In Sect. 5.1, we prove two vanishing results in relative Gromov–Witten theory of Hirzebruch surfaces. In Sect. 5.2, we define the relative Gromov–Witten invariants \(N_{g,\mathrm {rel}}^{\Delta ,n}\) of h-transverse toric sufaces. In Sect. 5.3, for \(\Delta =\Delta _{h,d}^{{\mathbb {F}}_k}\), we apply the degeneration formula in Gromov–Witten theory to express the invariants \(N_{g,\mathrm {rel}}^{\Delta ,n}\) in terms of the invariants \(N_{g,\mathrm {rel}}^{\mu \nu \emptyset \emptyset }\) defined in Sect. 4. The unrefined, that is, \(g=g_{\Delta ,n}\), version of this degeneration argument can be found for example in the proof of Theorem 4.9 of [15], or, in the Fock space language, in §2.5 of [16]. We adapt this degeneration argument to the refined, that is, \(g \ge g_{\Delta ,n}\), case using the vanishing results proved in Sect. 5.1. Finally, we prove in Sect. 5 our main result, Theorem 5.12, computing the invariants \(N_{g,\mathrm {rel}}^{\Delta ,n}\) for \(\Delta =\Delta _d^{{\mathbb {P}}^2}\) and \(\Delta =\Delta _{h,d}^{{\mathbb {F}}_k}\) in terms of q-refined counts of floor diagrams. The proof relies on the explicit calculation of the invariants \(N_{g,\mathrm {rel}}^{\mu \nu \rho \sigma }\) given by Theorem 4.4.
Dimension constraints
In this section, we prove two vanishing results for relative Gromov–Witten invariants of Hirzebruch surfaces. We use the notations introduced in Sect. 3.1 for relative Gromov–Witten invariants.
Lemma 5.1
Let \(h, d\in {\mathbb {Z}}_{\ge 0}\), and \(\beta =hD_k+dF \in H_2({\mathbb {F}}_k,{\mathbb {Z}}){\setminus } \{0\}\). Let \(\mu =(\mu _j)_{1\le j\le \ell (\mu )}\), \(\nu =(\nu _j)_{1\le j\le \ell (\nu )}\) be two partitions of \(\beta \cdot D_{-k}=d\) and \(\beta \cdot D_k=d+kh\) respectively. Let \(\delta ^1=(\delta ^1_j)_{1\le j\le \ell (\mu )}\) be \(\ell (\mu )\) elements of \(H^{*}(D_{-k},{\mathbb {Z}})\) and \(\delta ^2=(\delta _j^2)_{1\le j \ell (\nu )}\) be \(\ell (\nu )\) elements of \(H^{*}(D_{k},{\mathbb {Z}})\). Assume that among these \(\ell (\mu )+\ell (\nu )\) cohomology classes, s of them are equal to 1 and \(\ell (\mu )+\ell (\nu )-s\) of them are Poincaré dual to a point. Then, for every \(g \in {\mathbb {Z}}_{\ge 0}\) and \(0\le j \le g\), we have
$$\begin{aligned} \langle \mu ,\delta ^2|\, (-1)^j \lambda _j|\,\nu ,\delta ^1 \rangle _{g,\beta }^{{\mathbb {F}}_k/D_k \cup D_{-k}} =0 , \end{aligned}$$
(5.1)
unless the following conditions hold:
-
(i)
\(h=0\), that is \(\beta =dF\). In such case, we have \(\beta \cdot D_{-k}=\beta \cdot D_k = d\).
-
(ii)
\(\mu \) and \(\nu \) are both the trivial 1-part partition (d) of d, that is, \(\ell (\mu )=\ell (\nu )=1\).
-
(ii)
\(s=1\), that is among the \(\ell (\mu )+\ell (\nu )=2\) cohomology classes \(\delta ^1_1\) and \(\delta ^2_1\), exactly one of them is equal to 1 and the other is Poincaré dual to a point.
-
(iv)
\(g=j=0\).
If these conditions are satisfied, then
$$\begin{aligned} \langle \mu ,\delta ^2|\,\nu ,\delta ^1 \rangle _{0,\beta }^{{\mathbb {F}}_k/D_k \cup D_{-k}} =\frac{1}{d} . \end{aligned}$$
(5.2)
Proof
By (3.3), the virtual dimension of the moduli stack of relative stable maps used to define
$$\begin{aligned} \langle \mu ,\delta ^2|\, (-1)^j \lambda _j|\,\nu ,\delta ^1 \rangle _{g,\beta }^{{\mathbb {F}}_k/D_k \cup D_{-k}} \end{aligned}$$
(5.3)
is
$$\begin{aligned}&g-1+\beta \cdot (D_k+D_{-k}+2F)-(\beta \cdot D_{-k} -\ell (\mu ))-(\beta \cdot D_k -\ell (\nu )) \nonumber \\&\quad =g-1+2h+\ell (\mu )+\ell (\nu ). \end{aligned}$$
(5.4)
On the other hand, we integrate in 5.3 over the virtual dimension class a cohomology class of complex degree
$$\begin{aligned} j+\ell (\mu )+\ell (\nu )-s. \end{aligned}$$
(5.5)
(5.3) is 0 unless (5.4) = (5.5), that is \(2h+s+(g-j)=1\). As h, s and \((g-j)\) are nonnegative integers, this is only possible if either \((h,s,j)=(0,0,g-1)\) or \((h,s,j)=(0,1,g)\).
If \((h,s,j)=(0,0,g-1)\), then \(\beta =d F\) and we fix the position of the \(\ell (\mu )+\ell (\nu )\) contact points with \(D_k \cup D_{-k}\). Note that \(\ell (\mu )\), \(\ell (\nu ) \ge 1\) because the assumption \(\beta \ne 0\) implies that \(d>0\). As \(\ell (\mu )+\ell (\nu )\ge 2\), we can choose the position of two contact points in two different fibers of \(p :{\mathbb {F}}_k \rightarrow {\mathbb {P}}^1\). But a curve of class \(\beta =dF\) is contained in a \({\mathbb {P}}^1\)-fiber of p, so the set of curves matching the constraints is empty and so (5.3) is zero.
If \((h,s,j)=(0,1,g)\), then \(\beta =d F\) and we fix the position of \(\ell (\mu )+\ell (\nu )-1\) of the \(\ell (\mu )+\ell (\nu )\) contact points with \(D_k \cup D_{-k}\). If \(\ell (\mu )+\ell (\nu )-1\ge 2\), we can choose the position of two contact points in two different fibers of p and as a curve of class \(\beta =dF\) is contained in a \({\mathbb {P}}^1\)-fiber of p, the set of curves matching the constraints is empty. Hence, (5.3) is still zero unless \(\ell (\mu )=\ell (\nu )=1\). Finally, under the assumptions (i)-(iv), (5.2) follows from Lemma 3.3(ii). \(\square \)
Lemma 5.2
Let \(h,d \in {\mathbb {Z}}_{\ge 0}\), and \(\beta =hD_k+dF \in H_2({\mathbb {F}}_k,{\mathbb {Z}}){\setminus } \{0\}\). Let \(\mu =(\mu _j)_{1\le j\le \ell (\mu )}\), \(\nu =(\nu _j)_{1\le j\le \ell (\nu )}\) be two partitions of \(\beta \cdot D_{-k}=d\) and \(\beta \cdot D_k=d+kh\) respectively. Let \(\delta ^1=(\delta ^1_j)_{1\le j\le \ell (\mu )}\) be \(\ell (\mu )\) elements of \(H^{*}(D_{-k},{\mathbb {Z}})\) and \(\delta ^2=(\delta _j^2)_{1\le j \ell (\nu )}\) be \(\ell (\nu )\) elements of \(H^{*}(D_{k},{\mathbb {Z}})\). Assume that among these \(\ell (\mu )+\ell (\nu )\) cohomology classes, s of them are equal to 1 and \(\ell (\mu )+\ell (\nu )-s\) of them are Poincaré dual to a point. Then, for every \(g \in {\mathbb {Z}}_{\ge 0}\) and \(0\le j\le g\), denoting by \(\alpha _1\in H^4(X,Z)\) the class Poincaré dual to a point, we have
$$\begin{aligned} \langle \mu ,\delta ^2|\, (-1)^j \lambda _j; \alpha _1|\,\nu ,\delta ^1 \rangle _{g,\beta }^{{\mathbb {F}}_k/D_k \cup D_{-k}} =0 , \end{aligned}$$
(5.6)
unless we are in one of the following two situations.
-
(i)
\(h=0\), that is \(\beta =dF\), \(\mu \) and \(\nu \) are both the trivial 1-part partition (d) of d, that is, \(\ell (\mu )=\ell (\nu )=1\), \(s=1\), that is both of the \(\ell (\mu )+\ell (\nu )=2\) cohomology classes \(\delta ^1_1\) and \(\delta ^2_1\) are equal to 1, and \(g=j=0\). In this case, we have
$$\begin{aligned} \langle \mu ,\delta ^2|\, \alpha _1|\,\nu ,\delta ^1 \rangle _{0,\beta }^{{\mathbb {F}}_k/D_k \cup D_{-k}} =1 . \end{aligned}$$
(5.7)
-
(ii)
\(h=1\), that is \(\beta =D_k+dF\), \(s=0\), that is all of the \(\ell (\mu )+\ell (\nu )\) cohomology classes \(\delta ^1_j\) and \(\delta ^2_j\) are Poincaré dual to a point, and \(j=g\). In this case, we have
$$\begin{aligned} \langle \mu ,\delta ^2|\, (-1)^g \lambda _g; \alpha _1|\,\nu ,\delta ^1 \rangle _{g,\beta }^{{\mathbb {F}}_k/D_k \cup D_{-k}} =N_{g,\mathrm {rel}}^{\mu \nu \emptyset \emptyset } , \end{aligned}$$
(5.8)
where \(N_{g,\mathrm {rel}}^{\mu \nu \emptyset \emptyset }\) is the specialization for \(\rho =\sigma =\emptyset \) of the invariants \(N_{g,\mathrm {rel}}^{\mu \nu \rho \sigma }\) defined in (4.5).
Proof
By (3.3), the virtual dimension of the moduli stack of relative stable maps used to define
$$\begin{aligned} \langle \mu ,\delta ^2|\, (-1)^g \lambda _g;\alpha _1|\,\nu ,\delta ^1 \rangle _{g,\beta }^{{\mathbb {F}}_k/D_k \cup D_{-k}} \end{aligned}$$
(5.9)
is
$$\begin{aligned}&g-1+\beta \cdot (D_k+D_{-k}+2F)+1-(\beta \cdot D_{-k} -\ell (\mu ))-(\beta \cdot D_k -\ell (\nu )) \nonumber \\&\quad =g+2h+\ell (\mu )+\ell (\nu ). \end{aligned}$$
(5.10)
On the other hand, we integrate in 5.3 over the virtual dimension class a cohomology class of complex degree
$$\begin{aligned} j+2+\ell (\mu )+\ell (\nu )-s. \end{aligned}$$
(5.11)
(5.9) is 0 unless (5.10) = (5.11), that is \(2h+s+(g-j)=2\). As h, s, and \((g-j)\) are nonnegative integers, there are only four possibilities: \((h,s,j)=(0,0,g-2)\), \((h,s,j)=(0,1,g-1)\), \((h,s,j)=(0,2,g)\) and \((h,s,j)=(1,0,g)\).
If \((h,s,j)=(0,2-r,g-r)\) for some \(r \in \{0,1,2\}\), then \(\beta =d F\) and we fix the position of \(\ell (\mu )+\ell (\nu )+r-2\) of the \(\ell (\mu )+\ell (\nu )\) contact points with \(D_k \cup D_{-k}\). If \(\ell (\mu )+\ell (\nu )+r-2\ge 1\), we can choose the positions of one of the contact point and of the interior marked point in two different fibers of \(p :{\mathbb {F}}_k \rightarrow {\mathbb {P}}^1\), and so, as a curve of class \(\beta =dF\) is contained in a \({\mathbb {P}}^1\)-fiber of p, the set of curves matching the constraints is empty. Hence, (5.9) is still zero unless \(r=0\) and \(\ell (\mu )=\ell (\nu )=1\). Thus, we are in the case (i) and (5.7) follows from Lemma 3.3(i).
If \((h,s,j)=(1,0,g)\), then we are in the case (ii) and (5.8) follows from the definition (4.5) of \(N_{g,\mathrm {rel}}^{\mu \nu \rho \sigma }\). \(\square \)
Gromov–Witten invariants of h-transverse toric surfaces
Let \(\Delta \) be a h-transverse balanced collection of vectors in \({\mathbb {Z}}^2\) as in Definition 2.1, and n a nonnegative integer. We assume that the corresponding toric surface \(X_\Delta \) given by Definition 2.2 is smooth. We defined in Sect. 2.1 a curve class \(\beta _{\Delta }\) and two smooth disjoint divisors \(D_b\) and \(D_t\). We define the relative Gromov–Witten invariants \(N_{g,\mathrm {rel}}^{\Delta ,n}\) of \(X_\Delta /D_b \cup D_t\) by
$$\begin{aligned} N_{g,\mathrm {rel}}^{\Delta ,n} :=\langle \mu ,\delta ^2|(-1)^{g-g_\Delta } \lambda _{g-g_\Delta }; \alpha _1,\dots ,\alpha _n|\nu ,\delta ^1\rangle _{g,\beta _\Delta }^{X_\Delta /D_b\cup D_t} , \end{aligned}$$
(5.12)
where we apply the general definition (3.8) of relative Gromov–Witten invariants to \(X=X_\Delta \), \(D_1=D_b\), \(D_2=D_t\), \(\beta =\beta _\Delta \), and where:
-
(i)
\(\mu \) is the partition of \(d_t=\beta \cdot D_t\) whose all parts are equal 1, that is \(\ell (\mu )=d_t\), and \(\delta ^2=(\delta ^2_j)_{1\le j\le d_t}\), where \(\delta ^2_j=1 \in H^0(D_t,{\mathbb {Z}})\) for all j,
-
(ii)
\(\nu \) is the partition of \(d_b=\beta \cdot D_b\) whose all all parts are equal to 1, that is \(\ell (\nu )=d_b\), and \(\delta ^1=(\delta ^1_j)_{1\le j\le d_b}\), where \(\delta _j^1=1 \in H^0(D_b,{\mathbb {Z}})\) for all j,
-
(iii)
the cohomology classes \(\alpha _1,\dots ,\alpha _n\) inserted at the n interior marked points are all equal to the Poincaré dual class of a point in \(H^4(X_\Delta ,{\mathbb {Z}})\)
-
(iv)
the class \(\gamma \) in (3.8) on the moduli stack of relative stable maps is taken to be \((-1)^{g-g_{\Delta ,n}}\lambda _{g-g_{\Delta ,n}}\), where \(g_{\Delta ,n}=n+1-|\Delta |\) (see Definition 2.6(ii)) and \(\lambda _{g-g_{\Delta ,n}}\) the lambda class of complex degree \(g-g_{\Delta ,n}\) as in (3.9).
In other words, \(N_{g,\mathrm {rel}}^{\Delta ,n}\) is a virtual count of genus g class \(\beta _\Delta \) curves in \(X_\Delta \), with transversal intersections with the divisors \(D_b\) and \(D_t\) and passing through n given points in \(X_\Delta \).
The degeneration formula for the invariants \(N_{g,\mathrm {rel}}^{\Delta ,n}\)
In this section, we fix \(\Delta =\Delta _{h,d}^{{\mathbb {F}}_k}\), as in Sect. 2.5, so that \(X_\Delta ={\mathbb {F}}_k\) and we state in Lemma 5.5 a precise version of the degeneration formula [23] computing the relative Gromov–Witten invariants \(N_{g,\mathrm {rel}}^{\Delta ,n}\) defined in (5.12).
By successive degeneration of \({\mathbb {F}}_k\) to the normal cone of the divisor \(D_{-k}\), we construct a degeneration \({\mathcal {X}}\rightarrow {\mathbb {A}}^1\) of \({\mathbb {F}}_k\), whose special fiber \({\mathcal {X}}_0\) is a chain of n copies \({\mathbb {F}}_k^{(j)}\) of \({\mathbb {F}}_k\) \(1 \le j \le n\). For every \(1 \le j \le n-1\), the divisor \(D_{-k}^{(j)}\) of \({\mathbb {F}}_k^{(j)}\) is transversally glued with the divisor \(D_k^{(j+1)}\) of \({\mathbb {F}}_k^{(j+1)}\). We denote
$$\begin{aligned} D^0 :=D_{k}^{(0)},\,\,\,\, D^j :=D_{-k}^{(j)}\simeq D_k^{j+1}\, \text {for} \,\,1 \le j\le n-1, \,\,\,\,\text {and}\,\,\, D^{n} :=D_{-k}^{(n)}.\nonumber \\ \end{aligned}$$
(5.13)
Our goal is to state precisely the degeneration formula in relative Gromov–Witten theory [23] applied to \({\mathcal {X}}\rightarrow {\mathbb {A}}^1\) to compute \(N_{g,\mathrm {rel}}^{\Delta ,n}\), where we distribute the n point conditions \(\alpha _1,\dots ,\alpha _n\) appearing in (5.12) by placing one on each of the n components \({\mathbb {F}}_k^{(1)},\dots ,{\mathbb {F}}_k^{(n)}\) (see Fig. 5). To do that, we need to introduce some combinatorial notations, that might seem heavy but are geometrically completely natural. We first introduce in Definition 5.3 below a set of weighted decorated graphs which will index the terms of the degeneration formula and describe the possible degeneration types of curves in the special fiber \({\mathcal {X}}_0\). The geometric meaning of each condition is explained in the proof of Lemma 5.5 below.
Definition 5.3
We denote by \({\mathcal {G}}^{k,n}_{h,d,g}\) the set of decorated weighted graphs \(\Gamma \) which are as follows.
-
(i)
\(\Gamma \) is a weighted graph as in Sect. 2.2, with a set \(V(\Gamma )\) of vertices and a set \(E(\Gamma )\) of edges, which are either bounded or unbounded. Every edge \(E \in E(\Gamma )\) has a weight \(w_E \in {\mathbb {Z}}_{\ge 1}\), which is equal to 1 if E is unbounded.
-
(ii)
Every vertex \(V \in V(\Gamma )\) has a genus decoration \(g_V \in {\mathbb {Z}}_{\ge 0}\), and the first Betti number \(g_\Gamma \) of \(\Gamma \) is such that \(g_\Gamma + \sum _{V \in V(\Gamma )} g_V=g\).
-
(iii)
Every vertex \(V \in V(\Gamma )\) is decorated by an index \(j_V \in \{1,\dots ,n\}\) and a class \(\beta _V =h_V D_k^{(j_V)}+d_V F^{(j_V)} \in H_2({\mathbb {F}}_k^{(j_V)},{\mathbb {Z}})\).
-
(iv)
For every \(1 \le j \le n\), there is a distinguished vertex, denoted by \(V_j\), among the vertices with \(j_V=j\).
-
(v)
Every edge \(E \in E(\Gamma )\) is decorated by an index \(j_E\in \{0,\dots ,n\}\). If E is a bounded edge, then \(j_E \in \{1,\dots ,n-1\}\) and there is a labeling V, \(V'\) of the two vertices incident to E such that \(j_V=j_E\) and \(j_{V'}=j_E+1\). If E is an unbounded edge, then \(j_E\in \{0,n\}\), and there are exactly \(d+kh\) unbounded edges E with \(j_E=0\) and d unbounded edges E with \(j_E=n\). For every \(V \in V(\Gamma )\), there are exactly \(d_V\) edges E incident to V with \(j_E=j_V\) and \(d_V+kh_V\) egdes E incident to V with \(j_E=j_V-1\).
-
(vi)
Every half-edge, that is a pair (V, E) with \(V \in V(\Gamma )\) and \(E \in E(\Gamma )\) incident to V, is decorated by a cohomology class \(c_{(V,E)} \in \{1,p_{j_E}\}\), where \(1 \in H^0(D^{j_E},{\mathbb {Z}})\) and \(p_{j_E} \in H^2(D^{j_E},{\mathbb {Z}})\) is Poincaré dual to a point. If E is unbounded, with incident vertex V, then \(c_{(V,E)}=1\). If E is bounded, then for exactly one vertex \(V'\) incident to E we have \(c_{(V',E)}=1\), and for the other vertex \(V''\) incident to E, we have \(c_{(V'',E)}=p_{j_E}\).
-
(vii)
Every vertex \(V \in V(\Gamma )\) is decorated by an index \(0 \le m_V \le g_V\), and we have \(\sum _{V \in V(\Gamma )} m_V=g\).
Definition 5.4
For every \(\Gamma \in {\mathcal {G}}^{k,n}_{h,d,g}\) and \(V \in V(\Gamma )\), let \(E_{V,-}\) (resp. \(E_{V,+}\)) be the set of \(E \in E(\Gamma )\) incident to V such that \(j_E=j_V\) (resp. \(j_E=j_V-1\)). We denote
$$\begin{aligned} \mu _V :=(w_E)_{E \in E_{V,-}}, \,\,\, \nu _V :=(w_E)_{E \in E_{V,+}} \end{aligned}$$
(5.14)
and
$$\begin{aligned} \delta ^2_V :=(c_{(V,E)})_{E \in E_{V,-}} , \,\,\, \delta ^1_V :=(c_{(V,E)})_{E \in E_{V,+}}. \end{aligned}$$
(5.15)
We define a relative Gromov–Witten invariant of \({\mathbb {F}}_k^{(j_V)}\) relatively to \(D^{j_V-1} \cup D^{j_V}\) by
$$\begin{aligned} N_{\Gamma , V} :={\left\{ \begin{array}{ll} \langle \mu _V, \delta ^2_V| (-1)^{m_V} \lambda _{m_V}| \nu _V,\delta ^1_V \rangle _{g_V,\beta _V}^{{\mathbb {F}}_k/D^{j_V-1} \cup D^{j_V}} \,\textit{if} \,\,\, V \ne V_{j_V}\\ \langle \mu _V, \delta ^2_V| (-1)^{m_V} \lambda _{m_V};\alpha _{j_V}| \nu _V,\delta ^1_V \rangle _{g_V,\beta _V}^{{\mathbb {F}}_k/D^{j_V-1} \cup D^{j_V}} \,\textit{if} \,\,\, V=V_{j_V}. \end{array}\right. } \end{aligned}$$
(5.16)
Lemma 5.5
For \(\Delta =\Delta _{h,d}^{{\mathbb {F}}_k}\), the relative Gromov–Witten invariants \(N_{g,\mathrm {rel}}^{\Delta ,n}\) of \(X_{\Delta }={\mathbb {F}}_k\) defined in (5.12) are given by:
$$\begin{aligned} N_{g,\mathrm {rel}}^{\Delta _{h,d}^{{\mathbb {F}}_k},n} = \sum _{\Gamma \in {\mathcal {G}}_{h,d,g}^{k,n}} \frac{\prod _{E \in E(\Gamma )} w_E}{|\mathrm {Aut}(\Gamma )|} \prod _{V \in V(\Gamma )} N_{\Gamma , V}, \end{aligned}$$
(5.17)
where \(|\mathrm {Aut}(\Gamma )|\) is the order of the group of permutation symmetries of \(\Gamma \) as decorated weighted graph.
Proof
We claim that (5.17) is the degeneration formula in relative Gromov–Witten theory [23] applied to \({\mathcal {X}}\rightarrow {\mathbb {A}}^1\) to compute \(N_{g,\mathrm {rel}}^{\Delta _{h,d}^{{\mathbb {F}}_k},n}\), where we distribute the n point conditions \(\alpha _1,\dots ,\alpha _n\) appearing in (5.12) by placing one on each of the n components \({\mathbb {F}}_k^{(1)},\dots ,{\mathbb {F}}_k^{(n)}\). Indeed, a graph \(\Gamma \in {\mathcal {G}}_{h,d,g}^{k,n}\) as in Definition 5.3 indexes a moduli space of stable maps to the special fiber \({\mathcal {X}}_0\) that have virtually generically dual graph \(\Gamma \): a vertex V corresponds to a curve of genus \(g_V\) and class \(\beta _V\) contained in the component \({\mathbb {F}}_k^{(j_V)}\), the vertex \(V_j\) corresponds to the curve with the jth interior marked point where the point condition \(\alpha _j\) is imposed, a bounded edge E corresponds to a node on the divisor \(D^{j_E}\), and an unbounded edge E corresponds to a relative marking on either \(D^0\) or \(D^n\) (see Fig. 6). Furthermore, the cohomological decorations \(c_{(V,E)}\) implement the insertion of the diagonal class in the degeneration formula of [23], where we used the fact the the class of the diagonal \(D^j \simeq {\mathbb {P}}^1 \subset D^j \times D^j \simeq {\mathbb {P}}^1 \times {\mathbb {P}}^1\) is \(1 \times p_j +p_j \times 1\). Finally, in the definition 5.12 of \(N_{g,\mathrm {rel}}^{\Delta _{h,d}^{{\mathbb {F}}_k},n}\), there is an insertion of the class \((-1)^g \lambda _g\) and we used (3.11) to split the lambda class: the index \(m_V\) in Definition 5.3(vii) means that we insert the class \((-1)^{m_V} \lambda _{m_V}\) on the vertex V, and it is indeed what we did in the definition (5.16) of \(N_{\Gamma , V}\). \(\square \)
We use the following terminology in Sect. 5.4 below.
Definition 5.6
Given a decorated weighted graph \(\Gamma \in {\mathcal {G}}_{h,d,g}^{k,n}\), and a vertex \(V \in V(\Gamma )\), we say that:
-
(i)
V is of type A if \(V \ne V_{j_V}\), \(h_V=0\), \(\mu _V\) and \(\nu _V\) are both the trivial 1-part partition of \(d_V\) (in particular V is bivalent), one of the edges incident to V has \(c_{(V,E)}=1\) and the other has \(c_{(V,E')}=p_{j_{E'}}\), and \(m_V=g_V=0\).
-
(ii)
V is of type B if \(V=V_{j_V}\), \(h_V=0\), \(\mu _V\) and \(\nu _V\) are both the trivial 1-part partition of \(d_V\), (in particular V is bivalent), both edges incident to V have \(c_{(V,E)}=1\), and \(m_V=g_V=0\).
-
(iii)
V is of type C if \(V=V_{j_V}\), \(h_V=1\), all edges incident to V have \(c_{(V,E)} =p_{j_E}\), and \(m_V=g_V\).
We denote by \({\tilde{{\mathcal {G}}}}_{h,d,g}^{k,n}\) the set of graphs \(\Gamma \in {\mathcal {G}}_{h,d,g}^{k,n}\) whose vertices are all of type A, B or C.
Lemma 5.7
Let \(\Gamma \in {\tilde{{\mathcal {G}}}}_{h,d,g}^{k,n}\). Let c be a chain of edges in \(\Gamma \) connected by bivalent vertices of type A or B. Assume that the endpoints of c are vertices of type C. Then, the chain c contains exactly one vertex of type B.
Proof
We consider a chain c with edges \(E^{(1)},\dots ,E^{(m)}\), connected by bivalent vertices \(V^{(1)},\dots , V^{(m-1)}\) of type A and B. Assume first that the chain c has two endpoints: we denote by \(V^{0}\) and \(V^{(m)}\) the vertices of type C incident to \(E^{(0)}\) and \(E^{(m)}\) (see Fig. 7). We say that the edge \(E^{(\ell )}\) is of type [1, p] (resp. [p, 1]) if \(c_{(V^{(\ell -1)},E^{(\ell )})}=1\) and \(c_{(V^{(\ell )},E^{(\ell )})}=p_{j_{E^{(\ell )}}}\) (resp. \(c_{(V^{(\ell -1)},E^{(\ell )})}=p_{j_{E^{(\ell )}}}\) and \(c_{(V^{(\ell )},E^{(\ell )})}=1\)). By Definition 5.3, every edge is either of type [1, p] or of type [p, 1].
By Definition 5.6(i), the type of edges “propagates” through vertices of type A: if \(V^{(\ell )}\) is of type A and \(E^{(\ell )}\) is of type [p, 1] (resp. [1, p]), then \(E^{(\ell +1)}\) is of type [p, 1] (resp. [1, p]), whereas by Definition 5.6(ii) a vertex of type B “flip” an edge [p, 1] into an edge of type [1, p]: if \(V^{(\ell )}\) is of type B, then \(E^{(\ell )}\) is of type [p, 1] and \(E^{(\ell +1)}\) is of type [1, p]. On the other hand, by Definition 5.6(iii) of vertices of type C, \(E^{(1)}\) is of type [p, 1] and that \(E^{(m)}\) is of type [1, p], so the type of the edges needs to flip at some vertex from [p, 1] to [1, p] and this vertex is of type B. It is the unique vertex of type B as there is no type of vertex able to flip back [1, p] to [p, 1].
If the chain c has one or zero endpoints, that is if \(E^{(0)}\) or \(E^{(m)}\) are unbounded, the same argument applies using that \(c_{(V,E)}=1\) if E is unbounded by Definition 5.3(vi). \(\square \)
In Sect. 5.4 below, we use the following construction of a marked \((\Delta _{h,d}^{{\mathbb {F}}_k},n)\)-floor diagram \({\mathfrak {D}}_\Gamma \) starting from a decorated weighted graph \(\Gamma \in {\tilde{{\mathcal {G}}}}_{h,d,g}^{k,n}\).
Definition 5.8
Let \(\Gamma \in {\tilde{{\mathcal {G}}}}_{h,d,g}^{k,n}\). We define a marked oriented weighted graph \({\mathfrak {D}}_\Gamma \) as follows (see Fig. 6). The vertices of \({\mathfrak {D}}_\Gamma \) are the vertices of \(\Gamma \) of type C, and each such vertex V is marked by the index \(j_V \in \{1,\dots ,n\}\). Moreover, for each chain c of edges in \(\Gamma \) connected by bivalent vertices of type A or B and with endpoints of type C, there is an edge E in \({\mathfrak {D}}_\Gamma \) incident to the endpoints of c. We define the weight \(w_E\) of E as the common weight in \(\Gamma \) of the edges contained in c. The edge E is marked by \(j_{V} \in \{1,\dots ,n\}\), where V is the unique vertex of type B contained in c given by Lemma 5.7. Finally, we orient the edges of \({\mathfrak {D}}_\Gamma \) so that the marking is increasing.
Lemma 5.9
For every \(\Gamma \in {\tilde{{\mathcal {G}}}}_{h,d,g}^{k,n}\), the marked oriented weighted graph \({\mathfrak {D}}_\Gamma \) defined in Definition 5.8 is a marked \((\Delta _{h,d}^{{\mathbb {F}}_k},n)\)-floor diagram as in Definitions 2.6–2.8.
Proof
It is mostly a direct consequence of Definitions 5.3–5.6–5.8. The only part which requires an argument is why \({\mathfrak {D}}_\Gamma \) has first Betti number \(g_{\Delta ,n}=n+1-|\Delta |\). Let \(V({\mathfrak {D}}_\Gamma )\) (resp. \(E^b({\mathfrak {D}}_\Gamma )\), \(E^\infty ({\mathfrak {D}}_\Gamma )\)) be the set of vertices (resp. bounded and unbounded edges) of \({\mathfrak {D}}_\Gamma \). By construction, we have \(|E^b({\mathfrak {D}}_\Gamma )|+|E^\infty ({\mathfrak {D}}_\Gamma )|+|V({\mathfrak {D}}_\Gamma )|=n\) and \(2|V({\mathfrak {D}}_\Gamma )|+|E^{\infty }({\mathfrak {D}}_\Gamma |=|\Delta |\), and so the first Betti number of \({\mathfrak {D}}_\Gamma \) is \(|E^b({\mathfrak {D}}_\Gamma )|-|V({\mathfrak {D}}_\Gamma )|+1=n+1-|\Delta |\). \(\square \)
Lemma 5.10
For every \(\Gamma \in {\tilde{{\mathcal {G}}}}_{h,d,g}^{k,n}\), denoting by \(E(\Gamma )\) the set of edges of \(\Gamma \) and by \(V^A(\Gamma )\) the set of vertices of \(\Gamma \) of type A , we have
$$\begin{aligned} \prod _{E \in E(\Gamma )} w_E \prod _{V \in V^A(\Gamma )}\frac{1}{d_V} = \prod _{E \in E({\mathfrak {D}}_\Gamma )}w_E^2, \end{aligned}$$
(5.18)
where \(E({\mathfrak {D}}_\Gamma )\) is the set of edges of \({\mathfrak {D}}_\Gamma \).
Proof
Let \(E^{(1)},\dots , E^{(m)}\) be a chain of edges of \(\Gamma \) connected by bivalent vertices of type A or B and with endpoints of type C. All the edges of the chain have the same weight w and all the vertices of type A or B have \(d_V=w\). By Lemma 5.7, the chain contains one vertex of type B and \(m-2\) vertices of type A and so the contribution of the chain to the left-hand side of (5.18) is \(\frac{w^m}{w^{m-2}}=w^2\). \(\square \)
Main result
In Lemma 5.11 below, we express for \(\Delta =\Delta _d^{{\mathbb {P}}^2}\) and \(\Delta = \Delta _{h,d}^{{\mathbb {F}}_k}\) the relative Gromov–Witten invariants \(N_{g,\mathrm {rel}}^{\Delta ,n}\) defined in (5.12) in terms of floor diagrams whose vertices are weighted by the relative Gromov–Witten invariants \(N_{g,\mathrm {rel}}^{\mu \nu \emptyset \emptyset }\) of Hirzebruch surfaces defined in (4.5).
Lemma 5.11
Let \(\Delta \) be a h-transverse balanced collection of vectors in \({\mathbb {Z}}^2\) of the form \(\Delta _d^{{\mathbb {P}}^2}\) or \(\Delta _{h,d}^{{\mathbb {F}}_k}\) as in Examples 2.4–2.5. Let n be a nonnegative integer such that \(g_{\Delta ,n} \ge 0\). Then the relative Gromov–Witten invariants \(N_{g,\mathrm {rel}}^{\Delta ,n}\) of \(X_\Delta \) defined in (5.12) are given by:
$$\begin{aligned}&\sum _{g \ge g_{\Delta ,n}} N_{g,\mathrm {rel}}^{\Delta ,n} u^{2g-2+d_b+d_t} \nonumber \\&\quad =\sum _{{\mathfrak {D}}} \left( \prod _{E \in E({\mathfrak {D}})} w_E^2 \right) \left( \prod _{V \in V({\mathfrak {D}})} \sum _{g \ge 0} N_{g,\mathrm {rel}}^{\mu _V \nu _V \emptyset \emptyset } u^{2g-2+\ell (\mu _V)+\ell (\nu _V)} \right) ,\qquad \end{aligned}$$
(5.19)
where the sum over \({\mathfrak {D}}\) is over the isomorphism classes of marked \((\Delta ,n)\)-floor diagrams \({\mathfrak {D}}\) as in Definition 2.6, \(E({\mathfrak {D}})\) is the set of edges of \({\mathfrak {D}}\), \(w_E\) is the weight of the edge \(E \in E({\mathfrak {D}})\), \(V({\mathfrak {D}})\) is the set of vertices of \({\mathfrak {D}}\), and for every vertex \(V \in V({\mathfrak {D}})\), \(\mu _V\) (resp. \(\nu _V\)) is the partition whose parts are the weights of outgoing (resp. ingoing) edges of \({\mathfrak {D}}\) incident to V. Moreover, \(N_{g,\mathrm {rel}}^{\mu _V \nu _V\emptyset \emptyset }\) is the specialization \(\mu =\mu _V\), \(\nu =\nu _V\), \(\rho =\sigma =\emptyset \) of the relative Gromov–Witten invariants \(N_{g,\mathrm {rel}}^{\mu \nu \rho \sigma }\) defined in (4.5).
Proof
We first assume that \(\Delta = \Delta _{h,d}^{{\mathbb {F}}_k}\), and so in particular \(X_{\Delta }={\mathbb {F}}_k\). By Lemma 5.5, \(N_{g,\mathrm {rel}}^{\Delta ,n}\) is expressed by (5.17) in terms of the invariants \(N_{\Gamma ,V}\) defined in (5.16). Recall tha we introduced in Definition 5.6 the notions of vertices of type A, B or C. By Lemma 5.1, if \(V \ne V_{j_V}\), then \(N_{\Gamma ,V}=0\) unless V is of type A, and \(N_{\Gamma ,V}=\frac{1}{d_V}\) if V is of type A. By Lemma 5.2, if \(V =V_{j_V}\), then \(N_{\Gamma ,V}=0\) unless V is of type B or C, \(N_{\Gamma ,V}=1\) if V is of type B, and \(N_{\Gamma ,V}=N_{g_V,\mathrm {rel}}^{\mu _V\nu _V \emptyset \emptyset }\) if V is of type C. Hence, we can rewrite (5.17) as a sum over the set \({\tilde{{\mathcal {G}}}}_{h,d,g}^{k,n}\) of graphs \(\Gamma \in {\mathcal {G}}_{h,d,g}^{k,n}\) whose vertices are all of type A, B or C:
$$\begin{aligned} N_{g,\mathrm {rel}}^{\Delta ,n} = \sum _{\Gamma \in {\tilde{{\mathcal {G}}}}_{h,d,g}^{k,n}} \frac{\prod _{E \in E(\Gamma )} w_E}{|\mathrm {Aut}(\Gamma )|} \prod _{V \in V^A(\Gamma )} \frac{1}{d_V} \prod _{V \in V^C(\Gamma )}N_{g_V, \mathrm {rel}}^{\mu _V\nu _V \emptyset \emptyset }, \end{aligned}$$
(5.20)
where \(V^A(\Gamma )\) (resp. \(V^B(\Gamma )\), \(V^C(\Gamma )\)) is the set of vertices of \(\Gamma \) of type A (resp. B, C). In Definition 5.8-Lemma 5.9, we defined a marked \((\Delta ,n)\) floor diagram \({\mathfrak {D}}_\Gamma \) for every \(\Gamma \in {\tilde{{\mathcal {G}}}}_{h,d,g}^{k,n}\), and every marked \((\Delta ,n)\)-floor diagram can be uniquely obtained that way. Thus, (5.19) follows from Lemma 5.10.
For \(\Delta =\Delta _d^{{\mathbb {P}}^2}\), we follow exactly the same sequence of aguments by adapting the results of Sect. 5.3 to the degeneration of \({\mathbb {P}}^2\) to a chain made of one copy of \({\mathbb {P}}^2\) and n copies of \({\mathbb {F}}_1\), where each of the n point insertions is inserted in a copy of \({\mathbb {F}}_1\). \(\square \)
The following theorem is the main result of the present paper. It is a precise version of Theorem 1.1 stated in the introduction.
Theorem 5.12
Let \(\Delta \) be a h-transverse balanced collection of vectors in \({\mathbb {Z}}^2\) of the form \(\Delta ^{{\mathbb {P}}^2}_d\) or \(\Delta ^{{\mathbb {F}}_k}_{h,d}\), and let n be a nonnegative integer such that \(g_{\Delta ,n} \ge 0\). Then we have the equality
$$\begin{aligned} \sum _{g \ge g_{\Delta ,n}} N_{g,\mathrm {rel}}^{\Delta ,n} u^{2g-2+d_b+d_t}= u^{d_b+d_t-|\Delta |} N^{\Delta ,n}_{\mathrm {floor}}(q^{\frac{1}{2}}) \left( (-i) (q^{\frac{1}{2}} - q^{-\frac{1}{2}}) \right) ^{2 g_{\Delta ,n}-2+|\Delta |}\nonumber \\ \end{aligned}$$
(5.21)
of power series in u with rational coefficients, where
$$\begin{aligned} q=e^{iu}=\sum _{m \ge 0} \frac{(iu)^m}{m!} . \end{aligned}$$
(5.22)
Proof
Lemma 5.11 expresses \(N_{g,\mathrm {rel}}^{\Delta ,n}\) in terms of the invariants \(N_{g,\mathrm {rel}}^{\mu \nu \emptyset \emptyset }\), which are computed in Theorem 4.4. We obtain
$$\begin{aligned}&\sum _{g \ge g_{\Delta ,n}} N_{g,\mathrm {rel}}^{\Delta ,n} u^{2g-2+d_b+d_t} \end{aligned}$$
(5.23)
$$\begin{aligned}&\quad =u^{d_b+d_t-|\Delta |}\left( (-i) (q^{\frac{1}{2}} - q^{-\frac{1}{2}}) \right) ^{2 g_{\Delta ,n}-2+|\Delta |} \sum _{{\mathfrak {D}}} \prod _{E \in E({\mathfrak {D}})}w_E^2 \prod _{V \in V({\mathfrak {D}})}\nonumber \\&\qquad \prod _{j=1}^{\ell (\mu _V)}\frac{[\mu _{V,j}]_q}{\mu _{V,j}} \prod _{l=1}^{\ell (\nu _V)} \frac{[\nu _{V,l}]_q}{\nu _{V,l}} , \end{aligned}$$
(5.24)
where we used that \(\sum _{V}(\ell (\mu _V)+\ell (\nu _V))=2g_{\Delta ,n}-2+|\Delta |\). As every edge E of \({\mathfrak {D}}\) with \(w(E) \ne 1\) is connected to two vertices, we have
$$\begin{aligned} \prod _{E \in E({\mathfrak {D}})}w_E^2 \prod _{V \in V({\mathfrak {D}})} \prod _{j=1}^{\ell (\mu _V)}\frac{[\mu _{V,j}]_q}{\mu _{V,j}} \prod _{l=1}^{\ell (\nu _V)} \frac{[\nu _{V,l}]_q}{\nu _{V,l}} = \prod _{E \in E({\mathfrak {D}})} [w_E]_q^2 = m_{{\mathfrak {D}}}(q^{\frac{1}{2}}) ,\nonumber \\ \end{aligned}$$
(5.25)
where we used the Definition 2.12 of the q-refined multiplicity of a floor diagram \({\mathfrak {D}}\). We conclude using the Definition 2.13 of \(N^{\Delta ,n}_{\mathrm {floor}}(q^{\frac{1}{2}})\). \(\square \)
The relation between relative Gromov–Witten invariants and q-refined floor diagrams given by Theorem 5.12 can be read and exploited in both directions. First, the relative Gromov–Witten invariants \(N_{g,\mathrm {rel}}^{\Delta ,n}\) give an algebro-geometric realization of the q-refined counts of floor diagrams \(N^{\Delta ,n}_{\mathrm {floor}}(q^{\frac{1}{2}})\) which is independent of tropical geometry. An illustration of the use of this geometric point of view to solve a priori purely combinatorial questions about the q-refined counts of floor diagrams is given by our proof of the q-refined Abramovich–Bertram formula in Sect. 8. Conversely, Theorem 5.12 can be viewed as providing a convenient tool to compute the relative Gromov–Witten invariants \(N_{g,\mathrm {rel}}^{\Delta ,n}\). Indeed, the combinatorial enumeration of floor diagrams allows for efficient calculations, as shown for example in [2, 3, 9, 12, 13].
Theorem 5.12 is analogous to the main result of [8] which relates Block–Göttsche q-refined tropical curve counts and higher genus log Gromov–Witten invariants of toric surfaces with a lambda class insertion. Combining these two results we obtain in Sect. 7 a non-trivial comparison result (Theorem 7.1) between log and relative Gromov–Witten invariants for \({\mathbb {P}}^2\) and Hirzebruch surfaces.