# Heegaard Floer homology and rational cuspidal curves

Maciej Borodzik  Institute of Mathematics, University of Warsaw, ul. Banacha
2, 02-097 Warsaw, Poland<EMAIL_ADDRESS>and  Charles Livingston
Department of Mathematics, Indiana University, Bloomington, IN 47405
<EMAIL_ADDRESS>
###### Abstract.

We apply the methods of Heegaard Floer homology to identify topological
properties of complex curves in $\mathbb{C}P^{2}$. As one application, we
resolve an open conjecture that constrains the Alexander polynomial of the
link of the singular point of the curve in the case that there is exactly one
singular point, having connected link, and the curve is of genus 0.
Generalizations apply in the case of multiple singular points.

###### Key words and phrases:

rational cuspidal curve, $d$–invariant, surgery, semigroup density

###### 2010 Mathematics Subject Classification:

primary: 14H50, secondary: 14B05, 57M25, 57R58

The first author was supported by Polish OPUS grant No 2012/05/B/ST1/03195

The second author was supported by National Science Foundation Grant 1007196.

##  1\. Introduction

We consider irreducible algebraic curves $C\subset\mathbb{C}P^{2}$. Such a
curve has a finite set of singular points, $\\{z_{i}\\}_{i=1}^{n}$; a
neighborhood of each intersects $C$ in a cone on a link $L_{i}\subset S^{3}$.
A fundamental question asks what possible configurations of links
$\\{L_{i}\\}$ arise in this way. In this generality the problem is fairly
intractable and research has focused on a restricted case, in which each
$L_{i}$ is connected, and thus a knot $K_{i}$, and $C$ is a rational curve,
meaning that there is a rational surjective map $\mathbb{C}P^{1}\to C$. Such a
curve is called rational cuspidal. Being rational cuspidal is equivalent to
$C$ being homeomorphic to $S^{2}$.

Our results apply in the case of multiple singular points, but the following
statement gives an indication of the nature of the results and their
consequences.

######  Theorem 1.1.

Suppose that $C$ is a rational cuspidal curve of degree $d$ with one singular
point, a cone on the knot $K$, and the Alexander polynomial of $K$ is expanded
at $t=1$ to be
$\Delta_{K}(t)=1+\frac{(d-1)(d-2)}{2}(t-1)+(t-1)^{2}\sum_{l}k_{l}t^{l}$. Then
for all $j,0\leq j\leq d-3$, $k_{d(d-j-3)}=(j-1)(j-2)/2$.

There are three facets to the work here:

  1. (1)

We begin with a basic observation that a neighborhood $Y$ of $C$ is built from
the 4–ball by attaching a 2–handle along the knot $K=\\#K_{i}$ with framing
$d^{2}$, where $d$ is the degree of the curve. Thus, its boundary,
$S^{3}_{d^{2}}(K)$, bounds the rational homology ball
$\mathbb{C}P^{2}\setminus Y$. From this, it follows that the Heegaard Floer
correction term satisfies $d(S^{3}_{d^{2}}(K),{{\mathfrak{s}}}_{m})=0$ if
$d|m$, for properly enumerated Spincstructures ${{\mathfrak{s}}}_{m}$.

  2. (2)

Because each $K_{i}$ is an algebraic knot (in particular an $L$–space knot),
the Heegaard Floer complex $\operatorname{\it CFK}^{\infty}(S^{3},K_{i})$ is
determined by the Alexander polynomial of $K_{i}$, and thus the complex
$\operatorname{\it CFK}^{\infty}(S^{3},K)$ and the $d$–invariants are also
determined by the Alexander polynomials of the $K_{i}$.

  3. (3)

The constraints that arise on the Alexander polynomials, although initially
appearing quite intricate, can be reinterpreted in compact form using
semigroups of singular points. In this way, we can relate these constraints to
well-known conjectures.

###  1.1. The conjecture of Fernández de Bobadilla, Luengo, Melle-Hernandez
and Némethi

In [5] the following conjecture was proposed. It was also verified for all
known examples of rational cuspidal curves.

######  Conjecture 1.2 ([5]).

Suppose that the rational cuspidal curve $C$ of degree $d$ has critical points
$z_{1},\dots,z_{n}$. Let $K_{1},\dots,K_{n}$ be the corresponding links of
singular points and let $\Delta_{1},\dots,\Delta_{n}$ be their Alexander
polynomials. Let $\Delta=\Delta_{1}\cdot\ldots\cdot\Delta_{n}$, expanded as

$\Delta(t)=1+\frac{(d-1)(d-2)}{2}(t-1)+(t-1)^{2}\sum_{j=0}^{2g-2}k_{l}t^{l}.$

Then for any $j=0,\dots,d-3$, $k_{d(d-j-3)}\leq(j+1)(j+2)/2$, with equality
for $n=1$.

We remark that the case $n=1$ of the conjecture is Theorem 1.1. We will prove
this result in Section 4.4. Later we will also prove an alternative
generalization of Theorem 1.1 for the case $n>1$, stated as Theorem 5.4, which
is the main result of the present article. The advantage of this formulation
over the original conjecture lies in the fact that it gives precise values of
the coefficients $k_{d(d-j-3)}$. Theorem 6.5 provides an equivalent statement
of Theorem 5.4.

######  Acknowledgements.

The authors are grateful to Matt Hedden, Jen Hom and András Némethi for
fruitful discussions. The first author wants to thank Indiana University for
hospitality.

##  2\. Background: Algebraic Geometry and Rational Cuspidal Curves

In this section we will present some of the general theory of rational
cuspidal curves. Section 2.1 includes basic information about singular points
of plane curves. In Section 2.2 we discuss the semigroup of a singular point
and its connections to the Alexander polynomial of the link. We shall use
results from this section later in the article to simplify the equalities that
we obtain. In Section 2.3 we describe results from [5] to give some flavor of
the theory. In Section 2.4 we provide a rough sketch of some methods used to
study rational cuspidal curves. We refer to [13] for an excellent and fairly
up-to-date survey of results on rational cuspidal curves.

###  2.1. Singular points and algebraic curves

For a general introduction and references to this subsection, we refer to [3,
7], or to [12, Section 10] for a more topological approach. In this article we
will be considering algebraic curves embedded in $\mathbb{C}P^{2}$. Thus we
will use the word _curve_ to refer to a zero set of an irreducible homogeneous
polynomial $F$ of degree $d$. The _degree_ of the curve is the degree of the
corresponding polynomial.

Let $C$ be a curve. A point $z\in C$ is called _singular_ if the gradient of
$F$ vanishes at $z$. Singular points of irreducible curves in
$\mathbb{C}P^{2}$ are always isolated. Given a singular point and a
sufficiently small ball $B\subset\mathbb{C}P^{2}$ around $z$, we call
$K=C\cap\partial B$ the _link_ of the singular point. The singular point is
called _cuspidal_ or _unibranched_ if $K$ is a knot, that is a link with one
component, or equivalently, if there is an analytic map $\psi$ from a disk in
$\mathbb{C}$ onto $C\cap B$.

Unless specified otherwise, all singular points are assumed to be cuspidal.

Two unibranched singular points are called _topologically equivalent_ if the
links of these singular points are isotopic; see for instance [7, Definition
I.3.30] for more details. A unibranched singular point is topologically
equivalent to one for which the local parametrization $\psi$ is given in local
coordinates $(x,y)$ on $B$ by $t\mapsto(x(t),y(t))$, where $x(t)=t^{p}$,
$y(t)=t^{q_{1}}+\ldots+t^{q_{n}}$ for some positive integers
$p,q_{1},\ldots,q_{n}$ satisfying $p<q_{1}<q_{2}<\ldots<q_{n}$. Furthermore,
if we set $D_{i}=\gcd(p,q_{1},\ldots,q_{i})$, then $D_{i}$ does not divide
$q_{i+1}$ and $D_{n}=1$. The sequence $(p;q_{1},\ldots,q_{n})$ is called the
_characteristic sequence_ of the singular point and $p$ is called the
_multiplicity_. Sometimes $n$ is referred to as the _number of Puiseux pairs_
, a notion which comes from an alternative way of encoding the sequence
$(p;q_{1},\ldots,q_{n})$. We will say that a singular point is of type
$(p;q_{1},\ldots,q_{n})$ if it has a presentation of this sort in local
coefficients.

The link of a singular point with a characteristic sequence
$(p;q_{1},\ldots,q_{n})$ is an $(n-1)$–fold iterate of a torus knot
$T(p^{\prime},q^{\prime})$, where $p^{\prime}=p/D_{1}$ and
$q^{\prime}=q_{1}/D_{1}$; see for example [3, Sections 8.3 and 8.5] or [26,
Chapter 5.2]. In particular, if $n=1$, the link is a torus knot $T(p,q_{1})$.
In all cases, the genus of the link is equal to $\mu/2=\delta$, where $\mu$ is
the Milnor number and $\delta$ is the so-called $\delta$–invariant of the
singular point, see [7, page 205], or [12, Section 10]. The genus is also
equal to half the degree of the Alexander polynomial of the link of the
singular point. The Milnor number can be computed from the following formula,
see [12, Remark 10.10]:

$\mu=(p-1)(q_{1}-1)+\sum_{i=2}^{n}(D_{i}-1)(q_{i}-q_{i-1}).$

Suppose that $C$ is a degree $d$ curve with singular points
$z_{1},\ldots,z_{n}$ (and $L_{1},\ldots,L_{n}$ are their links). The genus
formula, due to Serre (see [12, Property 10.4]) states that the genus of $C$
is equal to

$g(C)=\frac{1}{2}(d-1)(d-2)-\sum_{i=1}^{n}\delta_{i}.$

If all the critical points are cuspidal, we have $\delta_{i}=g(L_{i})$, so the
above formula can be written as

(2.1) $g(C)=\frac{1}{2}(d-1)(d-2)-\sum_{i=1}^{n}g(L_{i}).$

In particular, $C$ is rational cuspidal (that is, it is a homeomorphic image
of a sphere) if and only $\sum g(L_{i})=\frac{1}{2}(d-1)(d-2)$.

###  2.2. Semigroup of a singular point

The notion of the semigroup associated to a singular point is a central notion
in the subject, although in the present work we use only the language of
semigroups, not the algebraic aspects. We refer to [26, Chapter 4] or [7, page
214] for details and proofs. Suppose that $z$ is a cuspidal singular point of
a curve $C$ and $B$ is a sufficiently small ball around $z$. Let
$\psi(t)=(x(t),y(t))$ be a local parametrization of $C\cap B$ near $z$; see
Section 2.1. For any polynomial $G(x,y)$ we look at the order at $0$ of an
analytic map $t\mapsto G(x(t),y(t))\in\mathbb{C}$. Let $S$ be the set
integers, which can be realized as the order for some $G$. Then $S$ is clearly
a semigroup of $\mathbb{Z}_{\geq 0}$. We call it the _semigroup of the
singular point_. The semigroup can be computed from the characteristic
sequence, for example for a sequence $(p;q_{1})$, $S$ is generated by $p$ and
$q_{1}$. The _gap sequence_ , $G:=\mathbb{Z}_{\geq 0}\setminus S$, has
precisely $\mu/2$ elements and the largest one is $\mu-1$, where $\mu$ is the
Milnor number.

We now assume that $K$ is the link of the singular point $z$. Explicit
computations of the Alexander polynomial of $K$ show that it is of the form

(2.2) $\Delta_{K}(t)=\sum_{i=0}^{2m}(-1)^{i}t^{n_{i}},$

where $n_{i}$ form an increasing sequence with $n_{0}=0$ and $n_{2m}=2g$,
twice the genus of $K$.

Expanding $t^{n_{2i}}-t^{n_{2i-1}}$ as
$(t-1)(t^{n_{2i}-1}+t^{n_{2i}-2}+\ldots+t^{n_{2i-1}})$ yields

(2.3) $\Delta_{K}(t)=1+(t-1)\sum_{j=1}^{k}t^{g_{j}},$

for some finite sequence $0<g_{1}<\ldots<g_{k}$. We have the following result
(see [26, Exercise 5.7.7]).

######  Lemma 2.4.

The sequence $g_{1},\ldots,g_{k}$ is the gap sequence of the semigroup of the
singular point. In particular $k=\\#G=\mu/2$, where $\mu$ is the Milnor
number, so $\\#G$ is the genus.

Writing $t^{g_{j}}$ as $(t-1)(t^{g_{j}-1}+t^{g_{j}-2}+\ldots+t+1)+1$ in (2.3)
yields the following formula

(2.5) $\Delta_{K}(t)=1+(t-1)g(K)+(t-1)^{2}\sum_{j=0}^{\mu-2}k_{j}t^{j},$

where $k_{j}=\\#\\{m>j\colon m\not\in S\\}$.

We shall use the following definition.

######  Definition 2.6.

For any finite increasing sequence of positive integers $G$, we define

(2.7) $I_{G}(m)=\\#\\{k\in G\cup\mathbb{Z}_{<0}\colon k\geq m\\},$

where $\mathbb{Z}_{<0}$ is the set of the negative integers. We shall call
$I_{G}$ the _gap function_ , because in most applications $G$ will be a gap
sequence of some semigroup.

######  Remark 2.8.

We point out that for $j=0,\ldots,\mu-2$, we have $I_{G}(j+1)=k_{j}$, where
the $k_{j}$ are as in (2.5).

######  Example 2.9.

Consider the knot $T(3,7)$. Its Alexander polynomial is

$\displaystyle\frac{(t^{21}-1)(t-1)}{(t^{3}-1)(t^{7}-1)}=$ $\displaystyle\
1-t+t^{3}-t^{4}+t^{6}-t^{8}+t^{9}-t^{11}+t^{12}$ $\displaystyle=$
$\displaystyle\ 1+(t-1)(t+t^{2}+t^{4}+t^{5}+t^{8}+t^{11})$ $\displaystyle=$
$\displaystyle\ 1+6(t-1)+$ $\displaystyle+(t-1)^{2}$
$\displaystyle\left(6+5t+4t^{2}+4t^{3}+3t^{4}+2t^{5}+2t^{6}+2t^{7}+t^{8}+t^{9}+t^{10}\right).$

The semigroup is $(0,3,6,7,9,10,12,13,14,\dots)$. The gap sequence is
$1,2,4,5,8,11$.

######  Remark 2.10.

The passage from (2.2) through (2.3) to (2.5) is just an algebraic
manipulation, and thus it applies to any knot whose Alexander polynomial has
form (2.2). In particular, according to [21, Theorem 1.2] it applies to any
$L$–space knot. In this setting we will also call the sequence
$g_{1},\dots,g_{k}$ the _gap sequence_ of the knot and denote it by $G_{K}$;
we will write $I_{K}(m)$ for the gap function relative to $G_{K}$. Even though
the complement $\mathbb{Z}_{\geq 0}\setminus G_{K}$ is not always a semigroup,
we still have $\\#G_{K}=\frac{1}{2}\deg\Delta_{K}$. This property follows
immediately from the symmetry of the Alexander polynomial.

###  2.3. Rational cuspidal curves with one cusp

The classification of rational cuspidal curves is a challenging old problem,
with some conjectures (like the Coolidge–Nagata conjecture [4, 14]) remaining
open for many decades. The classification of curves with a unique critical
point is far from being accomplished; the special case when the unique
singular point has only one Puiseux term (its link is a torus knot) is
complete [5], but even in this basic case, the proof is quite difficult.

To give some indication of the situation, consider two families of rational
cuspidal curves. The first one, written in projective coordinates on
$\mathbb{C}P^{2}$ as $x^{d}+y^{d-1}z=0$ for $d>1$, the other one is
$(zy-x^{2})^{d/2}-xy^{d-1}=0$ for $d$ even and $d>1$. These are of degree $d$.
Both families have a unique singular point, in the first case it is of type
$(d-1;d)$, in the second of type $(d/2;2d-1)$. In both cases, the Milnor
number is $(d-1)(d-2)$, so the curves are rational. An explicit
parametrization can be easily given as well.

There also exist more complicated examples. For instance, Orevkov [18]
constructed rational cuspidal curves of degree $\phi_{j}$ having a single
singular point of type $(\phi_{j-2};\phi_{j+2})$, where $j$ is odd and $j>5$.
Here the $\phi_{j}$ are the Fibonacci numbers, $\phi_{0}=0$, $\phi_{1}=1$,
$\phi_{j+2}=\phi_{j+1}+\phi_{j}$. As an example, there exists a rational
cuspidal curve of degree $13$ with a single singular point of type $(5;34)$.
Orevkov’s construction is inductive and by no means trivial. Another family
found by Orevkov are rational cuspidal curves of degree $\phi_{j-1}^{2}-1$
having a single singular point of type $(\phi_{j-2}^{2};\phi_{j}^{2})$, for
$j>5$, odd.

The main result of [5] is that apart of these four families of rational
cuspidal curves, there are only two sporadic curves with a unique singular
point having one Puiseux pair, one of degree $8$, the other of degree $16$.

###  2.4. Constraints on rational cuspidal curves.

Here we review some constraints for rational cuspidal curves. We refer to [13]
for more details and references. The article [5] shows how these constraints
can be used in practice. The fundamental constraint is given by (2.1). Next,
Matsuoka and Sakai [11] proved that if $(p_{1};q_{11},\ldots,q_{1k_{1}})$,
…,$(p_{n};q_{n1},\ldots,q_{nk_{n}})$ are the only singular points occurring on
a rational cuspidal curve of degree $d$ with $p_{1}\geq\ldots\geq p_{n}$, then
$p_{1}>d/3$. Later, Orevkov [18] improved this to
$\alpha(p_{1}+1)+1/\sqrt{5}>d$, where $\alpha=(3+\sqrt{5})/2\sim 2.61$ and
showed that this inequality is asymptotically optimal (it is related to the
curves described in Section 2.1). Both proofs use very deep algebro-geometric
tools. We reprove the result of [11] in Proposition 6.7 below.

Another obstruction comes from the semicontinuity of the spectrum, a concept
that arises from Hodge Theory. Even a rough definition of the spectrum of a
singular point is beyond the scope of this article. We refer to [1, Chapter
14] for a definition of the spectrum and to [5] for illustrations of its use.
We point out that recently (see [2]) a tight relation has been drawn between
the spectrum of a singular point and the Tristram–Levine signatures of its
link. In general, semicontinuity of the spectrum is a very strong tool, but it
is also very difficult to apply.

Using tools from algebraic geometry, such as the Hodge Index Theorem, Tono in
[25] proved that any rational cuspidal curve can have at most eight singular
points. An old conjecture is that a rational cuspidal curve can have at most
$4$ singular points; see [22] for a precise statement.

In [6] a completely new approach was proposed, motivated by a conjecture on
Seiberg–Witten invariants of links of surface singularities made by Némethi
and Nicolaescu; see [16]. Specifically, Conjecture 1.2 in the present article
arises from these considerations. Another reference for the general conjecture
on Seiberg–Witten invariants is [15].

##  3\. Topology, algebraic topology, and Spinc structures

Let $C\subset\mathbb{C}P^{2}$ be a rational cuspidal curve. Let $d$ be its
degree and $z_{1},\ldots,z_{n}$ be its singular points. We let $Y$ be a closed
manifold regular neighborhood of $C$, let $M=\partial Y$, and
$W=\overline{\mathbb{C}P^{2}-Y}$.

###  3.1.  Topological descriptions of $Y$ and $M$

The neighborhood $Y$ of $C$ can be built in three steps. First, disk
neighborhoods of the $z_{i}$ are selected. Then neighborhoods of $N-1$
embedded arcs on $C$ are adjoined, yielding a 4–ball. Finally, the remainder
of $C$ is a disk, so its neighborhood forms a 2–handle attached to the 4–ball.
Thus, $Y$ is a 4–ball with a 2–handle attached. The attaching curve is easily
seen to be $K=\\#K_{i}$. Finally, since the self-intersection of $C$ is
$d^{2}$, the framing of the attaching map is $d^{2}$. In particular,
$M=S^{3}_{d^{2}}(K)$.

One quickly computes that $H_{2}(\mathbb{C}P^{2},C)=\mathbb{Z}_{d}$, and
$H_{4}(\mathbb{C}P^{2},C)=\mathbb{Z}$, with the remaining homology groups 0.
Using excision, we see that the groups $H_{i}(W,M)$ are the same. Via
Lefschetz duality and the universal coefficient theorem we find that
$H_{0}(W)=\mathbb{Z}$, $H_{1}(W)=\mathbb{Z}_{d}$ and all the other groups are
0. Finally, the long exact sequence of the pair $(W,M)$ yields

$0\to H_{2}(W,M)\to H_{1}(M)\to H_{1}(W)\to 0$

which in this case is

$0\to\mathbb{Z}_{d}\to\mathbb{Z}_{d^{2}}\to\mathbb{Z}_{d}\to 0.$

This is realized geometrically by letting the generator of $H_{2}(W,M)$ be
${\it H}\cap W$, where $\it H\subset\mathbb{C}P^{2}$ is a generic line. Its
boundary is algebraically $d$ copies of the meridian of the attaching curve
$K$ in the 2–handle decomposition of $Y$.

Taking duals we see that the map $H^{2}(W)\to H^{2}(M)$, which maps
$\mathbb{Z}_{d}\to\mathbb{Z}_{d^{2}}$, takes the canonical generator to $d$
times the dual to the meridian in $M=S^{3}_{d^{2}}(K)$.

###  3.2.  Spinc structures

For any space $X$ there is a transitive action of $H^{2}(X)$ on Spinc($X$).
Thus, $W$ has $d$ Spincstructures and $M$ has $d^{2}$ such structures.

Since $\mathbb{C}P^{2}$ has a Spincstructure with first Chern class a dual to
the class of the line, its restriction to $W$ is a structure whose restriction
to $M$ has first Chern class equal to $d$ times the dual to the meridian.

For a cohomology class $z\in H^{2}(X)$ and a Spincstructure
${{\mathfrak{s}}}$, one has
$c_{1}(z\cdot{{\mathfrak{s}}})-c_{1}({{\mathfrak{s}}})=2z$. Thus for each
$k\in\mathbb{Z}$, there is a Spincstructure on $M$ which extends to $W$ having
first Chern class of the form $d+2kd$. Notice that for $d$ odd, all
$md\in\mathbb{Z}_{d^{2}}$ for $m\in\mathbb{Z}$ occur as first Chern classes of
Spincstructures that extend over $W$, but for $d$ even, only elements of the
form $md$ with $m$ odd occur. (Thus, for $d$ even, there are $d$ extending
structures, but only $d/2$ first Chern classes that occur.)

According to [20, Section 3.4], the Spincstructures on $M$ have an enumeration
${{\mathfrak{s}}}_{m}$, for $m\in[-d^{2}/2,d^{2}/2]$, which can be defined via
the manifold $Y$. Specifically, ${{\mathfrak{s}}}_{m}$ is defined to be the
restriction to $M$ of the Spincstructure on $Y$, ${{\mathfrak{t}}}_{m}$, with
the property that $\left<c_{1}({{\mathfrak{t}}}_{m}),C\right>+d^{2}=2m$. We
point out that if $d$ is even, ${{\mathfrak{s}}}_{d^{2}/2}$ and
${{\mathfrak{s}}}_{-d^{2}/2}$ denote the same structure; compare Remark 4.5
below.

It now follows from our previous observations that the structures
${{\mathfrak{s}}}_{m}$ that extend to $W$ are those with $m=kd$ for some
integer $k$, $-d/2\leq k\leq d/2$ if $d$ is odd. If $d$ is even, then those
that extend have $m=kd/2$ for some odd $k$, $-d\leq k\leq d$. For future
reference, we summarize this with the following lemma.

######  Lemma 3.1.

If $W^{4}=\overline{\mathbb{C}P^{2}-Y}$ where $Y$ is a neighborhood of a
rational cuspidal curve $C$ of degree $d$ (as constructed above), then the
Spincstructure ${{\mathfrak{s}}}_{m}$ on $\partial W^{4}$ extends to $W^{4}$
if $m=kd$ for some integer $k$, $-d/2\leq k\leq d/2$ if $d$ is odd. If $d$ is
even, then those that extend have $m=kd/2$ for some odd $k$, $-d\leq k\leq d$.
Here ${{\mathfrak{s}}}_{m}$ is the Spincstructure on $\partial W$ which
extends to a structure ${{\mathfrak{t}}}$ on $Y$ satisfying
$\left<c_{1}({{\mathfrak{t}}}_{m}),C\right>+d^{2}=2m$.

##  4\. Heegaard Floer theory

Heegaard Floer theory [19] associates to a 3–manifold $M$ with Spincstructure
${{\mathfrak{s}}}$, a filtered, graded chain complex
$CF^{\infty}(M,{{\mathfrak{s}}})$ over the field $\mathbb{Z}_{2}$. A
fundamental invariant of the pair $(M,{{\mathfrak{s}}})$, the correction term
or $d$–invariant, $d(M,{{\mathfrak{s}}})\in\mathbb{Q}$, is determined by
$CF^{\infty}(M,{{\mathfrak{s}}})$. The manifold $M$ is called an $L$–space if
certain associated homology groups are of rank one [21].

A knot $K$ in $M$ provides a second filtration on
$CF^{\infty}(M,{{\mathfrak{s}}})$ [19]. In particular, for $K\subset S^{3}$
there is a bifiltered graded chain complex $\operatorname{\it
CFK}^{\infty}(K)$ over the field $\mathbb{Z}_{2}$. It is known that for
algebraic knots the complex is determined by the Alexander polynomial of $K$.
More generally, this holds for any knot upon which some surgery yields an
$L$–space; these knots are called $L$–space knots.

The Heegaard Floer invariants of surgery on $K$, in particular the
$d$–invariants of $S^{3}_{q}(K)$, are determined by this complex, and for
$q>2(\textrm{genus}(K))$ the computation of $d(S^{3}_{q}(K),{{\mathfrak{s}}})$
from $CFK^{\infty}(K)$ is particularly simple. In this section we will
illustrate the general theory, leaving the details to references such as [9,
10].

###  4.1. $\operatorname{\it CFK}^{\infty}(K)$ for $K$ an algebraic knot

Figure 1 is a schematic illustration of a finite complex over
$\mathbb{Z}_{2}$. Each dot represents a generator and the arrows indicate
boundary maps. Abstractly it is of the form
$0\to\mathbb{Z}_{2}^{4}\to\mathbb{Z}_{2}^{5}\to 0$ with homology
$\mathbb{Z}_{2}$. The complex is bifiltered with the horizontal and vertical
coordinates representing the filtrations levels of the generators. We will
refer to the two filtrations levels as the $(i,j)$–filtrations levels. The
complex has an absolute grading which is not indicated in the diagram; the
generator at filtration level $(0,6)$ has grading 0 and the boundary map
lowers the grading by 1. Thus, there are five generators at grading level 0
and four at grading level one. We call the first set of generators type A and
the second type B.

We will refer to a complex such as this as a staircase complex of length $n$,
$\operatorname{St}(v)$, where $v$ is a $(n-1)$–tuple of positive integers
designating the length of the segments starting at the top left and moving to
the bottom right in alternating right and downward steps. Furthermore we
require that the top left vertex lies on the vertical axis and the bottom
right vertex lies on the horizontal axis. Thus, the illustration is of
$\operatorname{St}(1,2,1,2,2,1,2,1)$. The absolute grading of
$\operatorname{St}(v)$ is defined by setting the grading of the top left
generator to be equal to $0$ and the boundary map to lower the grading by $1$.

The vertices of $\operatorname{St}(K)$ will be denoted
$\operatorname{Vert}(St(K))$. We shall write
$\operatorname{Vert}_{A}(\operatorname{St}(K))$ to denote the set of type A
vertices and write $\operatorname{Vert}_{B}(\operatorname{St}(K))$ for the set
of vertices of type B.

If $K$ is a knot admitting an $L$–space surgery, in particular an algebraic
knot (see [8]), then it has Alexander polynomial of the form
$\Delta_{K}(t)=\sum_{i=0}^{2m}(-1)t^{n_{i}}$. To such a knot we associate a
staircase complex, $\operatorname{St}(K)=\operatorname{St}(n_{i+1}-n_{i})$,
where $i$ runs from 0 to $2m-1$. As an example, the torus knot $T(3,7)$ has
Alexander polynomial $1-t+t^{3}-t^{4}+t^{6}-t^{8}+t^{9}-t^{11}+t^{12}$. The
corresponding staircase complex is $\operatorname{St}(1,2,1,2,2,1,2,1)$.

0.6[subgriddiv=1,gridcolor=gray](-1,-1)(-1,-1)(7,7)  Figure 1. The staircase
complex $\operatorname{St}(K)$ for the torus knot $T(3,7)$.

Given any finitely generated bifiltered complex $S$, one can form a larger
complex $S\otimes\mathbb{Z}_{2}[U,U^{-1}]$, with differentials defined by
$\partial(x\otimes U^{i})=(\partial x)\otimes U^{i}$. It is graded by
$gr(x\otimes U^{k})=gr(x)-2k$. Similarly, if $x$ is at filtration level
$(i,j)$, then $x\otimes U^{i}$ is at filtration level $(i-k,j-k)$. If $K$
admits an $L$–space surgery, then
$\operatorname{St}(K)\otimes\mathbb{Z}_{2}[U,U^{-1}]$ is isomorphic to
$\operatorname{\it CFK}^{\infty}(K)$. Figure 2 illustrates a portion of
$\operatorname{St}(T(3,7))\otimes\mathbb{Z}_{2}[U,U^{-1}]$; that is, a portion
of the Heegaard Floer complex $\operatorname{\it CFK}^{\infty}(T(3,7))$.

0.6[subgriddiv=1,gridcolor=gray](-1,-1)(-1,-1)(7,7)  Figure 2. A portion of
$\operatorname{\it CFK}^{\infty}(T(3,7))$.

###  4.2. $d$–invariants from $\operatorname{\it CFK}^{\infty}(K)$.

We will not present the general definition of the $d$–invariant of a
3–manifold with Spincstructure; details can be found in [19]. However, in the
case that a 3–manifold is of the form $S^{3}_{q}(K)$ where $q\geq
2($genus($K$)), there is a simple algorithm (originating from [20, Section 4],
we use the approach of [9, 10]) to determine this invariant from
$\operatorname{\it CFK}^{\infty}(K)$.

If $m$ satisfies $-d/2\leq m\leq d/2$, one can form the quotient complex

$\operatorname{\it CFK}^{\infty}(K)/\operatorname{\it
CFK}^{\infty}(K)\\{i<0,j<m\\}.$

We let $d_{m}$ denote the least grading in which this complex has a nontrivial
homology class, say $[z]$, where $[z]$ must satisfy the added constraint that
for all $i>0$, $[z]=U^{i}[z_{i}]$ for some homology class $[z_{i}]$ of grading
$d_{m}+2i$.

In [20, Theorem 4.4], we find the following result.

######  Theorem 4.1.

For the Spincstructure ${{\mathfrak{s}}}_{m}$,
$d(S^{3}_{q}(K),{{\mathfrak{s}}}_{m})=d_{m}+\frac{(-2m+q)^{2}-q}{4q}$.

###  4.3. From staircase complexes to the $d$–invariants

Let us now define a distance function for a staircase complex by the formula

$J_{K}(m)=\min_{(v_{1},v_{2})\in\operatorname{Vert}(\operatorname{St}(K))}\max(v_{1},v_{2}-m),$

where $v_{1},v_{2}$ are coordinates of the vertex $v$. Observe that the
minimum can always be taken with respect to the set of vertices of type A. The
function $J_{K}(m)$ represents the greatest $r$ such that the region $\\{i\leq
0,j\leq m\\}$ intersects $\operatorname{St}(K)\otimes U^{r}$ nontrivially. It
is immediately clear that $J_{K}(m)$ is a non-increasing function. It is also
immediate that for $m\geq g$ we have $J_{K}(m)=0$.

0.6[subgriddiv=1,gridcolor=gray](0,0)(-7,-7)(7,7)  Figure 3. The function
$J(m)$ for the knot $T(3,7)$. When $(0,m)$ lies on the dashed vertical
intervals, the function $J(m)$ is constant; when it is on solid vertical
intervals the function $J(m)$ is decreasing. The dashed lines connecting
vertices to points on the vertical axis indicate how the ends of dashed and
solid intervals are constructed.

For the sake of the next lemma we define $n_{-1}=-\infty$.

######  Lemma 4.2.

Suppose $m\leq g$. We have $J_{K}(m+1)-J_{K}(m)=-1$ if $n_{2i-1}-g\leq
m<n_{2i}-g$ for some $i$, and $J_{K}(m+1)=J_{K}(m)$ otherwise.

###### Proof.

The proof is purely combinatorial. We order the type A vertices of
$\operatorname{St}(K)$ so that the first coordinate is increasing, and we
denote these vertices $v_{0},\ldots,v_{k}$. For example, for
$\operatorname{St}(T(3,7))$ as depicted on Figure 1, we have $v_{0}=(0,6)$,
$v_{1}=(1,4)$, $v_{2}=(2,2)$, $v_{3}=(4,1)$ and $v_{4}=(6,0)$. We denote by
$(v_{i1},v_{i2})$ the coordinates of the vertex $v_{i}$.

A verification of the two following facts is straightforward:

(4.3) $\begin{split}\max(v_{i1},v_{i2}-m)&=v_{i1}\textrm{ if and only if
$m\geq v_{i1}-v_{i2}$}\\\
\max(v_{i1},v_{i2}-m)&\geq\max(v_{i-1,1},v_{i-1,2}-m)\textrm{ if and only if
$m\leq v_{i1}-v_{i-1,2}$}.\end{split}$

By the definition of the staircase complex we also have
$v_{i1}-v_{i2}=n_{2i}-g$ and $v_{i1}-v_{i-1,2}=n_{2i-1}-g$. The second
equation of (4.3) yields

$J_{K}(m)=\max(v_{i1},v_{i2}-m)\text{ if and only if
}m\in[n_{2i-1},n_{2i+1}].$

Then the first equation of (4.3) allows to compute the difference
$J_{K}(m+1)-J_{K}(m)$. ∎

The relationship between $J_{K}$ and the $d$–invariant is given by the next
result.

######  Proposition 4.4.

Let $K$ be an algebraic knot, let $q>2g(K)$, and let $m\in[-q/2,q/2]$ be an
integer. Then

$d(S^{3}_{q}(K),\mathfrak{s}_{m})=\frac{(-2m+q)^{2}-q}{4q}-2J(m).$

###### Proof.

Denote by $S_{i}$ the subcomplex $\operatorname{St}(K)\otimes U^{i}$ in
$\operatorname{\it CFK}^{\infty}(K)$. The result depends on understanding the
homology of the image of $S_{i}$ in $\operatorname{\it
CFK}^{\infty}(K)/\operatorname{\it CFK}^{\infty}(K)\\{i<0,j<m\\}$. Because of
the added constraint (see the paragraph before Theorem 4.1), we only have to
look at the homology classes supported on images of the type A vertices.
Notice that if $i>J_{K}(m)$, then at least one of the type A vertices is in
$\operatorname{\it CFK}^{\infty}(K)\\{i<0,j<m\\}$. But all the type A vertices
are homologous in $S_{i}$, and since these generate $H_{0}(S_{i})$, the
homology of the image in the quotient is 0. On the other hand, if $i\leq
J_{K}(m)$, then none of the vertices of $S_{i}$ are in $\operatorname{\it
CFK}^{\infty}(K)\\{i<0,j<m\\}$ and thus the homology of $S_{i}$ survives in
the quotient.

It follows that the least grading of a nontrivial class in the quotient arises
from the $U^{J_{K}(m)}$ translate of one of type A vertices of
$S_{0}=\operatorname{St}(K)$. Since $U$ lowers grading by 2, the grading is
$-2J_{K}(m)$. The result follows by applying the shift described in Theorem
4.1. ∎

######  Remark 4.5.

Notice that in the case that $q$ is even, the integer values $m=-q/2$ and
$m=q/2$ are both in the given interval. One easily checks that Proposition 4.4
yields the same value at these two endpoints.

We now relate the $J$ function to the semigroup of the singular point. Let
$I_{K}$ be the gap function as in Definition 2.6 and Remark 2.10.

######  Proposition 4.6.

If $K$ is the link of an algebraic singular point, then for $-g\leq m\leq g$
$J_{K}(m)=I_{K}(m+g)$.

###### Proof.

In Section 2.2 we described the gap sequence in terms of the exponents
$n_{i}$. It follows immediately that the growth properties of $I_{K}(m+g)$ are
identical to those of $J_{K}(m)$, as described in Lemma 4.2. Furthermore,
since the largest element in the gap sequence is $2g-1$, we have
$I_{K}(2g)=J_{K}(g)=0$. ∎

###  4.4. Proof of Theorem 1.1

According to Lemma 3.1, the Spincstructures on $S^{3}_{d^{2}}(K)$ that extend
to the complement $W$ of a neighborhood of $C$ are precisely those
${{\mathfrak{s}}}_{m}$ where $m=kd$ for some $k$, where $-d/2\leq k\leq d/2$;
here $k\in\mathbb{Z}$ if $d$ odd, and $k\in\mathbb{Z}+\frac{1}{2}$ if $d$ is
even. Since $W$ is a rational homology sphere, by [19, Proposition 9.9] the
associated $d$–invariants are 0, so by Proposition 4.4, letting $q=d^{2}$ and
$m=kd$, we have

$2J_{K}(kd)=\frac{(-2kd+d^{2})^{2}-d^{2}}{4d^{2}}.$

By Proposition 4.6 we can replace this with

$8I_{G_{K}}(kd+g)=(d-2k-1)(d-2k+1).$

Now $g=d(\frac{d-3}{2})+1$, so by substituting $j=k+\frac{d-3}{2}$ we obtain

$8I_{K}(jd+1)=4(d-j+1)(d-j+2)$

and $j\in[-3/2,\ldots,d-3/2]$ is an integer regardless of the parity of $d$.
The proof is accomplished by recalling that $k_{jd}=I_{K}(jd+1)$, see Remark
2.8.

##  5\. Constraints on general rational cuspidal curves

###  5.1. Products of staircase complexes and the $d$–invariants

In the case that there is more than one cusp, the previous approach continues
to apply, except the knot $K$ is now a connected sum of algebraic knots.

For the connected sum $K=\\#K_{i}$, the complex $\operatorname{\it
CFK}^{\infty}(K)$ is the tensor product of the $\operatorname{\it
CFK}^{\infty}(K_{i})$. To analyze this, we consider the tensor product of the
staircase complexes $\operatorname{St}(K_{i})$. Although this is not a
staircase complex, the homology is still $\mathbb{Z}_{2}$, supported at
grading level 0. For the tensor product we shall denote by
$\operatorname{Vert}(\operatorname{St}(K_{1})\otimes\ldots\otimes\operatorname{St}(K_{n}))$
the set of vertices of the corresponding complex. These are of the form
$v_{1}+\ldots+v_{n}$, where $v_{j}\in\operatorname{Vert}(K_{j})$,
$j=1,\ldots,n$.

Any element of the form $a_{1q_{1}}\otimes a_{2q_{2}}\otimes\cdots\otimes
a_{nq_{n}}$ represents a generator of the homology of the tensor product,
where the $a_{iq_{i}}$ are vertices of type A taken from each
$\operatorname{St}(K_{i})$. Furthermore, if the translated subcomplex
$\text{St}(K)\otimes U^{i}\subset\text{St}(K)\otimes\mathbb{Z}_{2}[U,U^{-1}]$
intersects $\operatorname{\it CFK}^{\infty}(K)\\{i<0,j<m\\}$ nontrivially,
then the intersection contains one of these generators. Thus, the previous
argument applies to prove the following.

######  Proposition 5.1.

Let $q>2g-1$, where $g=g(K)$ and $m\in[-q/2,q/2]$. Then we have

$d(S^{3}_{q}(K),\mathfrak{s}_{m})=-2J_{K}(m)+\frac{(-2m+q)^{2}-q}{4q},$

where $J_{K}(m)$ is the minimum of $\max(\alpha,\beta-m)$ over all elements of
form $a_{1q_{1}}\otimes a_{2q_{2}}\otimes\ldots\otimes a_{nq_{n}}$, where
$(\alpha,\beta)$, is the filtration level of the corresponding element.

Since the $d$–invariants vanish for all Spincstructures that extend to $W$, we
have:

######  Theorem 5.2.

If $C$ is a rational cuspidal curve of degree $d$ with singular points $K_{i}$
and $K=\\#K_{i}$, then for all $k$ in the range $[-d/2,d/2]$, with
$k\in\mathbb{Z}$ for $d$ odd and $k\in\mathbb{Z}+\frac{1}{2}$ for $d$ even:

$J_{K}(kd)=\frac{(d-2k-1)(d-2k+1)}{8}.$

###### Proof.

We have from the vanishing of the $d$–invariants,
$d(S^{3}_{d^{2}}(K),{{\mathfrak{s}}}_{m})$ (for $m=kd$) the condition

$J_{K}(m)=\frac{(-2m+d^{2})^{2}-d^{2}}{8d^{2}}.$

The result then follows by substituting $m=kd$ and performing algebraic
simplifications. ∎

###  5.2. Restatement in terms of $I_{K_{i}}(m)$.

We now wish to restate Theorem 5.2 in terms of the coefficients of the
Alexander polynomial, properly expanded. As before, for the gap sequence for
the knot $K_{i}$, denoted $G_{K_{i}}$, let

$I_{i}(s)=\\#\\{k\geq s\colon k\in G_{K_{i}}\cup\mathbb{Z}_{<0}\\}.$

For two functions $I,I^{\prime}\colon\mathbb{Z}\to\mathbb{Z}$ bounded below we
define the following operation

(5.3) $I\diamond I^{\prime}(s)=\min_{m\in\mathbb{Z}}I(m)+I^{\prime}(s-m).$

As pointed out to us by Krzysztof Oleszkiewicz, in real analysis this
operation is sometimes called the _infimum convolution_.

The following is the main result of this article.

######  Theorem 5.4.

Let $C$ be a rational cuspidal curve of degree $d$. Let $I_{1},\dots,I_{n}$ be
the gap functions associated to each singular point on $C$. Then for any
$j\in\\{-1,0,\ldots,d-2\\}$ we have

$I_{1}\diamond I_{2}\diamond\ldots\diamond
I_{n}(jd+1)=\frac{(j-d+1)(j-d+2)}{2}.$

######  Remark 5.5.

  * •

For $j=-1$, the left hand side is $d(d-1)/2=d-1+(d-1)(d-2)/2$. The meaning of
the equality is that $\sum\\#G_{j}=(d-1)(d-2)/2$ which follows from (2.1) and
Lemma 2.4. Thus, the case $j=-1$ does not provide any new constraints.
Likewise, for $j=d-2$ both sides are equal to $0$.

  * •

We refer to Section 6.2 for a reformulation of Theorem 5.4.

  * •

We do not know if Theorem 5.4 settles Conjecture 1.2 for $n>1$. The passage
between the two formulations appears to be more complicated; see [17,
Proposition 7.1.3] and the example in Section 6.1.

Theorem 5.4 is an immediate consequence of the arguments in Section 4.4
together with the following proposition.

######  Proposition 5.6.

As in (5.3), let $I_{K}$ be given by $I_{1}\diamond\ldots\diamond I_{n}$, for
the gap functions $I_{1},\ldots,I_{n}$. Then $J_{K}(m)=I_{K}(m+g)$.

###### Proof.

The proof follows by induction over $n$. For $n=1$, the statement is
equivalent to Proposition 4.6. Suppose we have proved it for $n-1$. Let
$K^{\prime}=K_{1}\\#\ldots\\#K_{n-1}$ and let $J_{K^{\prime}}(m)$ be the
corresponding $J$ function. Let us consider a vertex
$v\in\operatorname{Vert}(\operatorname{St}_{1}(K)\otimes\ldots\otimes\operatorname{St}_{n}(K))$.
We can write this as $v^{\prime}+v_{n}$, where
$v^{\prime}\in\operatorname{Vert}(\operatorname{St}(K_{1})\otimes\cdots\otimes\operatorname{St}(K_{n-1}))$
and $v_{n}\in\operatorname{Vert}(\operatorname{St}(K_{n}))$. We write the
coordinates of the vertices as $(v_{1},v_{2})$,
$(v^{\prime}_{1},v^{\prime}_{2})$ and $(v_{n1},v_{n2})$, respectively. We have
$v_{1}=v^{\prime}_{1}+v_{n1}$, $v_{2}=v^{\prime}_{2}+v_{n2}$. We shall need
the following lemma.

######  Lemma 5.7.

For any four integers $x,y,z,w$ we have

$\max(x+y,z+w)=\min_{k\in\mathbb{Z}}\left(\max(x,z-k)+\max(y,w+k)\right).$

###### Proof of Lemma 5.7.

The direction ‘$\leq$’ is trivial. The equality is attained at $k=z-x$. ∎

_Continuation of the proof of Proposition 5.6._

Applying Lemma 5.7 to $v_{1}^{\prime},v_{2}^{\prime},v_{n1},v_{n2}-m$ and
taking the minimum over all vertices $v$ we obtain

$J_{K}(m)=\min_{v\in\operatorname{Vert}(\operatorname{St}(K_{1})\otimes\ldots\otimes\operatorname{St}(K_{n}))}\max(v_{1},v_{2}-m)=\\\
\min_{v^{\prime}\in\operatorname{Vert}^{\prime}}\min_{v_{n}\in\operatorname{Vert}_{n}}\min_{k\in\mathbb{Z}}\left(\max(v^{\prime}_{1},v_{2}^{\prime}-k)+\max(v_{n1},v_{n2}+k-m)\right),$

where we denote
$\operatorname{Vert}^{\prime}=\operatorname{Vert}(\operatorname{St}(K_{1})\otimes\cdots\otimes\operatorname{St}(K_{n-1}))$
and $\operatorname{Vert}_{n}=\operatorname{Vert}(\operatorname{St}(K_{n}))$.

The last expression is clearly
$\min_{k\in\mathbb{Z}}J_{K^{\prime}}(k)+J_{K_{n}}(m-k)$. By the induction
assumption this is equal to

$\min_{k\in\mathbb{Z}}I_{K^{\prime}}(k+g^{\prime})+I_{K_{n}}(m-k+g_{n})=I_{K}(m+g),$

where $g^{\prime}=g(K^{\prime})$ and $g_{n}=g(K_{n})$ are the genera, and we
use the fact that $g=g^{\prime}+g_{n}$. ∎

##  6\. Examples and applications

###  6.1. A certain curve of degree $6$

As described, for instance, in [5, Section 2.3, Table 1], there exists an
algebraic curve of degree $6$ with two singular points, the links of which are
$K=T(4,5)$ and $K^{\prime}=T(2,9)$. The values of $I_{K}(m)$ for
$m\in\\{0,\ldots,11\\}$ are $\\{6,6,5,4,3,3,3,2,1,1,1,1\\}$. The values of
$I_{K^{\prime}}(m)$ for $m\in\\{0,\ldots,7\\}$ are $\\{4,4,3,3,2,2,1,1\\}$. We
readily get

$I\diamond I^{\prime}(1)=10,\ I\diamond I^{\prime}(7)=6,\ I\diamond
I^{\prime}(13)=3,\ I\diamond I^{\prime}(19)=1,$

exactly as predicted by Theorem 5.4.

On the other hand, the computations in [5] confirm Conjecture 1.2 but we
sometimes have an inequality. For example $k_{6}=5$, whereas Conjecture 1.2
states $k_{6}\leq 6$. This shows that Theorem 5.4 is indeed more precise.

###  6.2. Reformulations of Theorem 5.4

Theorem 5.4 was formulated in a way that fits best with its theoretical
underpinnings. In some applications, it is advantageous to reformulate the
result in terms of the function counting semigroup elements in the interval
$[0,k]$. To this end, we introduce some notation.

Recall that for a semigroup $S\subset\mathbb{Z}_{\geq 0}$, the gap sequence of
$G$ is $\mathbb{Z}_{\geq 0}\setminus S$. We put $g=\\#G$ and for $m\geq 0$ we
define

(6.1) $R(m)=\\#\\{j\in S\colon j\in[0,m)\\}.$

######  Lemma 6.2.

For $m\geq 0$, $R(m)$ is related to the gap function $I(m)$ (see (2.7)) by the
following relation:

(6.3) $R(m)=m-g+I(m).$

###### Proof.

Let us consider an auxiliary function $K(m)=\\#\\{j\in[0,m):j\in G\\}$. Then
$K(m)=g-I(m)$. Now $R(m)+K(m)=m$, which completes the proof. ∎

We extend $R(m)$ by (6.3) for all $m\in\mathbb{Z}$. We remark that $R(m)=m-g$
for $m>\sup G$ and $R(m)=0$ for $m<0$. In particular, $R$ is a non-negative,
non-decreasing function.

We have the following result.

######  Lemma 6.4.

Let $I_{1},\dots,I_{n}$ be the gap functions corresponding to the semigroups
$S_{1},\ldots,S_{n}$. Let $g_{1},\dots,g_{n}$ be given by
$g_{j}=\\#{\mathbb{Z}_{\geq 0}\setminus S_{j}}$. Let $R_{1},\ldots,R_{n}$ be
as in (6.1). Then

$R_{1}\diamond R_{2}\diamond\ldots\diamond
R_{n}(m)=m-g+I_{1}\diamond\ldots\diamond I_{n}(m),$

where $g=g_{1}+\ldots+g_{n}$.

###### Proof.

To simplify the notation, we assume that $n=2$; the general case follows by
induction. We have

$\displaystyle R_{1}\diamond R_{2}(m)=$
$\displaystyle\min_{k\in\mathbb{Z}}R_{1}(k)+R_{2}(m-k)=$
$\displaystyle=\min_{k\in\mathbb{Z}}(k-g_{1}+I_{1}(k)+m-k-g_{2}+I_{2}(m-k))=$
$\displaystyle=m-g_{1}-g_{2}+I_{1}\diamond I_{2}(m).$

∎

Now we can reformulate Theorem 5.4:

######  Theorem 6.5.

For any rational cuspidal curve of degree $d$ with singular points
$z_{1},\dots,z_{n}$, and for $R_{1},\dots,R_{n}$ the functions as defined in
(6.1), one has that for any $j=\\{-1,\ldots,d-2\\}$,

$R_{1}\diamond R_{2}\diamond\ldots\diamond R_{n}(jd+1)=\frac{(j+1)(j+2)}{2}.$

This formulation follows from Theorem 5.4 by an easy algebraic manipulation
together with the observation that by (2.1) and Lemma 2.4, the quantity $g$
from Lemma 6.4 is given by $\frac{(d-1)(d-2)}{2}$.

The formula bears strong resemblance to [5, Proposition 2], but in that
article only the ‘$\geq$’ part is proved and an equality in case $n=1$ is
conjectured.

######  Remark 6.6.

Observe that by definition

$R_{1}\diamond\ldots\diamond
R_{n}(k)=\min_{\begin{subarray}{c}k_{1},\ldots,k_{n}\in\mathbb{Z}\\\
k_{1}+\ldots+k_{n}=k\end{subarray}}R_{1}(k_{1})+\ldots+R_{n}(k_{n}).$

Since for negative values $R_{j}(k)=0$ and $R_{j}$ is non-decreasing on
$[0,\infty)$, the minimum will always be achieved for
$k_{1},\ldots,k_{n}\geq-1$.

###  6.3. Applications

From Theorem 6.5 we can deduce many general estimates for rational cuspidal
curves. Throughout this subsection we shall be assuming that $C$ has degree
$d$, its singular points are $z_{1},\ldots,z_{n}$, the semigroups are
$S_{1},\ldots,S_{n}$, and the corresponding $R$–functions are
$R_{1},\ldots,R_{n}$. Moreover, we assume that the characteristic sequence of
the singular point $z_{i}$ is $(p_{i};q_{i1},\ldots,q_{ik_{i}})$. We order the
singular points so that that $p_{1}\geq p_{2}\geq\ldots\geq p_{n}$.

We can immediately prove the result of Matsuoka–Sakai, [11], following the
ideas in [5, Section 3.5.1].

######  Proposition 6.7.

We have $p_{1}>d/3$.

###### Proof.

Suppose $3p_{1}\leq d$. It follows that for any $j$, $3p_{j}\leq d$. Let us
choose $k_{1},\ldots,k_{n}\geq-1$ such that $\sum k_{j}=d+1$. For any $j$, the
elements $0,p_{j},2p_{j},\ldots$ all belong to the $S_{j}$. The function
$R_{j}(k_{j})$ counts elements in $S_{j}$ strictly smaller than $k_{j}$, hence
for any $\varepsilon>0$ we have

$R_{j}(k_{j})\geq
1+\genfrac{\lfloor}{\rfloor}{}{1}{k_{j}-\varepsilon}{p_{j}}.$

Using $3p_{j}\leq d$ we rewrite this as $R_{j}(k_{j})\geq
1+\genfrac{\lfloor}{\rfloor}{}{1}{3k_{j}-3\varepsilon}{d}$. Since
$\varepsilon>0$ is arbitrary, setting $\delta_{j}=1$ if $d|3k_{j}$, and $0$
otherwise, we write

$R_{j}(k_{j})\geq 1+\genfrac{\lfloor}{\rfloor}{}{1}{3k_{j}}{d}-\delta_{j}.$

We get

(6.8) $\sum_{j\colon
d|3k_{j}}R_{j}(k_{j})\geq\genfrac{\lfloor}{\rfloor}{}{1}{\sum 3k_{j}}{d}.$

Using the fact that
$\genfrac{\lfloor}{\rfloor}{}{1}{a}{d}+\genfrac{\lfloor}{\rfloor}{}{1}{b}{d}\geq\genfrac{\lfloor}{\rfloor}{}{1}{a+b}{d}-1$
for any $a,b\in\mathbb{Z}$, we estimate the other terms:

(6.9) $\sum_{j\colon d\not\;|\,3k_{j}}R_{j}(k_{j})\geq
1+\genfrac{\lfloor}{\rfloor}{}{1}{3\sum k_{j}}{d}.$

Since $\sum k_{j}=d+1$, there must be at least one $j$ for which $d$ does not
divide $3k_{j}$. Hence adding (6.8) to (6.9) we obtain

$R_{1}(k_{1})+\ldots+R_{n}(k_{n})\geq
1+\genfrac{\lfloor}{\rfloor}{}{1}{\sum_{j=1}^{n}3k_{j}}{d}=1+\genfrac{\lfloor}{\rfloor}{}{1}{3d+3}{d}=4.$

This contradicts Theorem 6.5 for $j=1$, and the contradiction concludes the
proof. ∎

We also have the following simple result.

######  Proposition 6.10.

Suppose that $p_{1}>\frac{d+n-1}{2}$. Then $q_{11}<d+n-1$.

###### Proof.

Suppose that $p_{1}>\frac{d+n-1}{2}$ and $q_{11}>d+n-1$. It follows that
$R_{1}(d+n)=2$. But then we choose $k_{1}=d+n$, $k_{2}=\ldots=k_{n}=-1$ and we
get $\sum_{j=1}^{n}R_{j}(k_{j})=2$, hence

$R_{1}\diamond R_{2}\diamond\ldots\diamond R_{n}(d+1)\leq 2$

contradicting Theorem 6.5. ∎

###  6.4. Some examples and statistics

We will now present some examples and statistics, where we compare our new
criterion with the semicontinuity of the spectrum as used in [5, Property
$(SS_{l})$] and the Orevkov criterion [18, Corollary 2.2]. It will turn out
that the semigroup distribution property is quite strong and closely related
to the semicontinuity of the spectrum, but they are not the same. There are
cases which pass one criterion and fail to another. Checking the semigroup
property is definitely a much faster task than comparing spectra; refer to [6,
Section 3.6] for more examples.

######  Example 6.11.

Among the 1,920,593 cuspidal singular points with Milnor number of the form
$(d-1)(d-2)$ for $d$ ranging between $8$ and $64$, there are only 481 that
pass the semigroup distribution criterion, that is Theorem 1.1. All of these
pass the Orevkov criterion $\overline{M}<3d-4$. Of those 481, we compute that
475 satisfy the semicontinuity of the spectrum condition and 6 them fail the
condition; these are: $(8;28,45)$, $(12;18,49)$, $(16;56,76,85)$,
$(24;36,78,91)$, $(24;84,112,125)$, $(36;54,114,133)$.

######  Remark 6.12.

The computations in Example 6.11 were made on a PC computer during one
afternoon. Applying the spectrum criteria for all these cases would take much
longer. The computations for degrees between $12$ and $30$ is approximately
$15$ times faster for semigroups; the difference seems to grow with the
degree. The reason is that even though the spectrum can be given explicitly
from the characteristic sequence (see [24]), it is a set of fractional numbers
and the algorithm is complicated.

######  Example 6.13.

There are $28$ cuspidal singular points with Milnor number equal to
$110=(12-1)(12-2)$. We ask, which of these singular points can possibly occur
as a unique singular point on a degree $12$ rational curve? We apply the
semigroup distribution criterion. Only 8 singular points pass the criterion,
as is seen on Table 1.

(3;56) | fails at $j=1$ | (6;9,44) | fails at $j=1$ | (8;12,14,41) | fails at $j=3$
---|---|---|---|---|---
(4;6,101) | fails at $j=1$ | (6;10,75) | fails at $j=1$ | (8;12,18,33) | fails at $j=4$
(4;10,93) | fails at $j=1$ | (6;14,59) | fails at $j=2$ | (8;12,22,25) | passes
(4;14,85) | fails at $j=1$ | (6;15,35) | fails at $j=2$ | (8;12,23) | passes
(4;18,77) | fails at $j=1$ | (6;16,51) | fails at $j=2$ | (8;14,33) | fails at $j=1$
(4;22,69) | fails at $j=1$ | (6;20,35) | fails at $j=4$ | (9;12,23) | passes
(4;26,61) | fails at $j=1$ | (6;21,26) | passes | (10;12,23) | passes
(4;30,53) | fails at $j=1$ | (6;22,27) | passes | (11;12) | passes
(4;34,45) | fails at $j=1$ | (6;23) | passes |  |
(6;8,83) | fails at $j=1$ | (8;10,57) | fails at $j=2$ |  |
Table 1. Semigroup property for cuspidal singular points with Milnor number
$12$. If a cuspidal singular point fails the semigroup criterion, we indicate
the first $j$ for which $I(12j+1)\neq\frac{(j-d+1)(j-d+2)}{2}$.

Among the curves in Table 1, all those that are obstructed by the semigroup
distribution, are also obstructed by the semicontinuity of the spectrum. The
spectrum also obstructs the case of $(8;12,23)$.

######  Example 6.14.

There are 2330 pairs $(a,b)$ of coprime integers, such that $(a-1)(b-1)$ is of
form $(d-1)(d-2)$ for $d=5,\ldots,200$. Again we ask if there exists a degree
$d$ rational cuspidal curve having a single singular point with characteristic
sequence $(a;b)$. Among these 2330 cases, precisely 302 satisfy the semigroup
distribution property. Out of these 302 cases, only one, namely $(2;13)$, does
not appear on the list from [5]; see Section 2.3 for the list. It is therefore
very likely that the semigroup distribution property alone is strong enough to
obtain the classification of [5].

######  Example 6.15.

In Table 2 we present all the cuspidal points with Milnor number
$(30-1)(30-2)$ that satisfy the semicontinuity of the spectrum. Out of these,
all but the three ($(18;42,65)$, $(18;42,64,69)$ and $(18;42,63,48)$) satisfy
the semigroup property. All three fail the semigroup property for $j=1$. In
particular, for these three cases the semigroup property obstructs the cases
which pass the semicontinuity of the spectrum criterion.

(15; 55, 69) | (18;42,64,69) | (20; 30, 59) | (25; 30, 59)
---|---|---|---
(15; 57, 71) | (18;42,63,68) | (24; 30, 57, 62) | (27; 30, 59)
(15;59) | (20; 30,55,64) | (24;30,58,63) | (28; 30,59)
(18;42,65) | (20; 30,58,67) | (24; 30,59) | (29; 30)
Table 2. Cuspidal singular points with Milnor number $752$ satisfying the
semicontinuity of the spectrum criterion.

######  Example 6.16.

The configuration of five critical points $(2;3)$, $(2;3)$, $(2;5)$, $(5;7)$
and $(5;11)$ passes the semigroup, the spectrum and the Orevkov criterion for
a degree $10$ curve. In other words, none of the aforementioned criteria
obstructs the existence of such curve. We point out that it is conjectured
(see [13, 22]) that a rational cuspidal curve can have at most $4$ singular
points. In other words, these three criteria alone are insufficient to prove
that conjecture.

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