than $\frac{3}{d}$.

####  3.5.3. Kodaira dimension $2$

In each case for $\kappa=2$, it is proven in [Ore02] that
$\widetilde{C}^{2}=-2$. In fact, [Ton12] proves that these are the only curves
with $\kappa=2$ and $\widetilde{C}^{2}\geq-2$, i.e. if there exist other
rational unicuspidal plane curves with $\kappa=2$, they must have
$\widetilde{C}^{2}<-2$.

Furthermore, these are the only known examples with ${\mathrm{lct}}>3/d$. We
summarize the known properties (which are verifiable by straightforward
computation):

######  Proposition 3.16.

For the unicuspidal rational curves with Kodaira dimension $2$, the log
canonical thresholds and self intersections are given in Table 6.

Table 6. Log canonical thresholds and self-intersection of Orevkov curves. type | degree | log canonical threshold | self intersection
---|---|---|---
$C_{4}$ | $8$ | $\frac{1}{3}+\frac{1}{22}$ | $-2$
$C_{4k}$, $k>1$ | $\phi_{4k+2}$ | $\frac{1}{\phi_{4k}}+\frac{1}{\phi_{4k+4}}$ | $-2$
$C_{4}^{*}$ | $16$ | $\frac{1}{6}+\frac{1}{43}$ | $-2$
$C_{4k}^{*}$, $k>1$ | $2\phi_{4k+2}$ | $\frac{1}{2\phi_{4k}}+\frac{1}{2\phi_{4k+4}}$ | $-2$

For every curve in this list, the log canonical threshold is greater than
$\frac{3}{d}$ (c.f Question 3.12).

##  4\. Remarks on prime degree curves

Motivated by [DS22], we collect remarks on prime degree rational unicuspidal
plane curves below.

######  Proposition 4.1.

Prime degree unicuspidal rational curves with $\kappa(\mathbb{P}^{2}-C)<2$ or
with cusp parameterized by at most two Newton pairs must be in the following
list:

  1. (1)

$d=p$ is any prime number, and the cusp of $C$ is parameterized by a single
Newton pair $(d-1,d)$ (as in Theorem 2.9.1),

  2. (2)

$d=\phi_{j}$ where $j\geq 5$ is an odd prime, and the cusp of $C$ is
parameterized by a single Newton pair $(\phi_{j-2},\phi_{j+2})$ (as in Theorem
2.9.4),

  3. (3)

$d=a^{2}s+1$, where $d$ is prime and $a\geq 3$, and the cusp of $C$ is
parameterized by two Newton pairs $(a-1,a),(a,(a+1)^{2})$ (as in Theorem
2.10.5) if $s=1$, or by three Newton pairs $(a-1,a),(s,as+1),(a,as+1)$ if
$s>1$,

  4. (4)

$d=8n^{2}+4n+1$, where $d$ is prime and $n\geq 2$, and the cusp of $C$ is
parameterized by two Newton pairs $(n,4n+1),(4n+1,(2n+1)^{2})$ (as in Theorem
2.10.6).

###### Proof.

Every curve with $\kappa(\mathbb{P}^{2}-C)$ is either an AMS curve, a
Kashiwara curve, or one of Tono’s curves with $\kappa=1$. The only AMS curves
of prime degree are those in Theorem 2.9.1, and the only Kashiwara curves of
prime degree are type $II_{sp}$, those in Theorem 2.9.4. The restriction that
$j$ must be prime in this case follows because the Fibonacci number $\phi_{j}$
is divisible by $\phi_{k}$ for any $k$ dividing $j$. Tono’s curves with
$\kappa=1$ of type I(a), I(b), and II(a) give the remaining items in the list,
noting that the curves of type II(b) have even degree.

To complete the proof, we must then justify that any curve with $\kappa=2$
parameterized by at most two Newton pairs does not have prime degree. The only
such curves are the degree $8$ or $16$ curve in Theorem 2.9.5 or .6, which
clearly have composite degree, or the degree $d=\phi_{4k+2}$ or
$d=2\phi_{4k+2}$ curves, where $k\geq 2$, in Theorem 2.10.7 or .8, which have
composite degree because $\phi_{4k+2}$ is divisible by $\phi_{2k+1}>1$. ∎

Speculating that the Orevkov curves are the only rational unicuspidal curves
with $\kappa=2$, this is a complete list of possibilities for prime degree
rational unicuspidal curves.

In fact, the possible ‘nontrivial’ prime degree curves (not isomorphic to
$x^{p-1}z=y^{d}$) conjecturally come in infinite families.

The first type of nontrivial prime degree curve are those of degree
$\phi_{j}$, where $\phi_{j}$ is a Fibonacci number. A famous conjecture in
number theory is:

######  Conjecture 4.2.

(Infinitude of Fibonacci primes.) There are infinitely many Fibonacci numbers
that are prime.

If true, this asserts that there are infinitely many prime degrees for which
there exists a unicuspidal rational curve with a cusp as in Proposition 4.1.2.

Similarly, another famous conjecture in number theory, posed by Landau in 1912
is:

######  Conjecture 4.3.

(Landau’s Fourth Problem.) There are infinitely many primes of the form
$n^{2}+1$.

If true, this asserts the infinitude of prime degrees for which there exists a
curve with cusp as in Proposition 4.1.3. While this conjecture is unsolved, by
[BZ06], there are infinitely many primes of the form $a^{2}s+1$ (when one
allows both $a$ and $s$ to vary). Therefore, there do exist infinitely many
prime degrees for which there is a nontrivial rational unicuspidal plane curve
of Type I(a) or Type I(b) in Tono’s classification. This is the content of the
following statement.

######  Theorem 4.4.

(= Theorem 1.9) There are infinitely many primes $p$ such that there exists a
unicuspidal rational curve $C$ of degree $p$ with Kodaira dimension
$\kappa(\mathbb{P}^{2}-C)=1$.

######  Example 4.5.

Two such curves are the degree $19=3^{2}\cdot 2+1$ curve and first degree
$28=3^{2}\cdot 3+1$ curve in Theorem 2.11. The theorem asserts that there are
infinitely many others.

More generally than the case $n^{2}+1$ or $sn^{2}+1$, given any polynomial
$f(n)$ with integer coefficients, one may ask how often $f(n)$ is prime for
integer input $n$. One conjecture in this direction is the Bunyakovsky
conjecture:

######  Conjecture 4.6.

(Bunyakovsky Conjecture.) Suppose $f(n)$ is a polynomial with integer
coefficients such that:

  1. (1)

the leading coefficient is positive;

  2. (2)

the polynomial is irreducible over $\mathbb{Q}$; and

  3. (3)

the values $\\{f(1),f(2),f(3),\dots\\}$ have no common factor.

Then, $f(n)$ is prime infinitely often.

There are in fact further conjectures on the asymptotic number of integers $n$
such that $f(n)$ is prime (part of the broader Bateman-Horn conjecture). For
our purposes, this conjecture would imply the infinitude of prime degrees
giving Tono curves of Type I(b) and Type II(a).

######  Lemma 4.7.

If $f(n)=sn^{2}+1$, the conditions of Bunyakovsky’s conjecture are satisfied.

###### Proof.

(1) and (2) are clear so we prove (3). If $s=1$, then $\gcd(f(1),f(2))=1$, so
(3) holds. If $s=2$, then $\gcd(f(1),f(3))=1$ so again (3) holds. Now, assume
$s\geq 3$. If $s\neq 2\pmod{3}$, then

$\gcd(f(1),f(2))=\gcd(s+1,4s+1)=\gcd(s+1,s-2)=\gcd(s-2,3)=1$

so (3) holds. If $s=2\pmod{3}$, then $\gcd(f(1),f(2))=3$, but

$\gcd(f(2),f(3))=\gcd(4s+1,9s+1)=\gcd(4s+1,s-1)=\gcd(s-1,5)$

so $\gcd(f(2),f(3))$ must equal $1$ or $5$. This implies that $f(1),f(2),$ and
$f(3)$ have no common factors, as desired. ∎

Therefore, if Bunyakovsky’s conjecture holds, this implies that there are
infinitely many primes of the form $a^{2}s+1$ for any fixed $s$, corresponding
to rational unicuspidal curves of degree $d=a^{2}s+1$ of Type I(a) and Type
I(b) in Tono’s classification.

######  Lemma 4.8.

If $f(n)=8n^{2}+4n+1$, the conditions of Bunyakovsky’s conjecture are
satisfied.

###### Proof.

(1) and (2) are clear, and $f(1)=13$ and $f(2)=41$, so $f(1)$ and $f(2)$ have
no common factors, thus (3) holds. ∎

Therefore, if Bunyakovsky’s conjecture holds, there are also infinitely many
primes of the form $8n^{2}+4n+1$ corresponding to rational unicuspidal curves
of degree $d=8n^{2}+4n+1$ of Type II(a) in Tono’s classification.

##  5\. Table of low degree rational unicuspidal plane curves

The results of the first three sections are combined in the following Theorem.

######  Theorem 5.1.

(= Theorem 1.7) For $d\leq 30$, the following table gives an exhaustive list
of all rational unicuspidal degree $d$ plane curves and their properties. See
footnote 1 for a description of what is meant by the parameterization column.

Table 7. All rational unicuspidal plane curves of degree $\leq 30$. degree | Newton pair(s) | parametrization | $\overline{m}$ | $\kappa$ | ${\mathrm{lct}}(C)$ | $\widetilde{C}^{2}$ | reference
---|---|---|---|---|---|---|---
$d\geq 3$ | $(d-1,d)$ | $(t^{d-1},t^{d})$ | $(d)$ | $-\infty$ | $\frac{1}{d-1}+\frac{1}{d}$ | $d$ |  2.9.1, §3.1.1
$\begin{array}[]{c}d=2n\\\ n\geq 2\end{array}$ | $(d/2,2d-1)$ | $(t^{d/2},t^{2d-1})$ | $(d/2_{3},d/2-1)$ | $-\infty$ | $\frac{2}{d}+\frac{1}{2d-1}$ | $d/2$ |  2.9.2, §3.1.1
5 | $(2,13)$ | $(t^{2},t^{13})$ | $(2_{6})$ | $-\infty$ | $\frac{1}{2}+\frac{1}{13}$ | -1 |  2.9.4, §3.1.2
6 | $(2,3),(2,5)$ | $(t^{4},t^{6}+t^{11})$ | $(4,2_{4})$ | $-\infty$ | $\frac{1}{4}+\frac{1}{6}$ | $2$ |  2.10.3, §3.1.1
8 | $(3,22)$ | $(t^{3},t^{22})$ | $(3_{7})$ | $2$ | $\frac{1}{3}+\frac{1}{22}$ | $-2$ |  2.9.5, §3.4
8 | $(2,7),(2,3)$ | $(t^{4},t^{14}+t^{17})$ | $(4_{3},2_{3})$ | $-\infty$ | $\frac{1}{4}+\frac{1}{14}$ | $2$ |  2.10.4, §3.1.1
8 | $(3,4),(2,7)$ | $(t^{6},t^{8}+t^{15})$ | $(6,2_{6})$ | $-\infty$ | $\frac{1}{6}+\frac{1}{8}$ | $2$ |  2.10.3, §3.1.1
9 | $(2,3),(3,8)$ | $(t^{6},t^{9}+t^{17})$ | $(6,3_{4},2)$ | $-\infty$ | $\frac{1}{6}+\frac{1}{9}$ | $3$ |  2.10.3, §3.1.1
10 | $(4,25)$ | $(t^{4},t^{25})$ | $(4_{6})$ | $-\infty$ | $\frac{1}{4}+\frac{1}{25}$ | $0$ |  2.9.3, §3.1.2
10 | $(2,3),(3,16)$ | $(t^{6},t^{9}+t^{25})$ | $(6,3_{7})$ | $1$ | $\frac{1}{6}+\frac{1}{9}$ | $-2$ |  2.10.5, §3.3
10 | $(4,5),(2,9)$ | $(t^{8},t^{10}+t^{19})$ | $(8,2_{8})$ | $-\infty$ | $\frac{1}{8}+\frac{1}{10}$ | $2$ |  2.10.3, §3.1.1
12 | $(2,7),(3,5)$ | $(t^{6},t^{21}+t^{26})$ | $(6_{3},3_{3},2)$ | $-\infty$ | $\frac{1}{6}+\frac{1}{21}$ | $3$ |  2.10.4, §3.1.1
12 | $(2,3),(4,11)$ | $(t^{8},t^{12}+t^{23})$ | $(8,4_{4},3)$ | $-\infty$ | $\frac{1}{8}+\frac{1}{12}$ | $4$ |  2.10.3, §3.1.1
12 | $(3,11),(2,7)$ | $(t^{6},t^{22}+t^{29})$ | $(6_{3},4,2_{5})$ | $-\infty$ | $\frac{1}{6}+\frac{1}{22}$ | $2$ |  2.10.4, §3.1.1
12 | $(3,4),(3,11)$ | $(t^{9},t^{12}+t^{23})$ | $(9,3_{6},2)$ | $-\infty$ | $\frac{1}{9}+\frac{1}{12}$ | $3$ |  2.10.3, §3.1.1
12 | $(5,6),(2,11)$ | $(t^{10},t^{12}+t^{23})$ | $(10,2_{10})$ | $-\infty$ | $\frac{1}{10}+\frac{1}{12}$ | $2$ |  2.10.3, §3.1.1
12 | $(2,3),(2,5),(2,3)$ | $(t^{8},t^{12}+t^{22}+t^{25})$ | $(8,4_{4},2_{3})$ | $-\infty$ | $\frac{1}{8}+\frac{1}{12}$ | $2$ |  2.11, §3.1.1
13 | $(5,34)$ | $(t^{5},t^{34})$ | $(5_{6},4)$ | $-\infty$ | $\frac{1}{5}+\frac{1}{34}$ | $-1$ |  2.9.4, §3.1.2
14 | $(6,7),(2,13)$ | $(t^{12},t^{14}+t^{27})$ | $(12,2_{12})$ | $-\infty$ | $\frac{1}{12}+\frac{1}{14}$ | $2$ |  2.10.3, §3.1.1
15 | $(2,3),(5,14)$ | $(t^{10},t^{15}+t^{29})$ | $(10,5_{4},4)$ | $-\infty$ | $\frac{1}{10}+\frac{1}{15}$ | $5$ |  2.10.3, §3.1.1
15 | $(4,5),(3,14)$ | $(t^{12},t^{15}+t^{29})$ | $(12,3_{8},2)$ | $-\infty$ | $\frac{1}{12}+\frac{1}{15}$ | $3$ |  2.10.3, §3.1.1
16 | $(6,43)$ | $(t^{6},t^{43})$ | $(6_{7})$ | $2$ | $\frac{1}{6}+\frac{1}{43}$ | $-2$ |  2.9.6, §3.4
16 | $(2,7),(4,7)$ | $(t^{8},t^{28}+t^{35})$ | $(8_{3},4_{3},3)$ | $-\infty$ | $\frac{1}{8}+\frac{1}{28}$ | $4$ |  2.10.4, §3.1.1
16 | $(3,4),(4,15)$ | $(t^{12},t^{16}+t^{31})$ | $(12,4_{6},3)$ | $-\infty$ | $\frac{1}{12}+\frac{1}{16}$ | $4$ |  2.10.3, §3.1.1
16 | $(4,15),(2,7)$ | $(t^{8},t^{30}+t^{37})$ | $(8_{3},6,2_{6})$ | $-\infty$ | $\frac{1}{8}+\frac{1}{30}$ | $2$ |  2.10.4, §3.1.1
16 | $(7,8),(2,15)$ | $(t^{14},t^{16}+t^{31})$ | $(14,2_{1}4)$ | $-\infty$ | $\frac{1}{14}+\frac{1}{16}$ | $2$ |  2.10.3, §3.1.1
16 | $(2,7),(2,3),(2,3)$ | $(t^{8},t^{28}+t^{34}+t^{37})$ | $(8_{3},4_{3},2_{3})$ | $-\infty$ | $\frac{1}{8}+\frac{1}{28}$ | $2$ |  2.11, §3.1.1
16 | $(3,4),(2,7),(2,3)$ | $(t^{12},t^{16}+t^{30}+t^{33})$ | $(12,4_{6},2_{3})$ | $-\infty$ | $\frac{1}{12}+\frac{1}{16}$ | $2$ |  2.11, §3.1.1
17 | $(3,4),(4,25)$ | $(t^{12},t^{16}+t^{41})$ | $(12,4_{9})$ | $1$ | $\frac{1}{12}+\frac{1}{16}$ | $-3$ |  2.10.5, §3.3
18 | $(2,3),(6,17)$ | $(t^{12},t^{18}+t^{35})$ | $(12,6_{4},5)$ | $-\infty$ | $\frac{1}{12}+\frac{1}{18}$ | $6$ |  2.10.3, §3.1.1
18 | $(3,11),(3,8)$ | $(t^{9},t^{33}+t^{41})$ | $(9_{3},6,3_{4},2)$ | $-\infty$ | $\frac{1}{9}+\frac{1}{33}$ | $3$ |  2.10.4, §3.1.1
18 | $(5,6),(3,17)$ | $(t^{15},t^{18}+t^{35})$ | $(15,3_{10},2)$ | $-\infty$ | $\frac{1}{15}+\frac{1}{18}$ | $3$ |  2.10.3, §3.1.1
18 | $(8,9),(2,17)$ | $(t^{16},t^{18}+t^{35})$ | $(16,2_{16})$ | $-\infty$ | $\frac{1}{16}+\frac{1}{18}$ | $2$ |  2.10.3, §3.1.1
18 | $(2,3),(2,5),(3,5)$ | $(t^{12},t^{18}+t^{33}+t^{38})$ | $(12,6_{4},3_{3},2)$ | $-\infty$ | $\frac{1}{12}+\frac{1}{18}$ | $3$ |  2.11, §3.1.1
18 | $(2,3),(3,8),(2,5)$ | $(t^{12},t^{18}+t^{34}+t^{39})$ | $(12,6_{4},4,2_{4})$ | $-\infty$ | $\frac{1}{12}+\frac{1}{18}$ | $2$ |  2.11, §3.1.1
19 | $(2,3),(2,7),(3,7)$ | $(t^{12},t^{18}+t^{39}+t^{46})$ | $(12,6_{5},3_{4})$ | $1$ | $\frac{1}{12}+\frac{1}{18}$ | $-2$ |  2.11, §3.3
20 | $(2,7),(5,9)$ | $(t^{10},t^{35}+t^{44})$ | $(10_{3},5_{3},4)$ | $-\infty$ | $\frac{1}{10}+\frac{1}{35}$ | $5$ |  2.10.4, §3.1.1
20 | $(3,4),(5,19)$ | $(t^{15},t^{20}+t^{39})$ | $(15,5_{6},4)$ | $-\infty$ | $\frac{1}{15}+\frac{1}{20}$ | $5$ |  2.10.3, §3.1.1
20 | $(4,5),(4,19)$ | $(t^{16},t^{20}+t^{39})$ | $(16,4_{8},3)$ | $-\infty$ | $\frac{1}{16}+\frac{1}{20}$ | $4$ |  2.10.3, §3.1.1
20 | $(5,19),(2,9)$ | $(t^{10},t^{38}+t^{47})$ | $(10_{3},8,2_{8})$ | $-\infty$ | $\frac{1}{10}+\frac{1}{38}$ | $2$ |  2.10.4 §3.1.1
20 | $(9,10),(2,19)$ | $(t^{18},t^{20}+t^{39})$ | $(18,2_{18})$ | $-\infty$ | $\frac{1}{18}+\frac{1}{20}$ | $2$ |  2.10.3, §3.1.1
20 | $(4,5),(2,9),(2,3)$ | $(t^{16},t^{20}+t^{38}+t^{41})$ | $(16,4_{8},2_{3})$ | $-\infty$ | $\frac{1}{16}+\frac{1}{20}$ | $2$ |  2.11, §3.1.1
21 | $(2,3),(7,20)$ | $(t^{14},t^{21}+t^{41})$ | $(14,7_{4},6)$ | $-\infty$ | $\frac{1}{14}+\frac{1}{21}$ | $7$ |  2.10.3, §3.1.1
21 | $(6,7),(3,20)$ | $(t^{18},t^{21}+t^{41})$ | $(18,3_{12},2)$ | $-\infty$ | $\frac{1}{18}+\frac{1}{21}$ | $3$ |  2.10.3, §3.1.1
22 | $(10,11),(2,21)$ | $(t^{20},t^{22}+t^{43})$ | $(20,2_{20})$ | $-\infty$ | $\frac{1}{20}+\frac{1}{22}$ | $2$ |  2.10.3, §3.1.1
24 | $(2,3),(8,23)$ | $(t^{16},t^{24}+t^{47})$ | $(16,8_{4},7)$ | $-\infty$ | $\frac{1}{16}+\frac{1}{24}$ | $8$ |  2.10.3, §3.1.1
24 | $(2,7),(6,11)$ | $(t^{12},t^{42}+t^{53})$ | $(12_{3},6_{3},5)$ | $-\infty$ | $\frac{1}{12}+\frac{1}{42}$ | $6$ |  2.10.4, §3.1.1
24 | $(3,4),(6,23)$ | $(t^{18},t^{24}+t^{47})$ | $(18,6_{6},5)$ | $-\infty$ | $\frac{1}{18}+\frac{1}{24}$ | $6$ |  2.10.3, §3.1.1
24 | $(3,11),(4,11)$ | $(t^{12},t^{44}+t^{55})$ | $(12_{3},8,4_{4},3)$ | $-\infty$ | $\frac{1}{12}+\frac{1}{44}$ | $4$ |  2.10.4, §3.1.1
24 | $(4,15),(3,11)$ | $(t^{12},t^{45}+t^{56})$ | $(12_{3},9,3_{6},2)$ | $-\infty$ | $\frac{1}{12}+\frac{1}{45}$ | $3$ |  2.10.4, §3.1.1
24 | $(5,6),(4,23)$ | $(t^{20},t^{24}+t^{47})$ | $(20,4_{10},3)$ | $-\infty$ | $\frac{1}{20}+\frac{1}{24}$ | $4$ |  2.10.3, §3.1.1
24 | $(6,23),(2,11)$ | $(t^{12},t^{46}+t^{57})$ | $(12_{3},10,2_{10})$ | $-\infty$ | $\frac{1}{12}+\frac{1}{46}$ | $2$ |  2.10.4, §3.1.1
24 | $(7,8),(3,23)$ | $(t^{21},t^{24}+t^{47})$ | $(21,3_{14},2)$ | $-\infty$ | $\frac{1}{21}+\frac{1}{24}$ | $3$ |  2.10.3, §3.1.1
24 | $(11,12),(2,23)$ | $(t^{22},t^{24}+t^{47})$ | $(22,2_{22})$ | $-\infty$ | $\frac{1}{22}+\frac{1}{24}$ | $2$ |  2.10.3, §3.1.1
24 | $(2,7),(2,3),(3,5)$ | $(t^{12},t^{42}+t^{51}+t^{56})$ | $(12_{3},6_{3},3_{3},2)$ | $-\infty$ | $\frac{1}{12}+\frac{1}{42}$ | $3$ |  2.11, §3.1.1
24 | $(2,7),(3,5),(2,5)$ | $(t^{12},t^{42}+t^{52}+t^{57})$ | $(12_{3},6_{3},4,2_{4})$ | $-\infty$ | $\frac{1}{12}+\frac{1}{42}$ | $2$ |  2.11, §3.1.1
24 | $(3,11),(2,5),(2,3)$ | $(t^{12},t^{44}+t^{54}+t^{57})$ | $(12_{3},8,4_{4},2_{3})$ | $-\infty$ | $\frac{1}{12}+\frac{1}{44}$ | $2$ |  2.11, §3.1.1
24 | $(2,3),(2,5),(4,7)$ | $(t^{16},t^{24}+t^{44}+t^{51})$ | $(16,8_{4},4_{3},3)$ | $-\infty$ | $\frac{1}{16}+\frac{1}{24}$ | $4$ |  2.11, §3.1.1
24 | $(2,3),(4,11),(2,7)$ | $(t^{16},t^{24}+t^{46}+t^{53})$ | $(16,8_{4},6,2_{6})$ | $-\infty$ | $\frac{1}{16}+\frac{1}{24}$ | $2$ |  2.11, §3.1.1
24 | $(3,4),(2,7),(3,5)$ | $(t^{18},t^{24}+t^{45}+t^{50})$ | $(18,6_{6},3_{3},2)$ | $-\infty$ | $\frac{1}{18}+\frac{1}{24}$ | $5$ |  2.11, §3.1.1
24 | $(3,4),(3,11),(2,5)$ | $(t^{18},t^{24}+t^{46}+t^{51})$ | $(18,6_{6},4,2_{4})$ | $-\infty$ | $\frac{1}{18}+\frac{1}{24}$ | $2$ |  2.11, §3.1.1
24 | $(5,6),(2,11),(2,3)$ | $(t^{20},t^{24}+t^{46}+t^{49})$ | $(20,4_{10},2_{3})$ | $-\infty$ | $\frac{1}{20}+\frac{1}{24}$ | $2$ |  2.11, §3.1.1
24 | $\begin{array}[]{c}(2,3),(2,5),\\\ (2,3),(2,3)\end{array}$ | $\begin{array}[]{c}(t^{16},\\\ t^{24}+t^{44}+t^{50}+t^{53})\end{array}$ | $(16,8_{4},4_{3},2_{2},3)$ | $-\infty$ | $\frac{1}{16}+\frac{1}{24}$ | $2$ |  2.12, §3.1.1
25 | $(4,5),(5,24)$ | $(t^{20},t^{25}+t^{49})$ | $(20,5_{8},4)$ | $-\infty$ | $\frac{1}{20}+\frac{1}{25}$ | $5$ |  2.10.3, §3.1.1
26 | $(4,5),(5,36)$ | $(t^{20},t^{25}+t^{61})$ | $(20,5_{11})$ | $1$ | $\frac{1}{20}+\frac{1}{25}$ | $-4$ |  2.10.5, §3.3
26 | $(12,13),(2,25)$ | $(t^{24},t^{26}+t^{41})$ | $(24,2_{19})$ | $-\infty$ | $\frac{1}{24}+\frac{1}{26}$ | $2$ |  2.10.3, §3.1.1
27 | $(2,3),(9,26)$ | $(t^{18},t^{27}+t^{53})$ | $(18,9_{4},8)$ | $-\infty$ | $\frac{1}{18}+\frac{1}{27}$ | $9$ |  2.10.3, §3.1.1
27 | $(8,9),(3,26)$ | $(t^{24},t^{27}+t^{53})$ | $(24,3_{16},2)$ | $-\infty$ | $\frac{1}{24}+\frac{1}{27}$ | $3$ |  2.10.3, §3.1.1
27 | $(2,3),(3,8),(3,8)$ | $(t^{18},t^{27}+t^{51}+t^{59})$ | $(18,9_{4},6,3_{6})$ | $-\infty$ | $\frac{1}{18}+\frac{1}{27}$ | $3$ |  2.11, §3.1.1
28 | $(2,7),(7,13)$ | $(t^{14},t^{49}+t^{62})$ | $(14_{3},7_{3},6)$ | $-\infty$ | $\frac{1}{14}+\frac{1}{49}$ | $7$ |  2.10.4, §3.1.1
28 | $(3,4),(7,27)$ | $(t^{21},t^{28}+t^{55})$ | $(21,7_{6},6)$ | $-\infty$ | $\frac{1}{21}+\frac{1}{28}$ | $3$ |  2.10.3, §3.1.1
28 | $(6,7),(4,27)$ | $(t^{24},t^{28}+t^{55})$ | $(24,4_{12},3)$ | $-\infty$ | $\frac{1}{24}+\frac{1}{28}$ | $4$ |  2.10.3, §3.1.1
28 | $(7,27),(2,11)$ | $(t^{14},t^{54}+t^{65})$ | $(14_{3},12,2_{11})$ | $-\infty$ | $\frac{1}{14}+\frac{1}{54}$ | $2$ |  2.10.4, §3.1.1
28 | $(13,14),(2,27)$ | $(t^{26},t^{28}+t^{55})$ | $(26,2_{26})$ | $-\infty$ | $\frac{1}{26}+\frac{1}{28}$ | $2$ |  2.10.3, §3.1.1
28 | $(2,3),(3,10),(3,10)$ | $(t^{18},t^{27}+t^{57}+t^{67})$ | $(18,9_{5},3_{6})$ | $1$ | $\frac{1}{18}+\frac{1}{27}$ | $-2$ |  2.11, §3.3
28 | $(6,7),(2,13),(2,3)$ | $(t^{24},t^{28}+t^{54}+t^{57})$ | $(24,4_{12},2_{3})$ | $-\infty$ | $\frac{1}{24}+\frac{1}{28}$ | $2$ |  2.11, §3.1.1
30 | $(2,3),(10,29)$ | $(t^{20},t^{30}+t^{59})$ | $(20,10_{4},9)$ | $-\infty$ | $\frac{1}{20}+\frac{1}{30}$ | $10$ |  2.10.3, §3.1.1
30 | $(3,11),(5,14)$ | $(t^{15},t^{55}+t^{69})$ | $(15_{3},10,5_{4},4)$ | $-\infty$ | $\frac{1}{15}+\frac{1}{55}$ | $5$ |  2.10.4, §3.1.1
30 | $(4,5),(6,29)$ | $(t^{24},t^{30}+t^{59})$ | $(24,6_{8},5)$ | $-\infty$ | $\frac{1}{24}+\frac{1}{30}$ | $6$ |  2.10.3, §3.1.1
30 | $(5,6),(5,29)$ | $(t^{25},t^{30}+t^{59})$ | $(25,5_{10},4)$ | $-\infty$ | $\frac{1}{25}+\frac{1}{30}$ | $5$ |  2.10.3, §3.1.1
30 | $(5,19),(3,14)$ | $(t^{15},t^{57}+t^{71})$ | $(15_{3},12,3_{8},2)$ | $-\infty$ | $\frac{1}{15}+\frac{1}{57}$ | $3$ |  2.10.4, §3.1.1
30 | $(9,10),(3,29)$ | $(t^{27},t^{30}+t^{59})$ | $(27,3_{18},2)$ | $-\infty$ | $\frac{1}{27}+\frac{1}{30}$ | $3$ |  2.10.3, §3.1.1
30 | $(14,15),(2,29)$ | $(t^{28},t^{30}+t^{59})$ | $(28,2_{28})$ | $-\infty$ | $\frac{1}{28}+\frac{1}{30}$ | $2$ |  2.10.3, §3.1.1
30 | $(2,3),(2,5),(5,9)$ | $(t^{20},t^{30}+t^{55}+t^{64})$ | $(20,10_{4},5_{3},4)$ | $-\infty$ | $\frac{1}{20}+\frac{1}{30}$ | $5$ |  2.11, §3.1.1
30 | $(2,3),(5,14),(2,9)$ | $(t^{20},t^{30}+t^{58}+t^{67})$ | $(20,10_{4},8,2_{8})$ | $-\infty$ | $\frac{1}{20}+\frac{1}{30}$ | $2$ |  2.11, §3.1.1
30 | $(4,5),(2,9),(3,5)$ | $(t^{24},t^{30}+t^{57}+t^{62})$ | $(24,6_{8},3_{3},2)$ | $-\infty$ | $\frac{1}{24}+\frac{1}{30}$ | $3$ |  2.11, §3.1.1
30 | $(4,5),(3,14),(2,5)$ | $(t^{24},t^{30}+t^{58}+t^{63})$ | $(24,6_{8},4,2_{4})$ | $-\infty$ | $\frac{1}{24}+\frac{1}{30}$ | $2$ |  2.11, §3.1.1

##  Appendix A Computer code

Here is the program used to produce a list of candidate curves with cusp
parameterized by three Newton pairs.

    def tripleNewtonPairs(degree):
        ab_max = int(((degree - 1) * (degree - 2)) / 2)
        return generate_mn_list(ab_max, degree)

    def generate_ab_list(ab_max):
        ab_list = []
        for a in range(2, ab_max+1):
            for b1 in range(a+1, ab_max+1):
                for b2 in range(b1+1, ab_max+1):
                    for b3 in range(b2+1, ab_max+1):
                        ab_list.append((a, b1, b2, b3))
        return ab_list

    def generate_mn_list(ab_max, degree):
        mn_list = []
        ab_list = generate_ab_list(ab_max)
        for nums in ab_list:
            #parameterization
            a = nums[0]
            b1 = nums[1]
            b2 = nums[2]
            b3 = nums[3]

            # uppercase: check delta invariant
            m1 = a # m_1 * m_2 * m_3
            m2 = math.gcd(m1, b1) # m_2 * m_3
            m3 = math.gcd(m2, b2) # m_3
            n1 = b1 # n_1 * m_2 * m_3
            n2 = b2 - b1 # n_2 * m_3
            n3 = b3 - b2 # n_3

            # lowercase: values in newton pairs (m, n)
            m_3 = m3
            m_2 = int(m2 / m3)
            m_1 = int(m1 / m2)
            n_3 = n3
            n_2 = int(n2 / m3)
            n_1 = int(n1 / m2)

            # semigroup generators
            w1 = m1
            w2 = n1
            w3 = (m_1 * w2) + n2
            w4 = (m_2 * w3) + n3

            # delta invariant
            d_inv = int(((degree - 1) * (degree - 2)))

            # check constraints
                if all(i > 1 for i in (m1, m2, m3, m_1, m_2, m_3, n_1, n_2)):
                    if (math.gcd(m_1, n_1) == 1 and math.gcd(m_2, n_2) == 1
                    and math.gcd(m_3, n_3) == 1):
                        pair1 = (m1, n1)
                        pair2 = (m2, n2)
                        pair3 = (m3, n3)
                        # check for duplicate pairs
                        if (not pair1 == pair2 and not
                        pair1 == pair3 and not pair2 == pair3):
                            d_inv_check = int(((m1 - 1) * (n1 - 1)) +
                            ((m2 - 1) * n2) + ((m3 - 1) * n3))
                            # check delta invariant
                            if(d_inv == d_inv_check):
                                valid = True
                                semigroup = [0]
                                # check semigroup property
                                for l in range(1, degree-2):
                                    semigroup = getSemigroupTriple(w1, w2, w3, w4,
                                    l, degree, semigroup)
                                    compare = (l+1)*(l+2)/2
                                    if(len(semigroup) != compare):
                                        valid = False
                                if(valid):
                                    mn_list.append((pair1, pair2, pair3))
        return mn_list

    def getSemigroupTriple(w1, w2, w3, w4, l, degree, semigroup):
        maxVal = l * degree
        for num in semigroup:
            num1 = num + w1
            num2 = num + w2
            num3 = num + w3
            num4 = num + w4
            if(num1 not in semigroup and num1 <= maxVal):
                semigroup.append(num1)
                semigroup.sort()
            if(num2 not in semigroup and num2 <= maxVal):
                semigroup.append(num2)
                semigroup.sort()
            if(num3 not in semigroup and num3 <= maxVal):
                semigroup.append(num3)
                semigroup.sort()
            if(num4 not in semigroup and num4 <= maxVal):
                semigroup.append(num4)
                semigroup.sort()
            if (num1 > maxVal and num2 > maxVal and num3 > maxVal and num4 > maxVal):
                return semigroup

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