2. $r^{th}$ roots of the canonical bundle and generalized spin structures

2.1. Plane curves and $\mathcal{P}_d$

A general reference for this paragraph is [ [Reference Dolgachev and Libgober12], Section 2].

By definition, a (projective) plane curve of degree $d$ is the vanishing locus $V(f)$ in ${\mathbb{C}} P^2$ of a nonzero homogeneous polynomial $f(x,y,z)$ of degree $d$ . The space of all plane curves is identified with ${\mathbb{C}} P^N$ , where $N =\binom{d+2}{2} - 1$ . A plane curve $X$ of degree $d$ is smooth if $X \cong \Sigma _g$ with $g =\binom{d-1}{2}$ , and otherwise $X$ is said to be singular.

We define the discriminant as the set

\begin{equation*} \mathcal {D}_d = \{f \in {\mathbb {C}} P^N \mid V(f) \mbox { is singular.} \}. \end{equation*}

The discriminant $\mathcal{D}_d$ is the vanishing locus of a polynomial $p_d$ known as the discriminant polynomial and is therefore a hypersurface in ${\mathbb{C}} P^N$ . The space of smooth plane curves is then defined as

\begin{equation*} \mathcal {P}_d = {\mathbb {C}} P^N \setminus \mathcal {D}_d. \end{equation*}

The universal family of plane curves is the space $\mathfrak{X}_d \subset \mathcal{P}_d \times{\mathbb{C}} \mathbb{P}^2$ defined via

\begin{equation*} \mathfrak {X}_d = \{(f,[x:y:z]) \in \mathcal {P}_d \times {\mathbb {C}} \mathbb {P}^2 \mid f(x,y,z) = 0 \}. \end{equation*}

The projection $\pi : \mathfrak{X}_d \to \mathcal{P}_d$ is the projection map for a $C^\infty$ fiber bundle structure on $\mathfrak{X}_d$ with fibers diffeomorphic to $\Sigma _g$ .

2.2. $r$ -spin structures

Let $X$ be a smooth projective algebraic curve over $\mathbb{C}$ and let $K \in \mathrm{Pic}(X)$ denote the canonical bundle.Footnote ² Recall that a spin structure on $X$ is an element $L \in \mathrm{Pic}(X)$ satisfying $L^{\otimes 2} = K$ . This admits an obvious generalization.

Definition 2.1. An $r$ -spin structure is a line bundle $L \in \mathrm{Pic}(X)$ satisfying $L^{\otimes r} = K$ .

Let $T^{*,1}X$ denote the unit cotangent bundle of $X$ , relative to an arbitrary Riemannian metric on $X$ . Just as ordinary spin structures are closely related to $H^1(T^{*,1}X;{\mathbb{Z}}/2{\mathbb{Z}})$ , there is an analogous picture of $r$ -spin structures.

Proposition 2.2. Let $L$ be an $r$ -spin structure on $X$ . Associated to $L$ are

1. 1. a regular $r$ -sheeted covering space $\widetilde{T^{*,1}X} \to T^{*,1}X$ with deck group ${\mathbb{Z}}/n{\mathbb{Z}}$ , and

2. 2. a cohomology class $\phi _L \in H^1(T^{*,1}X;{\mathbb{Z}}/r{\mathbb{Z}})$ restricting to a generator of the cohomology $H^1(S^1;{\mathbb{Z}}/r{\mathbb{Z}})$ of the fiber of $T^{*,1}X \to X$ .

Proof. In view of the equality $L^{\otimes r} = K$ in $\mathrm{Pic}(X)$ , taking $r^{th}$ powers in the fiber induces a map $\mu : L \to K$ . Let $L^\circ$ denote the complement of the zero section in $L$ and define $K^\circ$ similarly. Then $\mu : L^\circ \to K^\circ$ is an $r$ -sheeted covering space with deck group ${\mathbb{Z}}/r{\mathbb{Z}}$ induced from the multiplicative action of the $r^{th}$ roots of unity. The covering space $\widetilde{T^{*,1}X} \to T^{*,1}X$ is obtained from $L^\circ \to K^\circ$ by restriction.

As $\widetilde{T^{*,1}X} \to T^{*,1}X$ is a regular cover with deck group ${\mathbb{Z}}/r{\mathbb{Z}}$ , the Galois correspondence for covering spaces asserts that $\widetilde{T^{*,1}X}$ is associated to some homomorphism $\phi _L: \pi _1(T^{*,1}X) \to{\mathbb{Z}}/r{\mathbb{Z}}$ . This gives rise to a class, also denoted $\phi _L$ , in $H^1(T^{1,*}X;{\mathbb{Z}}/r{\mathbb{Z}})$ . On a given fiber of $T^{*,1}X \to X$ , the covering $\widetilde{T^{*,1}X}\to T^{*,1}X$ restricts to an $r$ -sheeted cover $S^1 \to S^1$ ; this proves the assertion concerning the restriction of $\phi _L$ to $H^1(S^1;{\mathbb{Z}}/r{\mathbb{Z}})$ .

Our interest in $r$ -spin structures arises from the fact that degree- $d$ plane curves are equipped with a canonical $(d-3)$ -spin structure.

Fact 2.3. Let $X$ be a smooth degree- $d$ plane curve, $d \ge 3$ . The canonical bundle $K \in \mathrm{Pic}(X)$ is induced from the restriction of $\mathcal{O}(d-3) \in \mathrm{Pic}({\mathbb{C}}\mathbb{P}^2)$ . Consequently, $\mathcal{O}(1)$ determines a $(d-3)$ -spin structure on $X$ for $d \ge 4$ .

Let $\varpi : \mathfrak{X}_d \to{\mathbb{C}} \mathbb{P}^2$ denote the projection onto the second factor. Then $\varpi ^*(\mathcal{O}(d-3)) \in \mathrm{Pic}(\mathfrak{X}_d)$ restricts to the canonical bundle on each fiber, and $\varpi ^*(\mathcal{O}(1))$ determines a $(d-3)$ -spin structure. Let $T^{*,1}\mathfrak{X}_d$ denote the $S^1$ -bundle over $\mathfrak{X}_d$ for which the fiber over $(X,x)$ consists of the unit cotangent vectors $T^{*,1}_x X$ .

Definition 2.4. The cohomology class

\begin{equation*} \phi _d \in H^1(T^{*,1}\mathfrak {X}_d; {\mathbb {Z}}/(d-3){\mathbb {Z}}) \end{equation*}

is obtained by applying the construction of Proposition 2.2 to the pair of line bundles $\varpi ^*(\mathcal{O}(1))$ , $\varpi ^*(\mathcal{O}(d-3)) \in \mathrm{Pic}(\mathfrak{X}_d)$ .

3. Generalized winding numbers and obstructions to surjectivity

In this section, we prove that the existence of $\phi _d$ gives rise to an obstruction for a mapping class $f \in \textrm{Mod}(\Sigma _g)$ to be contained in $\mathrm{Im}(\rho _d)$ . For any system of coefficients $V$ , there is a natural action of $\textrm{Mod}(\Sigma _g)$ on $H^1(T^{*,1}\Sigma _g; V)$ , which extends the action of $\textrm{Mod}(\Sigma _g)$ on $H^1(\Sigma _g; V)$ via $\Psi$ . To prove Theorem1.1, it therefore suffices to prove that the stabilizer $\textrm{Mod}(\Sigma _g)[\phi _d]$ of each nonzero element of $H^1(T^{*,1}\Sigma _g;{\mathbb{Z}}/(d-3){\mathbb{Z}})$ is not the full group $\Psi ^{-1}(\Psi (\mathrm{Im}(\rho _d)))$ .

The natural setting for what follows is in the unit tangent bundle of a surface, which we write $T^1\Sigma$ . Of course, a choice of Riemannian metric on $\Sigma$ identifies $T^1 \Sigma$ with $T^{*,1} \Sigma$ , and a choice of metric in each fiber identifies $T^{*,1} \mathfrak{X}_d$ with the “vertical unit tangent bundle” $T^1 \mathfrak{X}_d$ ; we will make no further comment on these matters.

The basis for our approach is the work of Humphries–Johnson [Reference Humphries and Johnson17], which gives an interpretation of $H^1(T^1 \Sigma _g; V)$ as the space of “ $V$ -valued generalized winding number functions”. A basic notion here is that of a Johnson lift. For our purposes, a simple closed curve is a $C^1$ -embedded $S^1$ -submanifold.

Definition 3.1. Let $a$ be a simple closed curve on the surface $\Sigma$ given by a unit-speed $C^1$ embedding $a: S^1 \to \Sigma$ . A choice of orientation on $S^1$ induces an orientation on $a$ , as well as providing a coherent identification $T^1_xS^1 = \{-1,1\}$ for each $x \in S^1$ . The Johnson lift of $a$ , written $\vec a$ , is the map $\vec a: S^1 \to T^1 \Sigma$ given by

\begin{equation*} \vec a (t) = (a(t), D_ta(1)). \end{equation*}

That is, the Johnson lift of $a$ is simply the curve in $T^1\Sigma$ induced from $a$ by tracking the tangent vector.

The Johnson lift allows for the evaluation of elements of $H^1(T^1\Sigma ;V)$ on simple closed curves. Let $\Sigma$ be a surface, $V$ an Abelian group, and $\alpha \in H^1(T^1 \Sigma ; V)$ a cohomology class. Let $a$ be a simple closed curve. By an abuse of notation, we write $\alpha (a)$ for the evaluation of $\alpha$ on the $1$ -cycle determined by the Johnson lift $\vec a$ . In this context, we call $\alpha$ a “generalized winding number function”.Footnote ³ In [Reference Humphries and Johnson17], it is shown that this pairing satisfies the following properties:

Proof of Theorem 1.1. Consider the class $\phi _d \in H^1(T^{*,1}\mathfrak{X}_d;{\mathbb{Z}}/(d-3){\mathbb{Z}})$ . The above discussion implies that on a given fiber $X$ of $\mathfrak{X}_d \to \mathcal{P}_d$ , the restriction of $\phi _d$ determines a generalized winding number function; we write $\alpha _d \in H^1(T^1X;{\mathbb{Z}}/(d-3){\mathbb{Z}})$ for this class. Since $\alpha _d$ is induced from the globally defined form $\phi _d$ , it follows that $\alpha _d$ is monodromy-invariant: if $f \in \mathrm{Im}(\rho _d)$ , then for any simple closed curve $a$ on $X$ , the equation

(3) \begin{equation} \alpha _d(f(a)) = \alpha _d(a) \end{equation}

must hold. Consequently,

\begin{equation*} \mathrm {Im}(\rho _d) \subseteq \textrm {Mod}(\Sigma _g)[\phi _d] \end{equation*}

as claimed.

We wish to exhibit a nonseparating simple closed curve $b$ for which $\alpha _d(b) \ne 0$ . Given such a $b$ , there is another simple closed curve $a$ satisfying $\langle{a,b} \rangle = 1$ . Then the twist-linearity condition (1) will show that

\begin{equation*} \alpha _d(T_b(a)) = \alpha _d(a) + \alpha _d(b) \ne \alpha _d(a); \end{equation*}

this contradicts (3). It follows that the Dehn twist $T_b$ for such a curve cannot be contained in $\textrm{Mod}(\Sigma _g)[\phi _d]$ .

In the case when $d$ is even, when $\Psi ^{-1}(\Psi (\mathrm{Im}(\rho _d))) = \textrm{Mod}(\Sigma _g)$ , this will prove Theorem1.1. For $d$ odd, there is an additional complication. Here, the class $\frac{d-3}{2} \phi _d \in H^1(T^{*,1}\mathfrak{X}_d;{\mathbb{Z}}/2{\mathbb{Z}})$ determines an ordinary spin structure, and according to Beauville, the group $\Psi (\textrm{Mod}(\Sigma _g)[\phi _d])$ is the stabilizer of $\frac{d-3}{2}\phi _d$ in $\mathrm{Sp}(2g,{\mathbb{Z}})$ . We must, therefore, exhibit a curve $b$ for which $\alpha _d(b)$ is nonzero and $\frac{d-3}{2}$ -torsion. Equation (1) shows that such a curve does stabilize the spin structure $\frac{d-3}{2} \phi _d$ , but not the refinement to a $(d-3)$ -spin structure $\phi _d$ .

It remains to exhibit a suitable curve $b$ . It follows easily from the twist-linearity condition (1) that given any subsurface $S \subset X$ of genus $1$ with one boundary component, there is some (necessarily nonseparating) curve $c$ contained in $S$ with $\alpha _d(c) = 0$ . Let $S_1,S_2,S_3$ be a collection of mutually disjoint subsurfaces of genus $1$ with one boundary component, and let $c_1, c_2, c_3$ be curves satisfying $\alpha _d(c_i) = 0$ , and for which $c_i$ is contained in $S_i$ (recall that $d \ge 6$ and so the genus of $X$ is $g \ge 10$ ). Choose $b$ disjoint from all $c_i$ so that the collection of curves $b, c_1,c_2, c_3$ encloses a subsurface $\Sigma$ homeomorphic to a sphere with $4$ boundary components. From (2) and the construction of the $c_i$ , it follows that when $b$ is suitably oriented, it satisfies

\begin{equation*} \alpha _d(b) = \chi (\Sigma )\alpha _d(\zeta ) = -2\alpha _d(\zeta ). \end{equation*}

Recall that by Proposition 2.2.2, the element $\alpha _d(\zeta ) \in{\mathbb{Z}}/(d-3){\mathbb{Z}}$ is primitive. Thus, $\alpha _d(b) \ne 0$ for any $d$ , but is $\frac{d-3}{2}$ -torsion when $d$ is odd, as required.

5. The Lönne presentation

In this section, we recall Lönne’s work [Reference Lönne21] computing $\pi _1(\mathcal{P}_d)$ and apply this to derive some first properties of the monodromy map $\rho _d: \pi _1(\mathcal{P}_d) \to \textrm{Mod}(\Sigma _g)$ .

5.1. Picard–Lefschetz theory

Picard–Lefschetz theory concerns the problem of computing monodromies attached to singular points of holomorphic functions $f:{\mathbb{C}}^n \to{\mathbb{C}}$ . This then serves as the local theory underpinning more global monodromy computations. Our reference is [Reference Arnold, Gusein-Zade and Varchenko1].

Let $U \subset{\mathbb{C}}^2$ and $V \subset{\mathbb{C}}$ be open sets for which $0 \in V$ . Let $f(u,v): U \to V$ be a holomorphic function. Suppose $f$ has an isolated critical value at $z = 0$ , and that there is a single corresponding critical point $p \in{\mathbb{C}}^2$ . Suppose that $p$ is of Morse type in the sense that the Hessian

\begin{equation*} \left (\begin {array}{c@{\quad}c} {\dfrac {\partial ^2 f}{\partial ^2 x}} & {\dfrac {\partial ^2 f}{\partial x \partial y}} \\[15pt] \dfrac {\partial ^2 f}{\partial y \partial x} & {\dfrac {\partial ^2 f}{\partial ^2 y}} \end {array}\right ) \end{equation*}

is non-singular at $p$ . Equivalently, $f$ is analytically equivalent to $f(u,v) = uv$ , the union of two lines crossing transversely.

In such a situation, the fiber $f^{-1}(z)$ for $z \ne 0$ is diffeomorphic to an open annulus. The core curve of such an annulus is called a vanishing cycle. Let $\gamma$ be a small circle in $\mathbb{C}$ enclosing only the critical value at $z = 0$ . Let $z_1 \in \gamma$ be a basepoint with corresponding core curve $c\subset f^{-1}(z_1)$ . The Picard–Lefschetz theorem describes the monodromy obtained by traversing $\gamma$ .

More generally, an algebraic curve $C$ is said to have a nodal singularity at $p \in C$ if there is an analytic local equation for $C$ near $p$ of the form $f(u,v) = uv$ . Perturbing $C$ slightly to a smooth $C'$ , the intersection of $C'$ with a small ball near $p$ is again an annulus, and the core curve is again called a vanishing cycle.

Theorem 5.1 (Picard–Lefschetz for $n = 2$ ). With reference to the preceding discussion, the monodromy $\mu \in \textrm{Mod}(f^{-1}(z_1))$ attached to traversing $\gamma$ counter-clockwise is given by a right-handed Dehn twist about the vanishing cycle:

\begin{equation*} \mu = T_c^{-1}. \end{equation*}

More generally, let $D^*$ denote the punctured unit disk

\begin{equation*} D^* = \{w \in {\mathbb {C}} \mid 0 \lt {\left | w \right |} \le 1\}, \end{equation*}

and write $D = \{w \in{\mathbb{C}} \mid \left | w \right | \le 1\}$ for the closed unit disk.

Let $f(x,y,z)$ be a homogeneous polynomial of degree $d$ with the following properties:

1. 1. For $c \in D$ , the plane curve $c z^d - f(x,y,z)$ is singular only for $c = 0$ .

2. 2. The only critical point for $f$ of the form $(x,y,0)$ is the point $(0,0,0)$ .

3. 3. The function $f(x,y,1)$ has a single critical point of Morse type at $(x,y) = (0,0)$ .

In this setting, the local theory of Theorem5.1 can be used to analyze the monodromy of the family

\begin{equation*} E = \{(c, [x:y:z]) \mid cz^d = f(x,y,z) \} \subset D^* \times {\mathbb {C}} P^2 \end{equation*}

around the boundary $\partial D^*$ .

Theorem 5.2 (Picard–Lefschetz for plane curve families). Let $f \in{\mathbb{C}} P^N$ satisfy the properties (1), (2), (3) listed above. Let $X = V(z^d - f(x,y,z))$ denote the fiber above $1 \in D^*$ . Then there is a vanishing cycle $c \subset X$ so that the monodromy $\mu \in \textrm{Mod}(X)$ obtained by traversing $\partial D^*$ counter-clockwise is given by a right-handed Dehn twist about the vanishing cycle:

\begin{equation*} \mu = T_c^{-1}. \end{equation*}

Proof. Condition (2) asserts that the monodromy can be computed by restricting attention to the affine subfamily obtained by setting $z = 1$ . Define

\begin{equation*} E^\circ = \{(c,x,y) \mid c = f(x,y,1) \} \subset D^* \times {\mathbb {C}}^2. \end{equation*}

Define $U = \{(x,y) \in{\mathbb{C}}^2 \mid \left |{f(x,y,1)}\right | \le 1\}$ and consider $f(x,y,1)$ as a holomorphic function $f: U \to D$ . The monodromy of this family then corresponds to the monodromy of the original family $E \to D^*$ . The result now follows from Condition (3) in combination with Theorem5.1 as applied to $f(x,y,1)$ .

5.2. Lönne’s theorem

There are some preliminary notions to establish before Lönne’s theorem can be stated. We begin by introducing the Lönne graphs $\Gamma _d$ . Lönne obtains his presentation of $\pi _1(\mathcal{P}_d)$ as a quotient a certain group constructed from $\Gamma _d$ .

Definition 5.3. [Lönne graph] Let $d \ge 3$ be given. The Lönne graph $\Gamma _d$ has vertex set

\begin{equation*} I_d = \{(a,b) \mid 1 \le a,b \le d-1 \}. \end{equation*}

Vertices $(a_1,b_1)$ and $(a_2, b_2)$ are connected by an edge if and only if both of the following conditions are met:

1. 1. $\left |{a_1-a_2}\right | \le 1$ and $\left |{b_1-b_2}\right | \le 1$ .

2. 2. $(a_1-a_2)(b_1 - b_2) \le 0$ .

The set of edges of $\Gamma _d$ is denoted $E_d$ .

Figure 3. The Lönne graph $\Gamma _5$ .

Vertices $i,j,k \in \Gamma _d$ are said to form a triangle when $i,j,k$ are mutually adjacent. The triangles in the Lönne graph are crucial to what follows. It will be necessary to endow them with orientations.

Definition 5.4 (Orientation of triangles). Let $i,j,k$ determine a triangle in $\Gamma _d$ .

1. 1. If

   \begin{equation*} i = (a,b),\quad j= (a,b+1), \quad k = (a+1,b), \end{equation*}
   then the triangle $i,j,k$ is positively-oriented by traversing the boundary clockwise.

2. 2. If

   \begin{equation*} i = (a,b), \quad j = (a,b+1), \quad k = (a-1,b+1), \end{equation*}
   then the triangle $i,j,k$ is positively oriented by traversing the boundary counterclockwise.

We say that the ordered triple $(i,j,k)$ of vertices determining a triangle is positively oriented if traversing the boundary from $i$ to $j$ to $k$ agrees with the orientation specified above.

Definition 5.5 (Artin group). Let $\Gamma$ be a graph with vertex set $V$ and edge set $E$ . The Artin group $A(\Gamma )$ is defined to be the group with generators

\begin{equation*} \sigma _i, i \in V, \end{equation*}

subject to the following relations:

1. 1. $\sigma _i \sigma _j = \sigma _j \sigma _i$ for all $(i,j) \not \in E$ .

2. 2. $\sigma _i \sigma _j \sigma _i = \sigma _j \sigma _i \sigma _j$ for all $(i,j) \in E$ .

Theorem 5.6 (Lönne). For $d \ge 3$ , the group $\pi _1(\mathcal{P}_d)$ is isomorphic to a quotient of the Artin group $A(\Gamma _d)$ , subject to the following additional relations:

1. 3. $\sigma _i \sigma _j \sigma _k \sigma _i = \sigma _j \sigma _k \sigma _i \sigma _j$ if $(i,j,k)$ forms a positively-oriented triangle in $\Gamma _d$ .

2. 4. An additional family of relations $R_i, i \in I_d$ .

3. 5. An additional relation $\widetilde R$ .

Remark 5.7. Define the group $B(\Gamma _d)$ as the quotient of the Artin group $A(\Gamma _d)$ by the family of relations ( 3 ) in Theorem 5.6 . As our statement of Lönne’s theorem indicates, the additional relations will be of no use to us, and our theorem really concerns the lift of the monodromy representation $\tilde \rho _d: B(\Gamma _d) \to \textrm{Mod}(\Sigma _g)$ .

For the analysis to follow, it is essential to understand the mapping classes $\rho _d(\sigma _i), i \in I_d$ .

Proposition 5.8. For each generator $\sigma _i$ of Theorem 5.6 , the image

\begin{equation*} \rho _d(\sigma _i) = T_{c_i}^{-1} \end{equation*}

is a right-handed Dehn twist about some vanishing cycle $c_i$ on a fiber $X \in \mathcal{P}_d$ .

Proof. The result will follow from Theorem 5.2, once certain aspects of Lönne’s proof are recalled.

The generators $\sigma _i$ of Theorem 5.6 correspond to specific loops in $\mathcal{P}_d$ known as geometric elements.

Definition 5.9 (Geometric element). Let $D = V(p)$ be a hypersurface in ${\mathbb{C}}^n$ defined by some polynomial $p$ . An element $x \in \pi _1({\mathbb{C}}^n \setminus D)$ that can be represented by a path isotopic to the boundary of a small disk transversal to $D$ is called a geometric element. If $\widetilde D$ is a projective hypersurface, an element $x \in \pi _1({\mathbb{C}} P^n \setminus \widetilde D)$ is said to be a geometric element if it can be represented by a geometric element in some affine chart.

In Lönne’s terminology, the generators $\sigma _i, i \in I_d$ arise as a “Hefez-Lazzeri basis”—this will require some explanation. Consider the linearly perturbed Fermat polynomial

\begin{equation*} f(x,y,z) = x^d + y^d + \nu _x x z^{d-1}+ \nu _y y z^{d-1} \end{equation*}

for well-chosen constants $\nu _x, \nu _y$ . Such an $f$ satisfies the conditions (1)–(3) of Theorem5.2 near each critical point. Moreover, there is a bijection between the critical points of $f(x,y,1)$ and the set $I_d$ of Definition 5.3. If $\nu _x, \nu _y$ are chosen carefully, each critical point lies above a distinct critical value - in this way we embed $I_d \subset{\mathbb{C}}$ .

Each $c \in{\mathbb{C}}$ determines a plane curve $V(cz^d-f)$ . The values of $c$ for which $V(c z^d-f)$ is not smooth are exactly the critical values of $f(x,y,1)$ . The family

\begin{equation*} H = \{V(c z^d-f) \mid V(c z^d-f) \mbox { is smooth}\} \end{equation*}

is a subfamily of $\mathcal{P}_{d}$ defined over ${\mathbb{C}} \setminus I_d$ . The Hefez–Lazzeri basis $\{\sigma _i \mid i \in I_d\}$ is a carefully chosen set of paths in ${\mathbb{C}} \setminus I_d$ with each $\sigma _i$ encircling an individual $i \in I_d$ . Lönne shows that the inclusions of these paths into $\mathcal{P}_d$ via the family $H$ generates $\pi _1(\mathcal{P}_d)$ . The result now follows from an application of Theorem5.2.

5.3. First properties of $\rho _d$

Proposition 5.8 establishes the existence of a collection $c_i, i \in I_d$ of vanishing cycles on $X$ . In this section, we derive some basic topological properties of this configuration arising from the fact that the Dehn twists $T_{c_i}^{-1}$ must satisfy the relations (1)-(3) of Lönne’s presentation.

Lemma 5.10.

1. 1. If the vertices $v_i, v_j$ are adjacent, then the curves $c_i, c_j$ satisfy $i(c_i, c_j) = 1$ .

2. 2. For $d \ge 4$ , the curves $c_i, i \in I_d$ are pairwise distinct, and all $c_i$ are non-separating.

3. 3. If the vertices $v_i, v_j$ in $\Gamma _d$ are non-adjacent, then the curves $c_i$ and $c_j$ are disjoint.

4. 4. For $d \ge 4$ , if the vertices $v_i, v_j, v_k$ form a triangle in $\Gamma _d$ , then the curves $c_i, c_j, c_k$ are supported on an essential subsurface Footnote ⁵ $S_{ijk} \subset X$ homeomorphic to $\Sigma _{1,2}$ . Moreover, if the triangle determined by $v_i,v_j,v_k$ is positively oriented, then $i(c_i, T^{-1}_{c_j}(c_k)) = 0$ .

Proof. (1): If $v_i$ and $v_j$ are adjacent, then the Dehn twists $T_{c_i}^{-1}$ and $T_{c_j}^{-1}$ satisfy the braid relation. It follows from Proposition 4.3 that $i(c_i,c_j) = 1$ .

(2): Suppose $v_i$ and $v_j$ are distinct vertices. For $d \ge 4$ , no two vertices have the same set of adjacent vertices, so that there is some $v_k$ adjacent to $v_i$ and not $v_j$ . By (1) above, it follows that $T_{c_i}^{-1}$ and $T_{c_k}^{-1}$ satisfy the braid relation, while $T_{c_j}^{-1}$ and $T_{c_k}^{-1}$ do not, showing that the isotopy classes of $c_i$ and $c_j$ are distinct. Since each $c_i$ satisfies a braid relation with some other $c_j$ , Proposition 4.3 shows that $c_i$ is non-separating.

(3): If $v_i$ and $v_j$ are non-adjacent, then the Dehn twists $T_{c_i}^{-1}$ and $T_{c_j}^{-1}$ commute. According to [ [Reference Farb and Margalit13], Section 3.5.2], this implies that either $c_i = c_j$ or else $c_i$ and $c_j$ are disjoint. By (2), the former possibility cannot hold.

(4): Via the change-of-coordinates principle, it can be checked that if $c_i, c_j, c_k$ are curves that pairwise intersect once, then $c_i \cup c_j \cup c_k$ is supported on an essential subsurface of the form $\Sigma _{1,b}$ for $1 \le b \le 3$ . In the case $b = 1$ , the curve $c_k$ must be of the form $c_k = T_{c_i}^{\pm 1}(c_j)$ . It follows that if $d$ is a curve such that $i(d,c_k) \ne 0$ , then at least one of $i(d, c_i)$ and $i(d,c_j)$ must also be nonzero. However, as $d \ge 4$ , there is always some vertex $v_l$ adjacent to exactly one of $c_i, c_j, c_k$ . The corresponding curve $c_l$ would violate the condition required of $d$ above (possibly after permuting the indices $i,j,k$ ).

It remains to eliminate the possibility $b = 3$ . In this case, the change-of-coordinates principle implies that up to homeomorphism, the curves $c_i, c_j, c_k$ must be arranged as in Figure 4. It can be checked directly (e.g., by examining the action on $H_1(\Sigma _{1,3})$ ) that for this configuration, the relation

\begin{equation*} T_{c_i}^{-1}T_{c_j}^{-1} T_{c_k}^{-1} T_{c_i}^{-1} = T_{c_j}^{-1}T_{c_k}^{-1} T_{c_i}^{-1} T_{c_j} ^{-1} \end{equation*}

does not hold. This violates relation (3) in Lönne’s presentation of $\pi _1(\mathcal{P}_d)$ . We conclude that necessarily $b = 2$ .

Having shown that $b = 2$ , it remains to show the condition $i(c_i, T^{-1}_{c_j}(c_k)) = 0$ for a positively oriented triangle. Let $(x,y,z)$ denote a $3$ -chain on $\Sigma _{1,2}$ . The change-of-coordinates principle implies that without loss of generality, $c_i=x, c_j=y$ , and $c_k = T_y^{\pm 1}(z)$ . We wish to show that necessarily $c_k = T_y(z)$ . It can be checked directly (e.g., by considering the action on $H_1(\Sigma _{1,2})$ ) that only in the case $c_k = T_y(z)$ does relation (3) in the Lönne presentation hold.

Figure 4. Lemma 5.10.4: the configuration of $c_i,c_j,c_k$ in the $b = 3$ case.

6. Configurations of vanishing cycles

The goal of this section is to derive an explicit picture of the configuration of vanishing cycles on a plane curve of degree $5$ . The main result of the section is Lemma 6.1.

Figure 5. The curves of Lemma 6.1. The bottom halves of curves $b,x,y,z,$ and $c_i$ for $i$ odd have been omitted for clarity; on the bottom half, each curve follows its mirror image on the top.

Proof. Lemma 6.1 will be proved in three steps.

Step 1: Uniqueness of Lönne configurations

A configuration of curves $c_i, i \in I_d$ as in Lemma 6.2 will be referred to as a Lönne configuration.

Proof. Let $a_{1,1},\dots, a_{d-1, d-1}$ determine a Lönne configuration on $\Sigma _g$ . We will exhibit a homeomorphism of $\Sigma _g$ taking each $a_{i,j}$ to a corresponding $b_{i,j}$ in a “reference” configuration $\{b_{i,j}\}$ to be constructed in the course of the proof. This will require three steps.

Step 1: A collection of disjoint chains

Each row in the Lönne graph determines a chain of length $d-1$ . The change of coordinates principle for chains of even length (Lemma 4.2) asserts that any two chains of length $d-1$ are equivalent up to homeomorphism. Considering the odd-numbered rows of $\Gamma _d$ , it follows that there is a homeomorphism $f_1$ of $\Sigma _g$ that takes each $a_{2i-1,j}$ for $1 \le i \le d-1$ to a curve $b_{2i-1,j}$ in a standard picture of a chain. We denote the subsurface of $\Sigma _g$ determined by the chain $a_{2i-1,1}, \dots, a_{2i-1,d-1}$ as $A_i$ , and similarly we define the subsurfaces $B_i$ of the reference configuration. Each $A_i, B_i$ is homeomorphic to $\Sigma _{(d-1)/2,1}$ .

Step 2: Arcs on $A_i$

The next step is to show that up to homeomorphism, there is a unique picture of what the intersection of the remaining curves $a_{2i,j}$ with $\bigcup A_i$ looks like. Consider a curve $a_{2i,j}$ . Up to isotopy, $a_{2i,j}$ intersects only the subsurfaces $A_i$ and $A_{i+1}$ . We claim that $a_{2i,j}$ can be isotoped so that its intersection with $A_i$ is a single arc, and similarly for $A_{i+1}$ . If $j = d-1$ , then $a_{2i, d-1}$ intersects only the curve $a_{2i-1, d-1}$ , and $i(a_{2i,d-1},a_{2i-1,d-1}) = 1$ . It follows that if $a_{2i,d-1} \cap A_i$ has multiple components, exactly one is essential, and the remaining components can be isotoped off of $A_i$ .

In the general case, where $a_{2i,j}$ intersects both $a_{2i-1,j}$ and $a_{2i-1,j+1}$ , an analogous argument shows that $a_{2i,j} \cap A_i$ consists of one or two essential arcs. Consider the triangle in the Lönne graph determined by $a_{2i,j}, a_{2i-1,j}, a_{2i-1, j+1}$ . According to Lemma 5.10.4, the union $a_{2i,j}\cup a_{2i-1,j}\cup a_{2i-1, j+1}$ is supported on an essential subsurface of the form $\Sigma _{1,2}$ . Figure 6 shows that if $a_{2i,j} \cap A_i$ consists of two essential arcs, then $a_{2i,j}\cup a_{2i-1,j}\cup a_{2i-1, j+1}$ is supported on an essential subsurface of the form $\Sigma _{1,3}$ , in contradiction with Lemma 5.10.4. Similar arguments establish that $a_{2i-2,j} \cap A_{i}$ is a single essential arc as well.

Figure 6. If $a_{2i,j}$ cannot be isotoped onto a single arc inside $A_i$ , then the curve enclosed by the inner strip (shaded) is essential in $\Sigma _g$ , causing $a_{2i,j}\cup a_{2i-1,j}\cup a_{2i-1, j+1}$ to be supported on a surface $\Sigma _{1,3}$ .

We next show that all points of intersection $a_{2i,j} \cap a_{2i, j +1}$ can be isotoped to occur on both $A_i$ and $A_{i+1}$ . This also follows from Lemma 5.10.4. If some point of intersection $a_{2i,j} \cap a_{2i, j +1}$ could not be isotoped onto $A_i$ , then the union $a_{2i, j} \cup a_{2i, j+1} \cup a_{2i-1, j+1}$ could not be supported on a subsurface homeomorphic to $\Sigma _{1,2}$ . An analogous argument applies with $A_{i+1}$ in place of $A_i$ . This is explained in Figure 7.

Figure 7. If the intersection $a_{2i,j} \cap a_{2i,j+1}$ cannot be isotoped to occur on $A_i$ , then both curves indicated by the shaded regions are essential in $\Sigma _g$ , causing $a_{2i,j}\cup a_{2i,j+1}\cup a_{2i-1, j+1}$ to be supported on a surface $\Sigma _{1,3}$ .

It follows from this analysis that all crossings between curves in row $2i$ can be isotoped to occur in a collar neighborhood of $\partial A_i$ . We define $A_i^+$ to be a slight enlargement of $A_i$ along such a neighborhood, so that all crossings between curves in row $2i$ occur in $A_i^+ \setminus A_i$ .

We can now understand what the collection of arcs $a_{2i,1} \cap A_i^+, \dots, a_{2i, d-1}\cap A_i^+$ looks like. To begin with, the change-of-coordinates principle asserts that up to a homeomorphism of $A_i$ fixing the curves $\{a_{2i-1,j}\}$ , the arc $a_{2i,1} \cap A_i$ can be drawn in one of two ways. The first possibility is shown in Figure 8(a), and the second is its mirror-image obtained by reflection through the plane of the page (i.e., the curve with the dotted and solid portions exchanged). In fact, $a_{2i,1} \cap A_i$ must look as shown. This follows from Lemma 5.10.4. The vertices $(a_{2i-1, 1}, a_{2i-1, 2}, a_{2i,1})$ form a positively oriented triangle, and so $i(a_{2i-1,1}, T_{a_{2i-1,2}}^{-1}(a_{2i,1})) = 0$ . This condition precludes the other possibility.

The pictures for $a_{2i,2}, \dots, a_{2i, d-1}$ are obtained by proceeding inductively. In each case, there are exactly two ways to draw an arc satisfying the requisite intersection properties, and Lemma 5.10.4 precludes one of these possibilities. The result is shown in Figure 8(b).

It remains to understand how the crossings between curves in row $2i$ are organized on $A_i^+$ . As shown, the arcs $a_{2i,j}\cap A_i$ and $a_{2i,j+1}\cap A_i$ intersect $\partial A_i$ twice each, and in both instances, the intersections are adjacent relative to the other arcs. There are thus apparently two possibilities for where the crossing can occur. However, one can see from Figure 8(c) that once a choice is made for one crossing, this enforces choices for the remaining crossings. Moreover, the two apparently distinct configurations are in fact equivalent: the cyclic ordering of the arcs along $\partial A_i^+$ is the same in either case, and the combinatorial type of the cut-up surface

\begin{equation*} A_i^\circ : = A_i^+ \setminus \bigcup \{a_{k, j} \mid 2i-1 \le k \le 2i, 1 \le j \le d-1 \} \end{equation*}

is the same in either situation. The change-of-coordinates principle then asserts the existence of a homeomorphism of $A_i^+$ sending each $a_{2i-1,j}$ to itself and taking one configuration of arcs to the other.

Having seen that the arcs $a_{2i,j} \cap A_i^+$ can be put into standard form, it remains to examine the other collection of arcs on $A_i^+$ , namely those of the form $a_{2i-2,j}$ . It is easy to see by induction on $d$ that the cut-up surface $A_i^\circ$ is a union of polyhedral disks for which the edges correspond to portions of the curves $a_{2i-1,j}$ , the arcs $a_{2i,j} \cap A_i^+$ , or else the boundary $\partial A_i^+$ . It follows that the isotopy class of an arc $a_{2i-2,j}\cap A_I^+$ is uniquely determined by its intersection data with the curves $a_{2i-1,j}$ and $a_{2i,j}$ .

For $j \ge 2$ , the curve $a_{2i-2,j}$ intersects $a_{2i-1,j-1}$ and $a_{2i-1,j}$ and is disjoint from all curves $a_{2i, k}$ . As $a_{2i,j-1}$ has the same set of intersections as $a_{2i-2,j}$ , it follows that $a_{2i-2,j} \cap A_i^+$ must run parallel to $a_{2i,j-1}$ . The curve $a_{2i-2,1}$ intersects only $a_{2i,1}$ ; consequently, $a_{2i-2,1}\cap A_i^+$ is uniquely determined. As can be seen from Figure 8(c), this forces each subsequent $a_{2i-2,j}$ onto a particular side of $a_{2i,j-1}$ .

Figure 8. The surface $A_i^+$ . (a): The correct choice for $a_{2i,1} \cap A_i$ . (b): The configuration $a_{2i,j} \cap A_i$ . (c) The configuration $a_{2i,j} \cap A_i^+$ .

Step 3: Arcs on the remainder of $\Sigma _g$

Consider now the subsurface

\begin{equation*} \Sigma _g^\circ := \Sigma _g \setminus \bigcup A_i. \end{equation*}

This has $(d-1)/2$ boundary components $\partial _{k}$ , indexed by the corresponding $A_k$ . The intersection $a_{2i,j} \cap \left ( \Sigma _g \setminus \bigcup A_i\right )$ consists of two arcs, each connecting $\partial _{i}$ with $\partial _{i+1}$ . The strategy for the remainder of the proof is to argue that when all these arcs are deleted from $\Sigma _g^\circ$ , the result is a union of disks. The change-of-coordinates principle will then assert the uniqueness of such a configuration of arcs, completing the proof.

For what follows, it will be convenient to refer to a product neighborhood $[0,1] \times [0,1] \subset \Sigma _g^\circ$ of some arc $a_{2i,j} \cap \Sigma _g^\circ$ as a strip. Our first objective is to compute the Euler characteristic $\chi$ of the surface $\Sigma _g^{\circ \circ }$ obtained by deleting strips for all arcs from $\Sigma _g^\circ$ . Then an analysis of the pattern by which strips are attached will determine the number of components of this surface.

To begin, we return to the setting of Figure 7. Above, it was argued that for $2i \lt (d-1)/2$ , the intersection $a_{2i,j} \cap a_{2i,j+1}$ can be isotoped onto either $A_{i}$ or $A_{i+1}$ . This means that there is a strip that contains both $a_{2i,j} \cap \Sigma _g^\circ$ and $a_{2i,j+1} \cap \Sigma _g^\circ$ . Grouping such strips together, it can be seen that for $1\le i \le (d-3)/2$ , the $2i^{th}$ row of the Lönne graph gives rise to $d$ strips. In the last row, there are $d-1$ strips. So in total there are $1/2(d+1)(d-2)$ strips, and each strip contributes $-1$ to the Euler characteristic.

Recall the relation $g = (d-1)(d-2)/2$ : this means that

\begin{equation*} \chi (\Sigma _g) = 2 - (d-1)(d-2). \end{equation*}

Each $A_i$ has Euler characteristic $\chi (A_i) = 2-d$ . It follows that

\begin{align*} \chi (\Sigma _g^\circ ) = \chi (\Sigma _g) - \sum _{i = 1}^{(d-1)/2} \chi (A_i) &= 2 - (d-1)(d-2) + (d-1)(d-2)/2\\ &= 2 - (d-1)(d-2)/2. \end{align*}

Therefore,

\begin{align*} \chi (\Sigma _g^{\circ \circ }) &= \chi (\Sigma _g^\circ ) + 1/2(d+1)(d-2) \\ &= d. \end{align*}

We claim that $\Sigma _g^{\circ \circ }$ has $d$ boundary components. This will finish the proof, as a surface of Euler characteristic $d$ and $b = d$ boundary components must be a union of $d$ disks. The claim can easily be checked directly in the case $d = 5$ of immediate relevance. For general $d$ , this follows from a straightforward, if notationally tedious, verification, proceeding by an analysis of the cyclic ordering of the arcs $a_{i,j}$ around the boundary components $\partial A_k^+$ .

Step 2: A convenient configuration

Figure 9. A Lönne configuration on $\Sigma _6$ . Only a portion of the figure has been drawn: the omitted curves are obtained by applying the involution $\iota$ to the depicted curves.

Figure 9 presents a picture of a Lönne configuration in the case of interest $d = 5$ . This was obtained by “building the surface” curve by curve, attaching one-handles in the sequence indicated by the numbering of the curves $a_1, \dots, a_{16}$ . There are other, more uniform depictions of Lönne configurations which arise from Akbulut–Kirby’s picture of a plane curve of degree $d$ derived from a Seifert surface of the $(d,d)$ torus link (see [Reference Akbulut and Kirby2] or [ [Reference Gompf and Stipsicz14], Section 6.2.7]), but the analysis to follow is easier to carry out using the model of Figure 9.

Step 3: Producing vanishing cycles

The bulk of this step will establish claim (1); claim (2) follows as an immediate porism. The set of vanishing cycles is invariant under the action of the monodromy group, since acting by a monodromy element on a nodal degeneration amounts to changing the path along which the nodal degeneration is performed. In particular, if $a$ and $b$ are vanishing cycles, then so is $T_a(b)$ . To begin with, curves $c_1,c_2, c_4, c_8,$ and $c_{12}$ are elements of the Lönne configuration and so are already vanishing cycles. The curve $c_3$ is obtained as

\begin{equation*} c_3 = T_{a_2}^{-1}(a_3); \end{equation*}

similarly,

\begin{equation*} c_{13} = T_{a_2}^{-1}(a_4). \end{equation*}

Curve $c_{10}$ is obtained as

\begin{equation*} c_{10} = T_{a_{15}}(a_{13}); \end{equation*}

$c_6$ is obtained from $a_{14}$ and $a_{16}$ analogously.

The curve $c_9$ is obtained as

\begin{equation*} c_9 = T_{c_{10}} T_{a_{10}}^{-1} (a_{11}); \end{equation*}

$c_7$ is obtained from $a_{10}, a_{12},$ and $c_6$ analogously.

To obtain $c_5$ , twist $a_{13}$ along the chain $c_6, \dots, c_{10}$ :

\begin{equation*} c_5 = T_{c_6}^{-1} T_{c_7}^{-1} T_{c_8}^{-1} T_{c_9}^{-1} T_{c_{10}}^{-1}(a_{13}). \end{equation*}

$c_{11}$ is obtained by an analogous procedure on $a_{14}$ .

The sequence of twists used to exhibit $x$ as a vanishing cycle is illustrated in Figure 10. Symbolically,

\begin{equation*} x = T_{c_6}^{-1} T_{c_7}^{-1} T_{c_8}^{-1}T_{c_9}^{-1}T_{c_5} T_{c_4} T_{c_6}^{-1}T_{c_7}^{-1}T_{c_8}^{-1}T_{a_{9}}^{-1}(a_7). \end{equation*}

$y$ is produced in an analogous fashion, starting with $a_{8}$ in place of $a_6$ .

To produce $z$ , we appeal to the genus- $2$ star relation. Applied to the surface bounded by $b,y,z$ , it shows that $T_b^2 T_y T_z \in \mathrm{Im}(\rho _5)$ , and hence $T_{b}^2 T_z \in \mathrm{Im}(\rho _5)$ since $T_y \in \mathrm{Im}(\rho _5)$ by above. Observe that $i(c_{10}, z) = 1$ , and that $T_{c_{10}} \in \mathrm{Im}(\rho _5)$ . Making use of the fact that $b$ is disjoint from both $z$ and $c_{10}$ , the braid relation gives

\begin{equation*} T_{c_{10}}T_b^2 T_z(c_{10}) = T_{c_{10}} T_z (c_{10}) = z. \end{equation*}

This exhibits $z$ as a vanishing cycle, establishing claim (1) of Lemma 6.1. As $T_b^2 T_z$ and $T_z$ are now both known to be elements of $\mathrm{Im}(\rho _5)$ , it follows that $T_b^2 \in \mathrm{Im}(\rho _5)$ as well, completing claim (2).

Figure 10. The sequence of twists used to obtain $x$ .

7. Proof of Theorem 1.2

In this final section, we assemble the work we have done so far in order to prove Theorem 1.2.

7.1. Step 1: Reduction to the Torelli group

The first step is to reduce the problem of determining $\mathrm{Im}(\rho _5)$ to the determination of $\mathrm{Im}(\rho _5) \cap \mathcal{I}_6$ . This will follow from [Reference Beauville4]. Recall that Beauville establishes that $\mathrm{Im}(\Psi \circ \rho _5)$ is the entire stabilizer of an odd-parity spin structure on $H_1(\Sigma _6;{\mathbb{Z}})$ . This spin structure was identified as $\phi _5$ in Section 2. Therefore, $\mathrm{Im}(\Psi \circ \rho _5) = \mathrm{Im}(\Psi \circ \operatorname{Mod}(\Sigma _6)[\phi _5])$ . It, therefore, suffices to show that

(7) \begin{equation} \mathrm{Im}(\rho _5) \cap \ker \Psi = \operatorname{Mod}(\Sigma _6)[\phi _5] \cap \ker \Psi = \mathcal{I}_6. \end{equation}

Figure 11. The cases of step 2.

7.2. Step 2: Enumeration of cases

Equation (7) will be derived as a consequence of Theorem4.6. Lemma 6.1.1 asserts that the curves $c_1, \dots, c_{12}$ in the Johnson generating set are contained in $\mathrm{Im}(\rho _5)$ , so that the first hypothesis of Theorem4.6 is satisfied. There are then eight cases to check: the four straight chain maps of the form $(c_1, \dots, c_k)$ for $k = 3,5,7,9$ and the four $\beta$ -chain maps of the form $(\beta, c_5, \dots, c_k)$ for $k = 6, 8, 10, 12$ . See Figure 11.

The verification of the $\beta$ -chain cases will be easier to accomplish after conjugating by the class $g = T_x T_{c_5}^{-1} T_{c_4}^{-1} \in \mathrm{Im}(\rho _5)$ . This has the following effect on the curves in the $\beta$ -chains (the curve $\gamma$ is indicated in Figure 11 in the picture for $k = 6$ ):

\begin{equation*} g(\beta ) = b, \qquad g(c_5) = c_4, \qquad g(c_6) = \gamma, \qquad g(c_k) = c_k \mbox { for } k \ge 7. \end{equation*}

7.3. Step 3: Producing bounding-pair maps

In this step, we explain the method by which we will obtain the necessary bounding-pair maps. This is an easy consequence of the chain relation.

Lemma 7.1. Let $C = (c_1, \dots, c_k)$ be a chain of odd length $k$ and boundary $\partial C = d_1 \cup d_2$ . Suppose that the mapping classes

\begin{equation*} T_{c_1}^2, T_{c_2}, \dots, T_{c_k}, T_{d_1}^2 \end{equation*}

are all contained in some subgroup $\Gamma \le \textrm{Mod}(\Sigma _g)$ . Then the chain map associated to $C$ (i.e., the bounding pair map $T_{d_1} T_{d_2}^{-1}$ ) is also contained in $\Gamma$ .

Proof. The chain relation (Proposition 4.4) implies that $T_{d_1} T_{d_2} \in \Gamma$ . By hypothesis, $T_{d_1}^2 \in \Gamma$ , so the bounding pair map $T_{d_1}T_{d_2}^{-1} \in \Gamma$ as well.