2.1 Resolution of singularities

Recall from [Reference WallWal04] that every curve singularity can be resolved by blowing up (sufficiently many times), and the diffeomorphism type of the link determines the topology of the resolution. If $\pi : \widetilde X \to X$ is a (multiple) blow-up at a point $x\in X$, and $C\subset X$ is a curve, we call $\overline {\pi ^{-1}(C\setminus \{x\})}$ the proper transform of $C$ under $\pi$, and we denote it by $\widetilde C$; we also call $\pi ^{-1}(C)$ the total transform of $C$ under $\pi$, and we denote it by $\overline C$.

For every singular curve $C\subset X$, there exists a composition of blow-ups $\pi : \widetilde X \to X$ such that $\widetilde C$ is smooth; we call any such $\pi$ a resolution of $C$.

Given a resolution $\pi$ of $C$, consider the restriction $n := \pi |_{\widetilde {C}}$ of $\pi$ to the proper transform $\widetilde {C}$ of $C$. We call $n: \widetilde {C} \to C$ (and, by abuse of notation, $\widetilde {C}$) the normalization of $C$. Note that this is essentially independent of the choice of the resolution: if $\pi '$ is another resolution of $C$ and we construct $n'$ accordingly, then there is an isomorphism $\phi$ between the sources of $n$ and $n'$ such that $n = n' \circ \phi$. We also observe that $C$ is cuspidal if and only if the normalization is one-to-one; in particular, in this case it is a homeomorphism onto its image.

There are two natural stopping points when resolving a singularity: the minimal resolution is the smallest resolution such that the proper transform $\widetilde C$ of $C$ is smooth; the normal crossing resolution is the smallest resolution such that the total transform $\overline C$ of $C$ is a normal crossing divisor, i.e. all singularities are double points.Footnote ¹

Figure 1. A handle diagram interpretation of the blow-up of a singularity of type $(p,q)$. (Recall that $p< q$ in a singularity of type $(p,q)$.) There are $p$ strands on each side, and the rectangular box with the $+1$ denotes a full twist.

Recall that the multiplicity $m_p$ of a singularity $p$ of $C \subset X$ is the minimal intersection of a germ of a divisor $D$ at $p$ with $C$; that is, $m_p = \min _D (C\cdot D)_p$; the multiplicity of a singularity has the following interpretation: blow up $X$ at $p$, obtaining $\widetilde X$, which contains the corresponding exceptional divisor $E$; then $m_p$ is the intersection number of $E$ and $\widetilde {C}$, i.e. $m_p = \widetilde {C}\cdot E$. In terms of homology, there is an orthogonal decomposition $H_2(\widetilde X) = H_2(X) \oplus \mathbb {Z}[E]$, and we have $[\widetilde C] = [C]-m_p[E]$. In particular, $[\widetilde C]^{2} = [C]^{2}-m_p^{2}$. See Figure 1 for a Kirby calculus interpretation of the blow-up at a cusp with multiplicity p.

We record the singularity types at each step in the minimal resolution into a sequence of integers, called the multiplicity sequence, as follows. Suppose that the minimal resolution of a singularity of $C$ is a sequence of $k$ blow-ups, and call $C_j$ the proper transform of $C$ after the first $j-1$ blow-ups, so that $C_1 = C$, and $C_{k+1}$ is the proper transform of $C$ in the minimal resolution. Then $C_j$ has a singularity of multiplicity $m_j$, and we write $[m_1,\ldots,m_k]$ for the multiplicity sequence of the singularity of $C$. Note that $m_{j+1} \le m_{j}$ for each $j$, but that not every sequence of integers correspond to the multiplicity sequence of a singularity; for instance, $[3,2,2]$ is not the multiplicity sequence of any singularity. The multiplicity sequence of an irreducible singularity determines the singularity [Reference WallWal04, Theorem 3.5.6]; however, the multiplicity sequence is defined both for irreducible and reducible singularities.

Given a cuspidal curve $C$, we say that the multiplicity multi-sequence of $C$ is the union of all multiplicity sequences of singularities of $C$. This is only well-defined up to choosing an order of the cusps. Instead of making the choice, we just sort all the entries decreasingly. We use double-bracket notation to denote the multiplicity multi-sequence: $[[m_1,\ldots, m_n]]$.

Here we describe the normal crossing resolution of singularities of type $(p,q)$. This is best described in terms of continued fraction expansions; given a rational number $r$, we write

\[ r = [a_1,\ldots,a_k]^- = a_1-\cfrac{1}{a_2-\cfrac{1}{\dots-\cfrac{1}{a_k}}} \]

for its (negative or Hirzebruch–Jung) continued fraction expansion, where $a_i \ge 2$ for each $i\ge 2$.

In what follows, we adopt the notation $a^{[\ell ]}$ to denote a string of $\ell$ entries, all equal to $a$; for instance, $[2^{[\ell ]}]$ will be the multiplicity sequence of the singularity of type $(2,2\ell +1)$.

It consists of a plumbing of spheres along a three-legged star-shaped graph decorated by Euler classes. The Euler class on the central vertex is $-1$; that is, the central vertex is the exceptional divisor corresponding to the last blow-up in the resolution. Two of the legs are linear chains whose decorations give the continued fraction expansions of

\[ \frac{p}{p-q^{*}} \quad \text{and}\quad \frac{q}{q-p^{*}}, \]

where $qq^{*}\equiv 1 \mod p$ and $0< q^{*}< p$, and $pp^{*} \equiv 1 \mod q$ and $0< p^{*}< q$.

More precisely, the resolution graph of the singularity of type $(p,q)$ looks like the following:

where $ {p}/({p-q^{*}}) = [a_1,\ldots,a_m]^{-}$ and $ {q}/({q-p^{*}}) = [b_1,\ldots,b_n]^{-}$, and the hollow dot represents the proper transform of $C$. (It might be helpful to recall that $ {p}/({p-q}) = [a_m,\ldots,a_1]^{-}$ and $ {q}/({q-r(p,q)}) = [b_n,\ldots,b_1]^{-}$, where $r(p,q)$ is the remainder of the division of $p$ by $q$.)

The following lemma is well-known among algebraic geometers; we include a proof both for completeness and to give an example of the techniques for low-dimensional topologists.

If $C$ is a curve of self-intersection $s$ with a unique singularity of type $(p,q)$, then the total transform of $C$ is the full plumbing graph, where the hollow dot is replaced by a vertex with Euler class $s-pq$. In our applications, in the case of a unique singularity, if $s-pq>1$, we will blow up further times along the edge between the central vertex and this short leg to change the coefficient on the central vertex to $-2$ and add a chain of $s-pq-2$ more $(-2)$-vertices, a single $(-1)$-vertex, and a single $+1$-vertex.

In general, we can resolve each singularity of $C$ independently of the others; furthermore, if all singularities are of type $(p_i,q_i)$, the total transform of $C$ will look like a plumbing tree, obtained by fusing the resolution graphs of the individual singularities along the hollow vertex, whose weight is $s-\sum _i p_iq_i$.

For instance, when $C$ is a rational curve with self-intersection $16$, with a singularity of type $(2,3)$ and one of type $(2,5)$, the total transform $\overline C$ of $C$ in the normal crossing resolution of $C$ is described by the following plumbing diagram:

Here the vertex labeled with $0 = 16 - 2\cdot 3 - 2\cdot 5$ is the proper transform $\widetilde C$ of $C$, and is obtained by fusing the corresponding hollow vertices in the resolution diagrams for the two singularities of $C$.

2.2 Singular symplectic curves

We are now ready to define the main objects of interest in the paper.

As the singularities are locally modeled on complex curves, we have the resolutions of § 2.1. In particular, $C$ can be parameterized as a locally injective smooth image of a parameterization $f:\Sigma \to X$ which is an embedding away from the singular points ($\Sigma$ is the normalization). When this normalization $\Sigma$ is connected, the curve $C$ is irreducible. When $\Sigma$ is disconnected, $C$ is reducible and we consider it as a curve with labeled components.

When we want to refer to the abstract topological type of the singular curve, namely the labeled components and their homology classes, and the topological types of the singularities (encoded by the links of the singularities up to isotopy), we call it a configuration and denote it with script font. A specific symplectic realization $C$ of a configuration $\mathcal {C}$ will be a singular symplectic curve with the homology and singularity data specified by $\mathcal {C}$.

We restrict to singularities modeled by complex curves because these are the singularity types that can occur in pseudoholomorphic curves by two results of McDuff, that we summarize in the following theorem.

As the symplectic condition is $C^{1}$-open, it is preserved by $C^{1}$-small isotopies. It follows that any $J$-holomorphic singular curve for any $\omega$-compatible $J$ is $C^{1}$-close to a curve $C_0$ whose singular points have complex models. Since $C^{1}$-small isotopies do not affect questions of existence or uniqueness, we model singular symplectic curves with complex Darboux charts as given in Definition 2.5.

There are two different notions of genus for a singular curve $C$. Recall that every singularity can be perturbed; that is, it can be replaced with a Milnor fiber, locally defined by a perturbation of the equation for the singularity. The genus of the Milnor fiber is also the 3-genus of its link, and it is a topological invariant of the singularity. If $q$ is a singular point of $C$, with $r$ branches, we define $\mu (q)$, the Milnor number of $q$, as the first Betti number of the Milnor fiber of the singularity of $C$ at $q$; we define $\delta (q)$ by $2\delta (q) = \mu (q) + r - 1$. If $q$ is a cusp point of $C$, then $\delta (q)$ is just the genus of the Milnor fiber of the singularity.

For the following definition, we borrow the terminology from algebraic geometry.

Definition 2.7 Let $C$ be a singular symplectic curve. We call $p_g(C) := g(\Sigma )$ where $\Sigma$ is the normalization of $C$, the geometric genus of $C$, and we will say that $C$ is rational if its geometric genus is zero.

We call

(2.1)\begin{equation} p_a(C) := p_g(C) + \sum_{p \in \operatorname{Sing}(C)} \delta(p) \end{equation}

the arithmetic genus of $C$.

As $p_a(C)$ is the genus of a symplectic smoothing of $C$, $C$ satisfies the adjunction formula

(2.2)\begin{equation} K_X\cdot C + C\cdot C + 2-2p_a(C) = 0. \end{equation}

It follows from the adjunction formula that for a singularity with multiplicity sequence $[m_1,\ldots, m_n]$

(2.3)\begin{equation} \delta(p) = \frac 12\sum m_j(m_j-1). \end{equation}

Our focus will be on singular symplectic curves in $\mathbb {C}\mathbb {P}^{2}$. In this case, because $H_2(\mathbb {C}\mathbb {P}^{2};\mathbb {Z})\cong \mathbb {Z}$, the homology class of a curve is determined by an integer. We use the convention that $[\mathbb {C}\mathbb {P}^{1}]$ with the complex orientation is the positive generator which we often denote by $h$. Note that $\langle [\omega ],[\mathbb {C}\mathbb {P}^{1}]\rangle = \int _{\mathbb {C}\mathbb {P}^{1}}\omega >0$. As all symplectic surfaces have positive symplectic area, their integral homology classes will correspond to positive integers. Extending the terminology from the algebraic case, we have the following definition.

We focus on classifying pairs $(X,C)$ which are relatively minimal, meaning there are no exceptional spheres in $X$ that are disjoint from $C$. The following definition will be convenient.

There is a natural equivalence relation between different singular curves.

We may sometimes drop the term equisingular for brevity and simply say that $C_0$ and $C_1$ are symplectically isotopic, with the understanding that all of the isotopies we consider in this article are equisingular.

Note that each singularity is locally uniquely determined up to equisingular isotopy by the smooth isotopy class of its link. Our definition of a symplectic singular curve required the existence of a local symplectomorphism to a complex curve singularity. Here we verify that complex curve singularities with the same topological type have the same symplectomorphism type. Thus, we can take one representative complex model for each topological type of curve singularity.

Proof. We can choose two complex charts around $(0,0)$ such that, for $i=0,1$, $C_i$ has a Puiseux expansions

\[ y = c^{i}_{b_1} x^{{b_1}/m} + c^{i}_{b_1+1} x^{(b_1+1)/m} + \cdots. \]

Two singularities are topologically equivalent if and only if they have the same Puiseux exponents $(m;b_1,\ldots,b_k)$. In terms of the Puiseux expansion, this means that $c^{i}_{b_j}\neq 0$ and that $c_\ell = 0$ for every $b_j < \ell < b_{j+1}$ such that $e_j \nmid \ell$. (Recall from the beginning of the section that the sequence $\{e_i\}$ is defined recursively as $e_1 = \gcd (m,b_1)$ and $e_i = \gcd (e_{i-1},b_i)$ for $i>1$.) As the space of sequences $\{c_\ell \}$ satisfying these constraints and the space of local charts are both connected, we can always isotope one singularity into the other without changing the singularity type.

2.3 The cuspidal contact structure and its caps

Note that the contact structure depends only on the topological types of the singularities of $C$, its geometric genus, and its self-intersection. In addition, observe that, in order to define $(Y_C,\xi _C)$, we are using that there exists a normal crossing resolution.

This is a generalization of the contact structures introduced by Chen (see [Reference ChenChe18, Section 4]) for rational curves. The focus in [Reference ChenChe18] was on curves in $\mathbb {C}\mathbb {P}^{2}$ (or in symplectic manifolds with the same algebraic topology). Chen described the contact structure using transverse symplectic handle attachment, instead of using the normal crossing resolution. The two contact structures are actually the same because they are both the canonical contact structure on the boundary of a small concave neighborhood of the curve, though we do not use this fact here.

2.4 Topological obstructions

We mention here two techniques used previously to obstruct rational cuspidal curves in $\mathbb {C}\mathbb {P}^{2}$ which have topological interpretations and can thus obstruct symplectic curves (not just complex algebraic curves).

The semigroup condition was first discovered by Borodzik and Livingston [Reference Borodzik and LivingstonBL14]. Recall that to each curve singularity $x$ there is an associated semigroup $\Gamma _x$, that records the local multiplicities of intersections of germs of curves with the singularity; for instance, for a singularity of type $(p,q)$, the semigroup is generated by $p$ and $q$. By convention, we declare that $0$ belongs to $\Gamma _x$, so $0$ is the first element of the semigroup, and the multiplicity is the second. We define the counting function $R_x$ of $\Gamma _x$ as $R_x(n) = \#(\Gamma _x \cap (-\infty,n-1))$, and the minimum convolution $F\diamond G$ of two functions as $(F\diamond G)(n) = \min _{k\in \mathbb {Z}} F(k) + G(n-k)$; given a rational cuspidal curve $C$ of degree $d$, with cusps $x_1,\ldots,x_k$, and associated $R$-functions $R_1,\ldots, R_k$, let $R = R_1\diamond \cdots \diamond R_k$. Then, [Reference Borodzik and LivingstonBL14, Theorem 6.5] asserts that, for each $j=-1,0,\ldots,d-2$,

(2.4)\begin{equation} R(jd+1) = \frac{(j+1)(j+2)}2. \end{equation}

Note that this is a smooth obstruction: it obstructs the existence of a rational homology 4-ball which glues by a diffeomorphism of the boundaries to a regular neighborhood of $C$.

We call spectrum semicontinuity an inequality on the Levine–Tristram signature and nullity functions of the links, $\sigma _\bullet, \eta _\bullet : S^{1}\to \mathbb {Z}$. It is related to the algebro-geometric spectrum semicontinuity, formulated in terms of Hodge-theoretic data, and then recast in more topological terms by Borodzik and Némethi [Reference Borodzik and NémethiBN12, Corollary 2.5.4]. From a curve $C$, by choosing a generic line $\ell$, we obtain a genus-$p_g(C)$ cobordism in $S^{3}\times [0,1]$ from the connected sum $K$ of links of all singularities of $C$, to the torus link $T(d,d)$; this cobordism is obtained from removing a neighborhood of a path connecting all singularities of $C$, and a neighborhood of the line $\ell$. Suppose that $C$ is rational and its singularities are $k$ cusps. Then $K$ is a knot, and for every $\zeta \in S^{1}_! \subset S^{1}$ we have

(2.5)\begin{equation} |\sigma_{T(d,d)}(\zeta) - \sigma_K(\zeta)| + |\eta_{T(d,d)}(\zeta)- \eta_{K}(\zeta)| \le d-1. \end{equation}

Here $S^{1}_!$ is the unit circle with all Knotennullstellen (roots of integer polynomials such that $p(1) = 1$) removed; that is,

\[ S^{1}_! = S^{1} \setminus \{\alpha \mid \exists p(t) \in \mathbb{Z}[t], p(1) = 1, p(\alpha) = 0\}. \]

This inequality was essentially proved by Nagel and Powell [Reference Nagel and PowellNP17], who also introduced the notation and terminology for $S^{1}_!$ (see also [Reference ConwayCon21, Theorem 2.12]).

To be concrete: all transcendental complex numbers of norm $1$ belong to $S^{1}_!$, so the set of Knotennullstellen has measure $0$. A root of unity belongs to $S^{1}_!$ if and only if its order is a prime power; for this it suffices to evaluate cyclotomic polynomials at $1$: $\Phi _{p^{r}}(1) = p$ for every prime $p$ and positive integer $r$, whereas $\Phi _n(1) = 1$ whenever $n$ has at least two distinct prime factors.

The obstruction (2.5) is topological, once we assume that there exists a locally flatly embedded sphere $\ell$ in the homology class $h$ that intersects $C$ transversely and positively in $d$ points. This assumption is satisfied when $C$ is a symplectic curve, because we can choose an almost complex structure such that $C$ is $J$-holomorphic (by Lemma 3.4) and then choose a generic $J$-holomorphic line. Then each $J$-holomorphic line intersects $C$ positively, and the space of $J$-holomorphic lines has (real) dimension $4$. The $J$-holomorphic lines which intersect $C$ at a singular point or tangentially has positive codimension (real codimension $2$) in the space of all $J$-holomorphic lines. Therefore there is a $J$-holomorphic line which intersects $C$ positively and transversally in $d$ points.

3.1 Proper transforms in symplectic blow-ups

In the symplectic category, blowing up and blowing down take on a slightly different character than they do in algebraic geometry [Reference McDuffMcD91]. To perform a symplectic blow-up at a point $p$ in a symplectic manifold $(X,\omega )$, we first choose a Darboux chart centered at $p$. Remove a ball inside this chart of radius $\lambda$, and collapse the boundary by the Hopf fibration to obtain the exceptional divisor. Alternatively, remove a ball of radius $\lambda +\varepsilon$ centered at $p$ and replace it with an $\varepsilon$-neighborhood of a copy of $\mathbb {C}\mathbb {P}^{1}$ of symplectic area $\lambda$ in $\mathcal {O}(-1)$. The symplectic form on the ring of the ball between radius $\lambda$ and $\lambda +\varepsilon$ agrees with the symplectic form on $\mathcal {O}(-1)$ when the zero section is a symplectic submanifold of area $\lambda$. A symplectic blow-down reverses the operation, replacing an $\varepsilon$-neighborhood of an exceptional divisor of symplectic area $\lambda$ with a ball of radius $\lambda +\varepsilon$ (or, equivalently, deleting the exceptional divisor of symplectic area $\lambda$ and replacing it by a closed ball of radius $\lambda$).

When we perform a symplectic blow-up at a point, we always choose the radius $\lambda +\varepsilon$ of the symplectic ball to be sufficiently small such that every sphere $S_r(p)$ of radius $0< r\leq \lambda +\varepsilon$ intersects the symplectic curves transversally. Note that using the radial vector field in the Darboux chart as a Liouville vector field, the spheres $S_r(p)$ are contact-type hypersurfaces.

Suppose now that we have a symplectic curve $C$ in $X$, and that we blow $X$ up at a (possibly singular) point $p$ on $C$. The transverse intersections of $C$ with the contact spheres are transverse links $T_r$ in the spheres $S_r(p)$. Note that if we identify spheres of different radii by a rescaling contactomorphism, the transverse links $T_r$ in the spheres of different radii $0< r<\lambda +\varepsilon$ are all transversally isotopic to each other. If $p$ is a smooth point, the transverse link $T_r$ will be the standard unknot of maximal self-linking number. If $p$ is a singularity, $T_r$ will be transversally isotopic to an algebraic link.

For the algebraic proper transform in the blow-up, the exceptional divisor $E$ replaces the point $p$, and the proper transform $\widetilde C$ intersects $E$ according to its tangent derivative information at that point. Thus, the proper transform is a finite-point compactification in the algebraic blow-up of the family of transverse links $T_r\subset S_r(p)$. We would like to have a similar proper transform in the symplectic blow-up, but in this case we delete a ball of radius $\lambda$ instead of only a point. Therefore, the links $T_r$ for $0< r\leq \lambda$ would be cut out. We cannot guarantee that the transverse links $T_r$ for $\lambda < r<\lambda +\varepsilon$ will finite-point compactify as $r\to \lambda$ with the same diffeomorphism type as the family $T_r$ for $0< r<\lambda +\varepsilon$ as $r\to 0$.

To overcome this issue, we squeeze the entire transverse isotopy $T_r$ for $0< r<\lambda +\varepsilon$ into the collar where $\lambda < r<\lambda +\varepsilon$ by reparametrizing the radial coordinate by a smooth, strictly increasing function $\rho$ such that $\rho (\widetilde {r})=\widetilde {r}-\lambda$ near $\widetilde {r}=\lambda$ and $\rho (\widetilde {r})=\widetilde r$ near $\lambda +\varepsilon$. In other words, in the symplectic blow-up, for $\lambda <\widetilde {r}<\lambda +\varepsilon$, the proper transform intersects the sphere $S_{\widetilde {r}}$ in a link $\widetilde {T}_{\widetilde {r}}$ defined by

\[ \widetilde{T}_{\widetilde{r}}:=T_{\rho(\widetilde{r})}. \]

As $\widetilde {r}\to \lambda$, $\rho (\widetilde {r})\to 0$. Therefore, this family extends via a finite-point compactification in the exceptional divisor in the same manner as the algebraic proper transform. Note that we do not change the symplectic structure on the ambient symplectic $4$-manifold, we only modify the symplectic curve.

Proof. We want to prove that the tangent spaces to the surface are still symplectic subspaces; the idea is that this is true because the bases for the tangent spaces only differ by a positive scaling factor $\rho '(r)$. Namely, if $\phi (r,t)$ denotes a polar parametrization of the original surface such that $\phi (r,t)=(T_r(t),r)$, then $\widetilde {\phi }(r,t) = (T_{\rho (r)}(t),\rho (r))$ is a parametrization for the new surface and

\[ \frac{\partial{\widetilde \phi}}{\partial r}(r,t) = \rho'(r)\frac{\partial \phi}{\partial r}(\rho(r),t), \quad \frac{\partial{\widetilde \phi}}{\partial t}(r,t) = \frac{\partial \phi}{\partial t}(\rho(r),t), \]

so the value of $\omega$ must be positive on an oriented basis for the curve parametrized by $\widetilde \phi$ because it is positive on an oriented basis for the curve parametrized by $\phi$.

A symplectic blow-down reverses this procedure. Deleting an $\varepsilon$-neighborhood of an exceptional sphere of weight $\lambda$ (i.e. the symplectic area is $\pi \lambda ^{2}$), we replace it with a symplectic ball of radius $\lambda +\varepsilon$. To define the image of a symplectic curve which intersects the exceptional divisor over $B_{\lambda +\varepsilon }(p)$ we reverse the squeezing procedure above to stretch the transverse link family back out. Namely we take the surface defined by the union of the center $p$ of the ball together with the transverse links $T_r\subset S_r(p)$ where

\[ T_r:=\widetilde{T}_{\rho^{-1}(r)}. \]

Note that the definition of the proper transform depends on the choice of the Darboux chart, of $\varepsilon$ and of $\rho$. However, the symplectic isotopy class is independent of all these choices.

3.2 Pseudoholomorphic curves

Here we prove two technical lemmas that we use throughout.

The strength of using $J$-holomorphic curves is that we have much greater control over their geometric intersections. Two general symplectic surfaces may intersect with a canceling pair of positive and negative intersections, but in this case they could not be realized as $J$-holomorphic simultaneously for the same almost complex structure $J$. When $(M,\omega )$ is a 4-manifold with a compatible almost complex structure $J$, the orientation induced by $\omega$ and $J$ agree. Two transversally intersecting $J$-holomorphic curves can be easily seen to have positive intersections because the almost complex structure induces complex orientations on each of the curves at an intersection point, which adds up to the positive orientation on the $4$-manifold induced by $J$. More generally, any (not necessarily transverse) intersection between simple $J$-holomorphic curves (possibly with singularities) contributes positively (see [Reference McDuff and SalamonMS12, Section 2.6, Appendix E.2]).

Blowing down exceptional spheres using this lemma, together with exceptional spheres appearing in a configuration, eventually we blow down spheres in all $e_i$ classes and reach a configuration in $\mathbb {C}\mathbb {P}^{2}$.

The order of the $e_i$ which we choose to $J$-holomorphically blow down does depend to some extent on the configuration $\mathcal {C}$ because if one of the surfaces in $\mathcal {C}$ represents the class $e_{i_0}-e_{i_1}-\cdots -e_{i_k}$, then we cannot blow down a $J$-holomorphic exceptional sphere in the class $e_{i_0}$ until we have blown down exceptional spheres in the classes $e_{i_1},\ldots, e_{i_k}$ first. This is because $e_{i_0}$ has negative algebraic intersection number with $e_{i_0}-e_{i_1}-\cdots -e_{i_k}$. If $k=0$, then the surface in $\mathcal {C}$ represents the class $e_{i_0}$ so it can itself act as the $J$-holomorphic exceptional sphere. However, this is the only restriction on the ordering because the positively intersecting exceptional classes can be geometrically realized disjointly.

3.3 Embeddings of plumbings into $\mathbb {C}\mathbb {P}^{2}\#N\overline {\mathbb {C}\mathbb {P}}\,\!^{2}$

Suppose $P$ is a plumbing of symplectic spheres, such that one of the spheres has self-intersection $+1$. In our context, this will typically be a neighborhood of the normal crossing resolution of a rational cuspidal curve (or possibly a further blow-up). A theorem of McDuff strongly restricts the closed symplectic manifolds in which $P$ can symplectically embed.

More generally, McDuff's result shows that if the plumbing contains a sphere of square $p\neq 4$, $p>0$, then the symplectic manifold is a blow-up of the $p$th Hirzebruch surface and the sphere is identified with the associated $0$-section. A sphere of square $4$ might be the conic in $\mathbb {C}\mathbb {P}^{2}$ or the sphere representing $2[S^{2}\times \{*\}]+[\{*\}\times S^{2}]$ in $S^{2}\times S^{2}$. A sphere of square $0$ is a fiber in a ruled surface. We typically work in cases where we can guarantee that our symplectic 4-manifold is a blow-up of $\mathbb {C}\mathbb {P}^{2}$ (or occasionally in $S^{2}\times S^{2}$). (Note that $S^{2}\times S^{2}\#\overline {\mathbb {C}\mathbb {P}}\,\!^{2} \cong \mathbb {C}\mathbb {P}^{2}\# 2 \overline {\mathbb {C}\mathbb {P}}\,\!^{2}$.)

We now discuss how to classify all symplectic embeddings of $P$ into $\mathbb {C}\mathbb {P}^{2}\#N\overline {\mathbb {C}\mathbb {P}}\,\!^{2}$. A symplectic embedding of a rational cuspidal curve is equivalent (by a sequence of blow-ups supported in a neighborhood of the rational cuspidal curve) to a symplectic embedding of its normal crossing resolution plumbing $(U,\omega _U)$.

We classify embeddings of $P$ into $\mathbb {C}\mathbb {P}^{2}\#N \overline {\mathbb {C}\mathbb {P}}\,\!^{2}$ in two steps (similar arguments appear in many classification of fillings results, starting with Lisca [Reference LiscaLis08]). First, we determine the possibilities for the map on second homology induced by the embedding. As the core spheres of the plumbing form a basis for $H_2(P)$, we just need to classify the possible classes in $H_2(\mathbb {C}\mathbb {P}^{2}\#N\overline {\mathbb {C}\mathbb {P}}\,\!^{2})$ that these symplectic spheres can represent. The restrictions here are given by the adjunction formula and the intersection form on $P$, as well as the constraint that the $+1$-sphere in $P$ is identified with $\mathbb {C}\mathbb {P}^{1}\subset \mathbb {C}\mathbb {P}^{2}\# N\overline {\mathbb {C}\mathbb {P}}\,\!^{2}$. Next, for each possible adjunctive embedding $H_2(P)\to H_2(\mathbb {C}\mathbb {P}^{2}\# N \overline {\mathbb {C}\mathbb {P}}\,\!^{2})$, we classify the geometric realizations of embeddings $P\to \mathbb {C}\mathbb {P}^{2}\# N\overline {\mathbb {C}\mathbb {P}}\,\!^{2}$ up to symplectic isotopy. Typically, we show that such a geometric embedding is unique or that it cannot exist. This will be done by supposing that we have such a geometric embedding into $\mathbb {C}\mathbb {P}^{2}\#N\overline {\mathbb {C}\mathbb {P}}\,\!^{2}$, keeping track of the configuration formed by the core symplectic spheres of the plumbing, and then blowing down using Lemma 3.5. We then look at the possible images of this configuration after blowing down to $\mathbb {C}\mathbb {P}^{2}$. This will generally be a reducible configuration, which we will then try to classify up to symplectic isotopy. This classification of reducible symplectic configurations in $\mathbb {C}\mathbb {P}^{2}$ is the subject of § 5.

Now we analyze the possible homology classes represented by the spheres in $P$. A symplectic surface $\Sigma$ in a symplectic 4-manifold $(X,\omega )$ satisfies the adjunction formula (2.2). When $X=\mathbb {C}\mathbb {P}^{2}\#N\overline {\mathbb {C}\mathbb {P}}\,\!^{2}$, $K_X=-c_1(X) = -3H+E_1+ \cdots +E_N$. Here our convention is that $h$ is the class of $\mathbb {C}\mathbb {P}^{1}$, with dual $H$ and $e_i$ is represented by the $i$th exceptional sphere with dual $E_i$. The following lemma is a generalization of [Reference LiscaLis08, Propositions 4.4]

Proof. As $\Sigma$ is a sphere ($g=0$), the adjunction formula gives

\[ 3a_0+a_1+ \cdots +a_N = 2+a_0^{2}-a_1^{2}-\cdots-a_N^{2}, \]

which can be rearranged to item (i). Note that $a_i^{2}+a_i\geq 0$ because $a_i\in \mathbb {Z}$, so the coefficient $a_0$ determines a bound on the possible coefficients $a_i$.

To prove items (ii) and (iii), we use positivity of intersections. Fix an index $0 < i_0 \le N$ and suppose first that $a_{i_0} > 0$. As $\Sigma$ and $\mathbb {C}\mathbb {P}^{1}$ are symplectic with non-negative intersections, there is an almost complex structure on $\mathbb {C}\mathbb {P}^{2}\#N\overline {\mathbb {C}\mathbb {P}}\,\!^{2}$ such that $\Sigma$ and $\mathbb {C}\mathbb {P}^{1}$ are $J$-holomorphic by Lemma 3.4. By Lemma 3.5, we can blow-down a $J$-holomorphic exceptional sphere in the class $e_i$ for some $i$. If we blow down a $J$-holomorphic sphere corresponding to $e_i$ for $i\neq i_0$, there is an induced $J'$ on the blow-down such that $\Sigma$ is a singular $J'$-holomorphic curve. Therefore, eventually we have a situation where both $\Sigma$ and $e_{i_0}$ are represented by $J$-holomorphic spheres, so if $a_{i_0} > 0$, then $[\Sigma ]\cdot e_{i_0}<0$. As $J$-holomorphic spheres must intersect non-negatively, this can only occur if $\Sigma$ is exactly equal to the $J$-holomorphic exceptional sphere representing $e_{i_0}$. In particular, if $a_{i_0} > 0$, then $a_0 = 0$, thus proving item (iii). If $a_0=0$, it implies $a_{i_0}=1$ and $\Sigma$ is a blow-up of the exceptional class representing $e_{i_0}$. In this case, for ${i_j}\neq {i_0}$ the same argument shows that $a_{i_j}$ cannot be positive so we get item (ii).

The particular cases follow from combining items (i) and (iii) and observing which integers give low values of $n^{2}+n$.

In the following lemmas, $C_i$ and $C_j$ are smooth symplectic spheres in a positive plumbing in $\mathbb {C}\mathbb {P}^{2}\#N\overline {\mathbb {C}\mathbb {P}}\,\!^{2}$ such that $[C_i]\cdot h=[C_j]\cdot h=0$.

Proof. It follows from Lemma 3.7(ii), that $[C_i]$ and $[C_j]$ have the form $[C_i]=e_{n_0}-e_{n_1}- \cdots -e_{n_k}$, $[C_j]=e_{m_0}-e_{m_1}- \cdots - e_{m_\ell }$. As the algebraic intersection is $+1$, either $e_{n_0}=e_{m_q}$ or $e_{m_0}=e_{n_p}$ for some $q,p$. To rule out the possibility that both of these occur, we consider the symplectic areas. As $C_i$ and $C_j$ are both symplectic spheres, $\langle \omega, [C_i]\rangle, \langle \omega, [C_j]\rangle >0$. Each of the exceptional classes also has positive symplectic area. Let $a := \langle \omega, e_{n_0}\rangle >0$ and $b := \langle \omega, e_{m_0}\rangle >0$. Then

\[ 0<\langle \omega, [C_i] \rangle < a-b \quad \text{and} \quad 0<\langle \omega, [C_j] \rangle < b-a, \]

which is a contradiction.

When there is a linear chain of such symplectic spheres with self-intersection $-2$ (consecutive spheres intersect once), there are few options for the homology classes of the spheres in that chain.

Often, only one of these options can occur. The following particular case will appear frequently in our homological classifications.

These lemmas, together with some arithmetic considerations, generally suffice to allow us to classify all possibilities for the homology classes of the spheres in a normal crossing resolution. Given these homology classes we can apply Lemma 3.5 to blow down to a configuration in $\mathbb {C}\mathbb {P}^{2}$. The way that the exceptional classes appeared in the homology classes of the plumbing spheres affects the intersections between the proper transforms. We record the data of these intersections, including the degree of tangency between surfaces at each intersection and when intersections between different components coincide. There may also be singularities in a single component which we record as well. Fixing this data of the singularities and intersections, we then try to classify symplectic configurations of surfaces in $\mathbb {C}\mathbb {P}^{2}$ up to equisingular symplectic isotopy. In the next section we solve this classification for families of singular reducible configurations of symplectic surfaces in $\mathbb {C}\mathbb {P}^{2}$ that we need.

3.4 Birational transformations

In complex dimension two, a birational transformation is a sequence of blow-ups and blow-downs. As blow-ups and blow-downs can be done symplectically [Reference McDuffMcD90], these transformations from algebraic geometry can be imported into the symplectic context. When we have a singular surface in $\mathbb {C}\mathbb {P}^{2}$, the birational transformation may change the singularities and self-intersection number of the surface. The blow-ups can begin to resolve singularities, and blow-downs may create new singularities.

There are two ways that we will relate singular symplectic surfaces in $\mathbb {C}\mathbb {P}^{2}$ using birational transformations. The first notion is weaker, but for two surfaces related in this way, the existence of one type of singular surface will imply the existence of another type of singular surface.

From this definition, if we birationally invert the blow-down $\pi$ by blowing-up, we see that the total transform of $\Sigma _2$ in $M_1\#N\overline {\mathbb {C}\mathbb {P}}\,\!^{2}$, contains the total transform of $\Sigma _1$, i.e. $\overline {\Sigma }_1\subseteq \overline {\Sigma }_2$. This follows from the fact that a set is contained in the preimage of its image, so $\overline {\Sigma }_2 = \pi ^{-1}(\Sigma _2) = \pi ^{-1}(\pi (\overline {\Sigma }_1)) \supseteq \overline {\Sigma }_1$. For this reason, we can also refer to the birational derivation relation as saying that $\mathcal {C}_1$ is birationally contained in $\mathcal {C}_2$.

Note that this relation is directional. If $\mathcal {C}_2$ is birationally derived from $\mathcal {C}_1$, it is not typically true that $\mathcal {C}_1$ is birationally derived from $\mathcal {C}_2$. To strengthen this notion, we impose a more restrictive condition that the exceptional spheres that are blown down by $\pi$ are actually contained in $\overline {\Sigma }_1$.

Equivalently, a birational equivalence is a birational derivation where in the blow-up $\overline {\Sigma }_1 = \overline {\Sigma }_2$. Reversing the sequence of blow-ups and blow-downs, we see that any realization $\Sigma _2$ of $\mathcal {C}_2$ in $(M_2,\omega _2)$ will have exceptional spheres contained in its total transform $\overline {\Sigma }_2$ which can be blown down to obtain a realization of $\mathcal {C}_1$ in $(M_1,\omega _1)$. Therefore, birational equivalence is a symmetric relation. If $\mathcal {C}_1$ and $\mathcal {C}_2$ are birationally equivalent, they are each birationally derived from the other.

Without the extra condition for birational equivalence, a birational derivation is not an equivalence relation. More precisely, suppose that a configuration $\mathcal {C}_2$ that is birationally derived from a configuration $\mathcal {C}_1$. Then for any realization $\Sigma _1$ of $\mathcal {C}_1$ it has a resolution such that $\overline {\Sigma }_1\subseteq \overline {\Sigma }_2$ for some realization $\Sigma _2$ of $\mathcal {C}_2$. However, $\overline {\Sigma }_2$ may contain some additional exceptional spheres that were not contained in $\overline {\Sigma }_1$. Therefore, when we blow down the exceptional spheres in $\overline {\Sigma }_1$, the image of $\overline {\Sigma }_2$ will only contain $\Sigma _1$ instead of being equal to $\Sigma _1$. A weaker equivalence relation could be obtained from the definition of birational derivation by replacing the condition $\Sigma _2=\pi (\overline {\Sigma }_1)$ by $\Sigma _2\subseteq \pi (\overline {\Sigma }_1)$, however this relation will not be particularly useful to us.

Note that the number of components of the configuration is preserved by a birational equivalence but not by a birational derivation.

Example 3.15 Let $\mathcal {C}_1$ be a configuration in $(\mathbb {C}\mathbb {P}^{2},\omega _{\rm {FS}})$ whose realizations consist of a single symplectic conic $C$ with three symplectic lines $L_1,L_2,L_3$ that intersect the conic tangentially at three distinct points $p_1,p_2,p_3$, respectively, and intersect each other transversally at three distinct points. Blow up at each of the three tangential intersection points, and denote the resulting proper transforms by $\widetilde {C}$ and $\widetilde {L}_i$. The resulting self-intersection numbers satisfy $[\widetilde {C}]^{2}=1$ and $[\widetilde {L_i}]^{2}=0$. The intersections between $\widetilde {C}$ and $\widetilde {L}_i$ are transverse and there are three new exceptional spheres $E_1, E_2, E_3$ which pass through those intersection points. See Figure 2. Because $[\widetilde {C}]^{2}=+1$ and $\widetilde {C}$ is a smooth sphere, we can find a symplectomorphism of $\mathbb {C}\mathbb {P}^{2}\# 3\overline {\mathbb {C}\mathbb {P}}\,\!^{2}$ which identifies $\widetilde {C}$ with $\mathbb {C}\mathbb {P}^{1}$. We calculate the possible homology classes of the other surfaces in the picture in terms of a standard basis which identifies $[\widetilde {C}]=h$ using Lemma 3.7 and intersection numbers. We find the only option is

\begin{gather*} [\widetilde{C}]=h, \quad [\widetilde{L}_1] = h-e_1, \quad [\widetilde{L}_2] = h-e_2, \quad [\widetilde{L}_3] = h-e_3,\\ [E_1] = h-e_2-e_3, \quad [E_2] = h-e_1-e_3, \quad [E_3] = h-e_1-e_2. \end{gather*}

Lemma 3.5 implies that there exist exceptional spheres in classes $e_1$, $e_2$, $e_3$ which intersect the labeled surfaces non-negatively. Blowing down such exceptional spheres results in a configuration of seven symplectic lines intersecting in six triple points (the three triple points that already existed between $C$, $L_i$, $E_i$ for $i=1,2,3$, and the three triple points that are created when blowing down $e_1$, $e_2$, $e_3$). Therefore, the configuration of seven lines with six triple points can be birationally derived from the configuration of a conic with three tangent lines. Note, the exceptional spheres which are blown down in the last step of the transformation are not included in the configuration (because none of the surfaces $\widetilde {C},\widetilde {L}_i$ or $E_i$ represented a class $e_i$). Therefore, this is not a birational equivalence.

Figure 2. Birational derivation. The lines forming the triangle on the left are $L_1,L_2,L_3$, and the circle represents the conic $C$. After blowing up as indicated in the center figure, we obtain three exceptional spheres $E_1,E_2,E_3$. There exist three other exceptional spheres which are not visible in the center diagram that can be blown down to result in the configuration shown on the right.

In order to get a birational equivalence, we would need to augment the original configuration by adding in components which will become the exceptional spheres that eventually will be blown down. We can reverse engineer to predict what must be added to our configuration to obtain a birational equivalence to the same final configuration. The exceptional sphere $\widetilde {S}_1$ in $\mathbb {C}\mathbb {P}^{2}\#3\overline {\mathbb {C}\mathbb {P}}\,\!^{2}$ representing $e_1$ intersects $E_2$ and $E_3$. Reversing the birational transformation by blowing down $E_2$ and $E_3$, we find that the image $S_1$ in $\mathbb {C}\mathbb {P}^{2}$ will be a sphere of self-intersection $+1$ which must pass transversally through the tangential intersections between $C$ with $L_2$ and $C$ with $L_3$. The other exceptional spheres descend similarly to $+1$-spheres passing through two of the tangential intersections. Replace our starting configuration by a configuration consisting of the original conic $C$ and lines $L_1,L_2,L_3$ with $L_i$ tangent to $C$ at $p_i$ together with three additional symplectic lines $S_1,S_2,S_3$ such that $S_i$ intersects $C$ transversally at the points $p_{i+1}$ and $p_{i+2}$ (where indices are taken mod $3$). Now the same birational transformation becomes a birational equivalence between a configuration consisting of a conic with three tangent lines and three lines through the three pairs of tangent points with a configuration of seven lines with six triple points and three double points. See Figure 3.

Figure 3. An example of a birational equivalence. The lines forming the outer triangle in the left figure represent $L_1,L_2,L_3$, the circle represents the conic $C$, and the lines forming the inscribed triangle represent $S_1,S_2,S_3$. After blowing up at the marked points to get the center figure, the proper transforms $\widetilde {S}_1,\widetilde {S}_2,\widetilde {S}_3$ are exceptional spheres. The short line segments in the center figure represent the exceptional spheres $E_1,E_2,E_3$. The right figure is obtained by blowing down $\widetilde {S}_1,\widetilde {S}_2,\widetilde {S}_3$. The images of $E_1,E_2,E_3$ after this blow-down are lines in the right figure.

Our use of birational derivations and equivalences arises from their implications to symplectic isotopy problems, which we state next. These implications are somewhat immediate from the definitions, so the mathematical power goes into proving such birational derivations exist as described in the previous remark.

Now we give the relations between symplectic isotopy problems and birational derivations and equivalences.

This statement is immediate from the definition, but the utility of the statement comes from its contrapositive. Namely, we show that certain configurations $\mathcal {C}_1$ cannot be symplectically realized in a closed symplectic manifold $(M_1,\omega _1)$, using the non-existence of a subconfiguration of symplectic curves that can be birationally derived from $\mathcal {C}_1$.

If $\mathcal {C}_1$ and $\mathcal {C}_2$ are birationally equivalent, they are each birationally derived from the other yielding the following corollary.

3.5 Riemann–Hurwitz

The Riemann–Hurwitz obstruction uses symplectic information in a more global way. Fix an almost complex structure $J$ on $\mathbb {C}\mathbb {P}^{2}$ such that $C$ is $J$-holomorphic. Fix a point $p\in \mathbb {C}\mathbb {P}^{2}$, and consider the pencil of $J$-holomorphic lines through $p$, and the associated projection $\pi _p: \mathbb {C}\mathbb {P}^{2} \setminus \{p\} \to \mathbb {C}\mathbb {P}^{1}$. Restricting this projection to $\pi : C\to \mathbb {C}\mathbb {P}^{1}$, and pre-composing with the normalization map $n:\widetilde C \to C$, gives a ramified covering map $\pi \circ n : \widetilde C \to \mathbb {C}\mathbb {P}^{1}$. The fact that the only singularities are ramification points follows from positivity of intersections between $J$-holomorphic curves. Ramification points arise from tangencies between lines in the pencil with $C$ and from singular points of $C$.

For a point $q\neq p$, the ramification index of $n^{-1}(q)$ is equal to the multiplicity of intersection of the $J$-line through $q$ with $C$. For a smooth point, this is just the order of tangency between the $J$-line and $C$ at $q$. At a singular point $q\neq p$ with multiplicity sequence starting with $[m_q,m_{q,2}]$ the ramification index of $x=n^{-1}(q)$ is $m_q$ if the $J$-line from $p$ to $q$ is transverse to $C$ at $q$. If the $J$-line from $p$ to $q$ is tangent to $C$ at $q$, then $x$ will have ramification index at least $m_q+m_{q,2}$.

A priori, if $p\in C$, the map is not well-defined at $p$, but, in fact, $\pi$ has a unique continuous extension defined by sending $p$ to the image of the $J$-holomorphic line through $p$ which is tangent to $C$ at $p$. In this case, the ramification index of $p$ is equal to the multiplicity of intersection of the $J$-line tangent to $C$ at $p$ minus the multiplicity of $C$ at $p$. Therefore, if $p$ is a singular point whose multiplicity sequence starts with $[m_p,m_{p,2}]$ (where we set $m_{p,2}=1$ if the multiplicity sequence has length one), then the ramification index of $\pi \circ n$ at $p$ is at least $m_{p,2}$.

The Riemann–Hurwitz formula is the calculation of the Euler characteristic of the branched covering in terms of the ramification indices and degree of the cover. If $\widetilde {\pi }:=\pi \circ n: \widetilde C \to \mathbb {C}\mathbb {P}^{1}$ is a $k$-fold ramified cover with ramification points $x_1,\ldots, x_\ell$ with corresponding ramification indices $e_{\widetilde {\pi }}(x_j)$, then

\[ \chi (\widetilde C) = k(2-\ell ) + \sum_{j=1}^{\ell} (k+1-e_{\widetilde{\pi}}(x_j)) = 2k-\sum_{j=1}^{\ell} (e_{\widetilde{\pi}}(x_j)-1). \]

In our case, $\widetilde C$ is a $2$-sphere so $\chi (\widetilde C)=2$. Suppose $d$ is the degree of $C$. If we choose $p\notin C$, then a generic line through $p$ intersects $C$ $d$ times, so the degree of the cover is $d$. Therefore, this equation specializes to

\[ 2d-2=\sum_{j=1}^{\ell} (e_{\widetilde{\pi}}(x_j)-1). \]

If instead, we choose $p\in C$, where $p$ has multiplicity $m_p$ (where $m_p$ is the first entry of the multiplicity sequence if $p$ is a singular point and is $1$ if $p$ is a smooth point of $C$), then a generic line through $p$ intersects $C$ at $d-m_p$ other points. Therefore, $\widetilde {\pi }$ is a $(d-m_p)$-fold cover. This gives the following equation and inequality. The inequality is particularly useful as an obstruction to symplectically realizing certain cuspidal curves in $\mathbb {C}\mathbb {P}^{2}$:

\[ 2(d-m_p)-2=\sum_{j=1}^{\ell} (e_{\widetilde{\pi}(x_j)}-1) \geq \sum_{q\neq p} (m_q-1) +(m_{p,2}-1), \]

which we re-write as follows,

(3.1)\begin{equation} 2d-2m_p \ge 2 + \sum_{q\neq p} (m_q-1) + (m_{p,2} - 1). \end{equation}

Example 3.20 We can apply Riemann–Hurwitz to exclude the following configurations of cusps for a quintic: $\{[3,2], [2], [2]\}$, $\{[3], [2], [2], [2]\}$, $\{[2,2], [2], [2], [2], [2]\}$, and $\{[2], [2], [2], [2], [2], [2]\}$. In the first case, we project from the $[3,2]$-cusp, and we obtain the following contradiction:

\[ 2\cdot 5-2\cdot 3 \geq 2+1+1+1. \]

In the second, we project from the $[3]$-cusp:

\[ 2\cdot 5-2\cdot 3 \geq 2+1+1+1. \]

In the third case, we project from the $[2,2]$-cusp:

\[ 2\cdot 5-2\cdot 2 \geq 2+1+1+1+1+1, \]

and in the last from any of the cusps, to obtain

\[ 2\cdot 5-2\cdot 2 \geq 2+1+1+1+1+1. \]

Thus, in each of the four cases, we get a contradiction.