# Genus $0$ logarithmic and tropical fixed-domain counts for Hirzebruch
surfaces

Alessio Cela and Aitor Iribar López

###### Abstract

For a non-singular projective toric variety $X$, the virtual logarithmic
Tevelev degrees are defined as the virtual degree of the morphism from the
moduli stack of logarithmic stable maps
$\overline{\mathcal{M}}_{\mathsf{\Gamma}}(X)$ to the product
$\overline{\mathcal{M}}_{g,n}\times X^{n}$. In this paper, after proving the
genus $0$ correspondence theorem in this setting, we use tropical methods to
provide closed formulas for the case in which $X$ is a Hirzebruch surface. In
order to do so, we explicitly list all the tropical curves contributing to the
count.

###### Contents

  1. 1 Introduction
    1. 1.1 Logarithmic curve counting with fixed domain
    2. 1.2 Genus $0$ Correspondence theorem
    3. 1.3 Genus $0$ counts for Hirzebruch surfaces
      1. 1.3.1 Description of the curves enumerated in $\mathsf{tropTev}^{\mathcal{H}_{a}}_{\mathsf{\Gamma}}$
    4. 1.4 Comparison of virtual fundamental classes for maps to Hirzebruch surfaces
    5. 1.5 Further directions
  2. 2 The correspondence theorem
  3. 3 Genus $0$ tropical count for Hirzebruch surfaces
    1. 3.1 Tropical curves and intersection theory
    2. 3.2 Proof of Theorem 7
      1. 3.2.1 Case $|\mu_{3}|+|\mu_{4}|\geq n-1$
      2. 3.2.2 Case $|\mu_{3}|+|\mu_{4}|<n-1$
      3. 3.2.3 Exclusion of further contributions
  4. 4 Absence of rational curves interpolating $n$ points on Hirzebruch surfaces
  5. 5 Proof of Theorem 13
    1. 5.0.1 Case $a=2j$
    2. 5.0.2 Case $a=2j+1$

##  1 Introduction

Let $X$ be a non-singular projective variety defined over $\mathbb{C}$ of
dimension $r$. Fix integers $g\geq 0$ and $n\geq 1$ such that $2g-3+n>0$,
ensuring that the moduli stack $\overline{\mathcal{M}}_{g,n}$ of stable curve
is well-defined. Fix also an effective curve class $\beta\in
H_{2}(X,\mathbb{Z})$. Curve counts on $X$ are formulated in Gromov-Witten
theory as intersection numbers on the moduli space of stable maps
$\overline{\mathcal{M}}_{g,n}(X,\beta)$ against

$[\overline{\mathcal{M}}_{g,n}(X,\beta)]^{\mathrm{vir}}\in
A_{*}(\overline{\mathcal{M}}_{g,n}(X,\beta))$

where $[\overline{\mathcal{M}}_{g,n}(X,\beta)]^{\mathrm{vir}}$ is the virtual
fundamental class constructed in [BF98]. The Tevelev degrees of $X$ are such
counts where in additions the domain curve is fixed (and general) and $n$
point insertions are imposed. More precisely, assume the dimensional
constraint

$\mathrm{vdim}(\overline{\mathcal{M}}_{g,n}(X,\beta))=\mathrm{dim}(\overline{\mathcal{M}}_{g,n}\times
X^{n})$

holds and let

$\tau^{\prime}:\overline{\mathcal{M}}_{g,n}(X,\beta)\to\overline{\mathcal{M}}_{g,n}\times
X^{n}$

be the natural map obtained from the stabilized domain curve and the
evaluation morphisms.

######  Definition 1.

[BP21, Definition 1.1] The Tevelev degree
$\mathsf{vTev}^{X}_{g,n,\beta}\in\mathbb{Q}$ of $X$ is defined by the equality

$\tau^{\prime}_{*}[\overline{\mathcal{M}}_{g,n}(X,\beta)]^{\mathrm{vir}}=\mathsf{vTev}^{X}_{g,n,\beta}[\overline{\mathcal{M}}_{g,n}\times
X^{n}]\in A_{*}(\overline{\mathcal{M}}_{g,n}\times X^{n}).$

Fixed-domain curve counts for Grassmanians have a beautiful story at the
intersection between algebraic geometry and physics. They are computed by the
celebrated Vafa-Intriligator formula, conjectured by the physicists Vafa and
Intriligator [Int91] and partially proved by Siebert-Tian [ST97] and by
Bertram-Daskalopoulos-Wentworth in [BDW96, Ber94], and fully proven by Marian-
Oprea in [MO05] using Quot-schemes. The equivalence with the formulation in
terms of stable maps was then proven by Marian-Oprea-Pandharipande in [MOP11].
The systematic study of Tevelev degrees for general targets started with
[CPS22], motivated by work of Tevelev [Tev23] on scattering amplitudes in
mathematical physics. The paper [CPS22] then stimulated a series of subsequent
studies [BLL+on, BP21, Cel22, CL23a, CL23b, FL22, Lia23a, Lia23b, LP20]

In this paper, our aim is to extend the notion of Tevelev degrees to the
situation where $X$ is a toric variety and any tangency condition with the
boundary $\partial X$ is imposed. This is achieved using the moduli stack of
logarithmic stable maps [GS13].

After Mikhalkin’s breakthrough [Mik05], a natural correspondence between
algebraic curves and tropical curves is expected in certain nice situations.
In the recent years, various versions of such correspondence have been proved
[NS06, Shu05, Tyo17, Ran17] and many tropical analogs of classical curve
counting problems have been proved [GM08, GM07]. Using [Tyo17, Gol21], we show
that the correspondence theorem holds in our context when the genus is $0$.
Furthermore, we provide simple closed formulas for Hirzebruch surfaces using
tropical methods. Our method can be applied to many other geometries.

###  1.1 Logarithmic curve counting with fixed domain

Assume further that $X$ is a toric variety with fan $\Sigma$. Fix integers
$g\geq 0$ and $n,m\geq 1$ and contact order $c$ along the toric buondary
$\partial X$ of $X$ (see [GS13, Definition 3.1]). We package the discrete data
$(g,n,m,c)$ in the symbol $\mathsf{\Gamma}$, while still assuming the
stability condition $2g-2+n>0$.

Let $\overline{\mathcal{M}}_{\mathsf{\Gamma}}(X)$ be the moduli space of genus
$g$ and $n+m$ marked logarithmic stable maps
$[f:(C,p_{1},...,p_{n},q_{1},...,q_{m})\to X]$ having contact order $c$ to the
toric boundary divisor along the $m$ marked points $q_{1},...,q_{m}$. This
space and its virtual fundamental class
$[\overline{\mathcal{M}}_{\mathsf{\Gamma}}(X)]^{\mathrm{vir}}$ were
constructed in [GS13] and shown to be proper in [ACMW17].

In this paper we deal with logarithmic fixed-domain curve count problems with
point insertions at the markings $p_{1},\ldots,p_{n}$.

We set up the discrete data so that the problem has finitely many solutions.
In order to make it precise we require some notation.

######  Notation 2.

Order the components $D_{1},\ldots,D_{k}$ of $\partial X$. Then, we can think
of $c$ as the following data:

  1. $\bullet$

a function $\varphi:\\{1,...,m\\}\to\\{1,\ldots,k\\}$ encoding to which
divisor the marking $q_{i}$ is sent for $i=1,\ldots,m$;

  2. $\bullet$

$k$ vectors $\mu_{i}\in\mathbb{N}_{\geq 0}^{m_{i}}$ for $i=1,\ldots,k$ defined
by

$\mu_{i,j}=\text{ multiplicity prescibed by }c\text{ of the $j$-th marked
point }q_{k}\text{ mapping to }D_{i}.$

We will denote by $|\mu_{i}|$ the length of $\mu_{i}$ for $i=1,\ldots,k$.

Assume the dimensional constraint

$\mathrm{vdim}(\overline{\mathcal{M}}_{\mathsf{\Gamma}}(X))=\mathrm{dim}(\overline{\mathcal{M}}_{g,n}\times
X^{n})$

holds or equivalently that

$m=r(n+g-1)$ (1)

and let

$\tau:\overline{\mathcal{M}}_{\mathsf{\Gamma}}(X)\rightarrow\overline{\mathcal{M}}_{g,n}\times
X^{n}$ (2)

be the canonical morphism obtained from the domain curve
$\pi:\overline{\mathcal{M}}_{g,n}(X,\beta)\rightarrow\overline{\mathcal{M}}_{g,n}$
and the evaluation maps
$\mathrm{ev}:\overline{\mathcal{M}}_{\mathsf{\Gamma}}(X)\rightarrow X^{n}.$

######  Definition 3.

The virtual logarithmic Tevelev degree
$\mathsf{vTev}^{X}_{\mathsf{\Gamma}}\in\mathbb{Q}$ of $X$ is defined by the
equality

$\tau_{*}[\overline{\mathcal{M}}_{\mathsf{\Gamma}}(X)]^{\mathrm{vir}}=\Bigg{(}\prod_{i=1}^{k}\prod_{u\geq
1}|\\{v\ |\ \mu_{i,v}=u\\}|!\Bigg{)}\hskip
2.84544pt\mathsf{vTev}^{X}_{\mathsf{\Gamma}}[\overline{\mathcal{M}}_{g,n}\times
X^{n}]\in A^{0}(\overline{\mathcal{M}}_{g,n}\times X^{n}).$

The factor $\prod_{i=1}^{k}\prod_{u\geq 0}|\\{v\ |\ \mu_{i,v}=u\\}|!$ reflects
the possible orderings of the markings $q_{j}$.

In Theorem 7 below, we compute all the genus $0$ virtual Tevelev degrees for
Hirzebruch surfaces using tropical methods.

###  1.2 Genus $0$ Correspondence theorem

Suppose that $g=0$ and let $M^{\mathrm{trop}}(\mathbb{R}^{r},\mathsf{\Gamma})$
be the moduli space of labelled tropical rational $n$ marked tropical curves
$[h:\mathsf{C}\to\mathbb{R}^{r}]$ of degree $\Delta$ prescribed by $c$. By
definition, $\Delta$ is an ordered list of $m$ vectors $v_{i}$ in
$\mathbb{R}^{r}$ each parallel to one ray of the fan $\Sigma$ and such that if
$c$ prescribes that $q_{k}$ is the $j$-th marking mapped to $D_{i}$ then the
lattice length of $v_{k}$ is $\mu_{i,j}$. Our definitions of tropical curves,
maps and their moduli spaces is that in [GKM09, Definition 3.2 and 4.1].

Note that, by Equation (1), we have

$|\Delta|=m=r(n-1).$

Let

$\mathrm{trop}(\tau):M^{\mathrm{trop}}(\mathbb{R}^{r},\mathsf{\Gamma})\to
M_{0,n}^{\mathrm{trop}}\times(\mathbb{R}^{r})^{n}$ (3)

be the canonical morphism obtained from the domain curve and the evaluation
map. The map $\mathrm{trop}(\tau)$ is a morphism of equidimensional tropical
fans with $M_{0,n}^{\mathrm{trop}}\times(\mathbb{R}^{r})^{n}$ [GKM09,
Definition 2.8]. Since $M_{0,n}^{\mathrm{trop}}\times(\mathbb{R}^{r})^{n}$ is
irreducible, by [GKM09, Corollary 2.26], we have a well-defined notion of
degree.

######  Definition 4.

Define

$\mathsf{tropTev}^{X}_{\mathsf{\Gamma}}=\frac{\mathrm{degree}(\mathrm{trop}(\tau))}{\prod_{i=1}^{k}\prod_{u\geq
1}|\\{v\ |\ \mu_{i,v}=u\\}|!}$

to be the tropical Tevelev degree of $X$ w.r.t. $\mathsf{\Gamma}$.

After Mikhalkin’s break-through [Mik05], various correspondence theorems have
been proved [NS06, Shu05, Tyo17, Ran17]. In our context, we have the
following:

######  Theorem 5.

Virtual logarithmic tevelev degrees and their corresponding tropical degrees
coincide in genus $0$, i.e.

$\mathsf{vTev}^{X}_{\mathsf{\Gamma}}=\mathsf{tropTev}^{X}_{\mathsf{\Gamma}}.$

The proof is given in §2 below.

###  1.3 Genus $0$ counts for Hirzebruch surfaces

In the following we specialize to the case when $X=\mathcal{H}_{a}$ is the
Hirzebruch surface $\mathbb{P}(\mathcal{O}\oplus\mathcal{O}(a))$ with $a\geq
1$ and provide closed formulas for the tropical (and so the virtual
logarithmic) Tevelev degrees of $\mathcal{H}_{a}$ with any tangency conditions
$c$.

######  Notation 6.

The fan $\Sigma$ of $\mathcal{H}_{a}$ has four rays, with associated unit
vectors

$n_{1}=(-1,a),\ n_{2}=(0,1),\ n_{3}=(1,0),\text{ and }n_{4}=(0,-1).$

Denote by $D_{1},D_{2},D_{3}$ and $D_{4}$ the corresponding toric divisors.

######  Theorem 7.

We have

  1. $\bullet$

if either $|\mu_{1}|>n-1$ or $|\mu_{3}|>n-1$, then

$\mathsf{tropTev}^{\mathcal{H}_{a}}_{\mathsf{\Gamma}}=0,$

  2. $\bullet$

otherwise

$\mathsf{tropTev}^{\mathcal{H}_{a}}_{\mathsf{\Gamma}}=\Bigg{(}\prod_{i=1}^{4}\frac{|\mu_{i}|!\prod_{j=1}^{|\mu_{i}|}\mu_{i,j}}{\prod_{u\geq
1}|\\{v\ |\
\mu_{i,v}=u\\}|!}\Bigg{)}a^{n-1-|\mu_{2}|-|\mu_{4}|}\binom{n-1-|\mu_{4}|}{|\mu_{2}|}$

The proof of this theorem is given in §3.

######  Remark 8.

The formula above gives zero whenever $|\mu_{2}|>n-1-|\mu_{4}|$. In
particular, suppose that $\mu_{i,j}=1$ for all $i,j$ and that $a\geq 2$. Then

$|\mu_{1}|=|\mu_{3}|\text{ and }|\mu_{4}|=|\mu_{2}|+(a+1)|\mu_{1}|$

so

$|\mu_{4}|+|\mu_{2}|>\frac{|\Delta|}{2}=n-1$

and $\mathsf{tropTev}^{\mathcal{H}_{a}}_{\mathsf{\Gamma}}=0$.

A geometric interpretation of this fact is given in §4.

Suppose $a=1$ and $\mu_{2}=\emptyset$. Formally, $\Sigma$ reduces to the fan
of $\mathbb{P}^{2}$ and we are counting curves in $\mathbb{P}^{2}$. Then (the
proof of) Theorem 7 also shows the following.

######  Theorem 9.

We have

$\mathsf{tropTev}^{\mathbb{P}^{2}}_{\mathsf{\Gamma}}=\prod_{i=1,3,4}\frac{|\mu_{i}|!\prod_{j=1}^{|\mu_{i}|}\mu_{i,j}}{\prod_{u\geq
1}|\\{v\ |\ \mu_{i,v}=u\\}|!}.$

####  1.3.1 Description of the curves enumerated in
$\mathsf{tropTev}^{\mathcal{H}_{a}}_{\mathsf{\Gamma}}$

When $r=2$, we can describe all the curves contributing to
$\mathsf{tropTev}^{X}_{\mathsf{\Gamma}}$. Fix general points
$x_{1},\ldots,x_{n}$ in $\mathbb{R}^{2}$ and fix the stabilized domain curve
$\bar{\mathsf{C}}$ in $M_{0,n}^{\mathrm{trop}}$ to have have all lengths equal
to $0$. Such a curve is not in the interior of a maximal cone of
$M_{0,n}^{\mathrm{trop}}$, but we are allowed to assume so by the intersection
theoretic point view presented in §3.1.

######  Proposition 10.

The curves $[h:\mathsf{C}\to\mathbb{R}^{2}]$ contributing to
$\mathsf{tropTev}^{X}_{\mathsf{\Gamma}}$ with point insertions
$x_{1},\ldots,x_{n}$ and stabilized domain curve $\bar{\mathsf{C}}$ are all
embeddings and the domain curve $\mathsf{C}$ has one of the two shapes in
Figure 1:

  1. A)

in type A there is a central vertex $V$ with the marking $p_{1}$ attached to
it and $n-1$ leaves from $V$ consisting of two bounded edges, one marking and
two ends,

  2. B)

in type B, there still is a central vertex $V$ and $n$ leaves from it of which
exactly two consist of one bounded edge, one marking and one unbounded edge
and the other $n-2$ leaves are as in type A.

$p_{1}$$V$$p_{i_{1}}$$p_{i_{2}}$$p_{i_{3}}$$\bullet$$\bullet$$\bullet$Type
(A)$V$$p_{i_{1}}$$p_{i_{2}}$$p_{i_{3}}$$p_{i_{4}}$$p_{i_{5}}$$\bullet$$\bullet$$\bullet$Type
(B) Figure 1: Shape of the domain curve

The proof of this proposition is also given in §3. For $X=\mathcal{H}_{a}$, we
will choose points in the following way:

  1. $\bullet$

if $|\mu_{3}|+|\mu_{4}|\geq n-1$, the point $x_{1}$ is in the origin $(0,0)$,
there are $n-1-|\mu_{4}|$ points in $\\{x>0,y>0\\}$,
$|\mu_{3}|+|\mu_{4}|-(n-1)$ in $\\{ax+y>0,y<0\\}$ and $n-1-|\mu_{3}|$ in
$\\{x<0,y<0\\}$. Note that if $|\mu_{3}|>n-1$ or $|\mu_{4}|>n-1$ then Theorem
7 prescribes $\mathsf{tropTev}^{\mathcal{H}_{a}}_{\mathsf{\Gamma}}=0$.

  2. $\bullet$

if instead $|\mu_{3}|+|\mu_{4}|<n-1$, then again $x_{1}$ is in the origin
$(0,0)$, there are $n-1-|\mu_{3}|-|\mu_{4}|$ points in $\\{x<0,ax+y>0\\}$,
$|\mu_{3}|$ in $\\{x>0,y>0\\}$ and $|\mu_{4}|$ in $\\{x<0,y<0\\}$.

We will then prove that there are

$\prod_{i=1}^{4}|\mu_{i}|!\binom{n-1-|\mu_{4}|}{|\mu_{2}|}$

tropical curves as in Proposition 10 and moreover that each of such curves
contributes with multiplicity

$a^{n-1-|\mu_{2}|-|\mu_{4}|}\prod_{j=1}^{|\mu_{i}|}\mu_{i,j}.$

to $\mathsf{tropTev}^{\mathcal{H}_{a}}_{\mathsf{\Gamma}}$.

######  Example 11.

Suppose $a=2$ and $\mu_{1}=(1,2)$, $\mu_{2}=(3)$, $\mu_{3}=(1,1,1)$ and
$\mu_{4}=(4,4)$. So in this case $|\mu_{3}|+|\mu_{4}|\geq n-1$. We list in
Figure 2 the $4$ contributing curves, all of which are of type A.

$3$$1$$1$$1$$1$$4$$4$$2$$\bullet$$x_{6}$$\bullet$$x_{1}$$\bullet$$x_{2}$$\bullet$$x_{3}$$\bullet$$x_{4}$$3$$1$$1$$1$$1$$4$$4$$2$$\bullet$$x_{6}$$\bullet$$x_{1}$$\bullet$$x_{2}$$\bullet$$x_{3}$$\bullet$$x_{4}$$3$$1$$1$$2$$1$$4$$4$$1$$\bullet$$x_{6}$$\bullet$$x_{1}$$\bullet$$x_{2}$$\bullet$$x_{3}$$\bullet$$x_{4}$$3$$1$$1$$2$$1$$4$$4$$1$$\bullet$$x_{6}$$\bullet$$x_{1}$$\bullet$$x_{2}$$\bullet$$x_{3}$$\bullet$$x_{4}$
Figure 2: The $4$ tropical curves contributing to
$\mathsf{tropTev}^{\mathcal{H}_{a}}_{\mathsf{\Gamma}}$ in Example 11

######  Example 12.

Suppose $a=1$ and $\mu_{1}=(1,1,1)$, $\mu_{2}=(1)$, $\mu_{3}=(3)$ and
$\mu_{4}=(4)$. In this case $|\mu_{3}|+|\mu_{4}|\geq n-1$. We list in Figure 3
below the $2$ contributing curves: one of type A and one of time B.

$1$$\bullet$$x_{2}$$1$$1$$\bullet$$x_{3}$$3$$1$$\bullet$$x_{4}$$4$$\bullet$$x_{1}$$3$$\bullet$$x_{2}$$1$$1$$1$$\bullet$$x_{3}$$1$$4$$\bullet$$x_{4}$$\bullet$$x_{1}$
Figure 3: The $2$ tropical curves contributing to
$\mathsf{tropTev}^{\mathcal{H}_{a}}_{\mathsf{\Gamma}}$ in Example 12

###  1.4 Comparison of virtual fundamental classes for maps to Hirzebruch
surfaces

We can use fixed-domain curve counts to distinguish the virtual fundamental
class of the moduli spaces of logarithmic and stable maps to Hirzebruch
surfaces.

More precisely, let $X=\mathcal{H}_{a}$ be a Hirzebruch surface and $\beta\in
H_{2}(X,\mathbb{Z})$ be an effective curve class. Let $c$ be defined by

$\mu_{i}=\underbrace{(1,\ldots,1)}_{\beta\cdot D_{i}\text{ times}}$

for $i=1,2,3,4$ and let $n\in\mathbb{N}$ be such that the dimensional
constraint (1) holds. Consider the natural (proper) morphism

$\alpha:\overline{\mathcal{M}}_{\mathsf{\Gamma}}(\mathcal{H}_{a})\to\overline{\mathcal{M}}_{0,n}(\mathcal{H}_{a},\beta)$

of virtually equidimensional Deligne-Mumford stacks.

######  Theorem 13.

In the following cases:

  1. 1.

$a=2j$ where $j\in\mathbb{Z}_{\geq 1}$ and $\beta=d[(j+1)D_{1}+D_{2}]$ for
$d>0$ such that $2d=n-1$; or

  2. 2.

$a=2j+1$ where $j\in\mathbb{Z}_{\geq 1}$ and
$\beta=[j(d-k)+d]D_{1}+(d-k)D_{2}$ for $d$ and $k$ integers such that $0\leq
k\leq d$, $0\leq k\leq n-1-d$ and $3d-k=2(n-1)$;

we have

$\alpha_{*}[\overline{\mathcal{M}}_{\mathsf{\Gamma}}(\mathcal{H}_{a})]^{\mathrm{vir}}\neq[\overline{\mathcal{M}}_{0,n}(\mathcal{H}_{a},\beta)]^{\mathrm{vir}}.$

This is achieved in §5 by comparing the corresponding Tevelev degrees and
using the results of [CL23b].

As already observed in [Spi02], Hirzebruch surfaces provide an excellent
example for the fact that in general Gromov–Witten invariants might well count
curves in the boundary components of the moduli spaces. The proof of theorem 7
and Theorem 13 show that logarithmic stable maps behave better from this point
of view.

###  1.5 Further directions

Our approach for computing $\mathsf{tropTev}^{X}_{\mathsf{\Gamma}}$ for
Hirzebruch surfaces should generalize to other geometries and higher
dimensional varieties.

Higher dimensional generalizations of $\mathcal{H}_{a}$ includes
$\mathbb{P}^{1}$-bundles
$\mathbb{P}(\mathcal{O}_{\mathbb{P}^{r}}\oplus\mathcal{O}_{\mathbb{P}^{r}}(a))$
over $\mathbb{P}^{r}$, for which we conjecture the following formula to hold.

######  Conjecture 14.

Let
$X=\mathbb{P}(\mathcal{O}_{\mathbb{P}^{r}}\oplus\mathcal{O}_{\mathbb{P}^{r}}(a))$
and let $D_{1},\ldots,D_{r+1}$ be the fibers over the invariant hyperplanes
$\\{x_{1}=0\\},\ldots,\\{x_{r+1}=0\\}$, and let $D_{r+2},D_{r+3}$ be the zero
section and the infinity section, respectively. Then, when
$\mathsf{vTev}^{X}_{\mathsf{\Gamma}}$ is not $0$,

$\mathsf{vTev}^{X}_{\mathsf{\Gamma}}=\mathsf{tropTev}^{X}_{\mathsf{\Gamma}}=\left(\prod_{i=1}^{r+3}\frac{|\mu_{i}|!\prod_{j=0}^{|\mu_{ij}|}\mu_{i,j}}{\prod_{u\geq
1}|\\{v\ |\
\mu_{i,v}=u\\}|!}\right)a^{(n-1)-|\mu_{r+2}|-|\mu_{r+3}|}{n-1-|\mu_{r+2}|\choose|\mu_{r+3}|}.$

The Hirzebruch surface $\mathcal{H}_{1}$ is isomorphic to the blow-up of
$\mathbb{P}^{2}$ at one point. In [CL23a], the authors computed the geometric
degrees with simple incidence conditions with the toric boundary for blowups
of $\mathbb{P}^{r}$ at up to $r+1$ points. We also conjecture the following
generalization of that formula ho hold.

######  Conjecture 15.

Let $X$ be the blowup of $\mathbb{P}^{r}$ at $r$ of the torus fixed points,
and let $D_{1},\ldots,D_{r}$ be the exceptional divisors of
$[0:1:\ldots:0],\ldots,[0:\ldots:1]$, and $D_{r+1},\ldots,D_{2r+1}$ the strict
transforms of the linear subspaces $\\{x_{1}=0\\},\ldots,\\{x_{r+1}=0\\}$ of
$\mathbb{P}^{r}$. Then if $\mathsf{vTev}^{X}_{\mathsf{\Gamma}}$ is nonzero,

$\mathsf{vTev}^{X}_{\mathsf{\Gamma}}=\mathsf{tropTev}^{X}_{\mathsf{\Gamma}}=\left(\prod_{i=0}^{2r+1}\frac{|\mu_{i}|!\prod_{j=0}^{|\mu_{ij}|}\mu_{i,j}}{\prod_{u\geq
1}|\\{v\ |\
\mu_{i,v}=u\\}|!}\right)\prod_{i=1}^{r}{n-1-|\mu_{i+r+1}|\choose|\mu_{i}|}.$

Finally, we expect a more complicated formula could be obtained with our
method for the blow-up of $\mathbb{P}^{r}$ at the $r+1$ torus fixed points
(with any tangencies with the toric boundary).

### Acknowledgments

This project began with the participation of the first author in the MSRI
summer school titled ”Tropical Geometry” at St. Mary’s College in Moraga,
California, in August 2022. The first author is deeply grateful to the
organizers, Renzo Cavalieri, Hannah Markwig, and Dhruv Ranganathan, for
teaching him tropical and logarithmic geometry. We would also like to thank
these three researchers for their invaluable assistance with this project
during their visit to ETH Zurich in the spring semester of 2023. Lastly, we
thank Gavril Farkas, Carl Lian, Rahul Pandharipande, and Johannes Schmitt for
several useful discussions regarding fixed domain curve counts. A.C. received
support from SNF-200020-182181. A.I.L. was supported by
ERC-2017-AdG-786580-MACI. The project received funding from the European
Research Council (ERC) under the European Union Horizon 2020 research and
innovation programme (grant agreement 786580).

##  2 The correspondence theorem

In this section, we assume familiarity with the intersection theory on
balanced fans (see [Gol21] for an introduction). The starting point to prove
the correspondence theorem 5 are [Tyo17, Gol21].

######  Lemma 16.

The natural maps
$\overline{\mathcal{M}}_{0,n}\to\prod_{i=4}^{n}\overline{\mathcal{M}}_{0,\\{1,2,3,i\\}}$
and
$M^{\mathrm{trop}}_{0,n}\to\prod_{i=4}^{n}M_{0,\\{1,2,3,i\\}}^{\mathrm{trop}}$
have degree $1$.

###### Proof.

The statement is clear for the first map and for the second follows from the
first map having degree $1$ and [Gro15, Theorem 4.1]. ∎

The degree of the map 2 is then equal to the degree of the map

$\bigg{(}\prod_{i=4}^{n}\mathrm{ft}_{i}\bigg{)}\times\mathrm{ev}:\overline{\mathcal{M}}_{\mathsf{\Gamma}}(X)\to\prod_{i=4}^{n}\overline{\mathcal{M}}_{0,\\{1,2,3,i\\}}\times
X^{n}$ (4)

obtained by the forgetful morphisms
$\mathrm{ft}_{i}:\overline{\mathcal{M}}_{\mathsf{\Gamma}}(X)\to\overline{\mathcal{M}}_{0,\\{1,2,3,i\\}}$
for $i=4,\ldots,n$ and the evaluation map.

Similarly, the degree of the map 3 is equal to the degree of

$\bigg{(}\prod_{i=4}^{n}\mathrm{ft}_{i}\bigg{)}\times\mathrm{ev}:M^{\mathrm{trop}}(\mathbb{R}^{r},\mathsf{\Gamma})\to\prod_{i=4}^{n}M_{0,\\{1,2,3,i\\}}^{\mathrm{trop}}\times(\mathbb{R}^{r})^{n}$
(5)

The correspondence theorem 5 will now follow from [Tyo17, Theorem 5.1], a
generalization of [Gol21, Proposition 3.5] to higher dimensions and next two
lemmas.

######  Lemma 17.

In genus $0$ the moduli spaces $\overline{\mathcal{M}}_{\mathsf{\Gamma}}(X)$
are irreducible, generically of expected dimension. Moreover,

$[\overline{\mathcal{M}}_{\mathsf{\Gamma}}(X)]=[\overline{\mathcal{M}}_{\mathsf{\Gamma}}(X)]^{\mathrm{vir}}\in
A_{*}(\overline{\mathcal{M}}_{\mathsf{\Gamma}}(X)).$

###### Proof.

Let $\mathfrak{M}_{0,n}$ be the moduli space of genus $0$ and $n$-marked
prestable curves, endowed with the logarithmic structure given by
$\partial\mathfrak{M}_{0,n+m}$. Let also
$\mathcal{L}og_{\mathfrak{M}_{0,n+m}}$ be the stack constructed in [Ols03]. It
has pure dimension $-3+n+m$. Consider the natural (strict) logarithmic map

$\varphi:\overline{\mathcal{M}}_{\mathsf{\Gamma}}(X)\to\mathcal{L}og_{\mathfrak{M}_{0,n+m}}$

The relative perfect obstruction theory (see [GS13, Section 5]) is

$E^{\bullet}=Rp_{*}(f^{*}T_{X}^{\mathrm{log}})^{\vee}\to
L^{\bullet}_{\overline{\mathcal{M}}_{\mathsf{\Gamma}}(X)/\mathcal{L}og_{\mathfrak{M}_{0,n+m}}}$

where $p:\mathcal{C}\to\overline{\mathcal{M}}_{\mathsf{\Gamma}}(X)$ is the
universal curve and $f:\mathcal{C}\to X$ is the universal map. In our
situation, we have $T_{X}^{\mathrm{log}}=\mathcal{O}_{X}^{r}$ and therefore
$E^{\bullet}\cong\Omega_{\overline{\mathcal{M}}_{\mathsf{\Gamma}}(X)/\mathcal{L}og_{\mathfrak{M}_{0,n+m}}}$
is a vector bundle with fiber over $[f:C\to X]$ given by
$(H^{0}(C,\mathcal{O})^{r})^{\vee}$. It follows that $\varphi$ is smooth (and
therefore log-smooth) and that
$[\overline{\mathcal{M}}_{\mathsf{\Gamma}}(X)]=[\overline{\mathcal{M}}_{\mathsf{\Gamma}}(X)]^{\mathrm{vir}}$.
The irreducibility statement is [Ran17, Proposition 3.3.5]. ∎

######  Remark 18.

The proof of [Gol21, Proposition 3.5] straightforwardly generalizes to the
higher-dimensional setting. In the proof of Theorem 5 below, we will use its
higher-dimensional generalization.

###### Proof of Theorem 5.

By Lemma 17, $\mathsf{vTev}^{X}_{\mathsf{\Gamma}}$ equals the geometric
numbers of logarithmic stable maps from a fixed general curve
$(C,p_{1},...,p_{n})$ to $X$ with contact orders prescribed by $c$. By Lemma
16, [Tyo17, Theorem 5.1] and the higher-dimensional generalization of [Gol21,
Proposition 3.5], the number of such maps where in addition the domain curve
is smooth and the image of the map does not meet any toric point of $X$ is
equal to the degree of $\mathrm{trop}(\tau)$. In order to conclude it is then
enough to exclude contributions in $\mathsf{vTev}^{X}_{\mathsf{\Gamma}}$ from
$\partial\overline{\mathcal{M}}_{\mathsf{\Gamma}}(X)$ and from the locus $B$
of maps containing toric points in their image. The boundary
$\partial\overline{\mathcal{M}}_{\mathsf{\Gamma}}(X)$ and $B$ are both proper
subsets of $\overline{\mathcal{M}}_{\mathsf{\Gamma}}(X)$ [Ran17, Proposition
3.3.3] so, by lemma 17, they cannot dominate
$\overline{\mathcal{M}}_{0,n}\times X^{n}$ for dimensional reasons. ∎

##  3 Genus $0$ tropical count for Hirzebruch surfaces

In this section we calculate $\mathsf{tropTev}^{X}_{\mathsf{\Gamma}}$ for
tropical maps to Hirzebruch surfaces.

###  3.1 Tropical curves and intersection theory

The moduli spaces $M^{\mathrm{trop}}_{0,n}$ and
$M^{\mathrm{trop}}(\mathbb{R}^{n},\Gamma)$ are fans and therefore the
machinery of tropical intersection theory developed in [All10], can be applied
to it.
All the results in this sections are valid for any toric surface, or in
general a toric variety of any dimension after appropriate modifications,
specially to Corollary 24.

######  Notation 19.

For a morphism of fans, there is a notion of pullback of Cartier divisors. For
a point $\bar{x}=(\bar{x}_{1},\bar{x}_{2})\in\mathbb{R}^{2}$ and any affine
cycle $Z$ of $M^{\mathrm{trop}}(\mathbb{R}^{n},\Gamma)$, we make the
abbreviation

$\mathrm{ev}^{*}(p)\cdot
Z=(\mathrm{pr}_{1}\circ\mathrm{ev})^{*}(\bar{x}_{1})\cdot(\mathrm{pr}_{2}\circ\mathrm{ev})^{*}(\bar{x}_{2})\cdot
Z,$

where $\mathrm{pr}_{i}:\mathbb{R}^{2}\to\mathbb{R}$ are the two projections
and $\bar{x}_{i}\in\mathbb{R}$ is regarded as a Cartier divisor for $i=1,2$.

######  Lemma 20.

The degree of $\operatorname{trop}(\tau)$ is equal to the degree of the
tropical $0$-cycle

$\prod_{i=4}^{n}\mathrm{ft}_{i}^{*}(0)\cdot\prod_{i=1}^{n}\mathrm{ev}_{i}^{*}(x_{i})\cdot
M^{\mathrm{trop}}(\mathbb{R}^{2},\Gamma),$ (6)

where $\mathrm{ft}_{i}:M^{\mathrm{trop}}(\mathbb{R}^{2},\Gamma)\to
M^{\mathrm{trop}}_{0,\\{1,2,3,i\\}}$, where $0$ represents the unique
$4$-valent curve in $M_{0,4}^{\mathrm{trop}}$, and $x_{i}\in\mathbb{R}^{2}$.

###### Proof.

Let
$((\lambda_{1},\ldots,\lambda_{n-3}),(x_{1},\ldots,x_{n}))\in\left(M^{\mathrm{trop}}_{0,4}\right)^{n-3}\times(\mathbb{R}^{2})^{n}$
be a general point and let
$g=\bigg{(}\prod_{i=4}^{n}\mathrm{ft}_{i}\bigg{)}\times\mathrm{ev}$ be the
morphism of tropical fans in 5. The equality

$\deg(g)=\deg\left(\prod_{i=4}^{n}\mathrm{ft}_{i}^{*}(\lambda_{i})\cdot\prod_{i=1}^{n}\mathrm{ev}_{i}^{*}(x_{i})\cdot
M^{\mathrm{trop}}(\mathbb{R}^{2},\Gamma)\right)$

follows essentially from [Rau09, Lemma 1.2.9], and the equality

$\deg\left(\prod_{i=4}^{n}\mathrm{ft}_{i}^{*}(\lambda_{i})\cdot\prod_{i=1}^{n}\mathrm{ev}_{i}^{*}(x_{i})\cdot
M^{\mathrm{trop}}(\mathbb{R}^{2},\Gamma)\right)=\deg\left(\prod_{i=4}^{n}\mathrm{ft}_{i}^{*}(0)\cdot\prod_{i=1}^{n}\mathrm{ev}_{i}^{*}(x_{i})\cdot
M^{\mathrm{trop}}(\mathbb{R}^{2},\Gamma)\right)$

follows from the fact that pullbacks, intersections and the degree are defined
up to rational equivalence, and all points in $M^{\mathrm{trop}}_{0,4}$ are
rationally equivalent. ∎

######  Lemma 21.

Consider the tropical affine cycle

$Z=\prod_{i=4}^{n}\mathrm{ft}_{i}^{*}(0)\cdot
M^{\mathrm{trop}}(\mathbb{R}^{2},\Gamma)$

Then the combinatorial type of any top-dimensional polyhedron of $Z$ is such
that one vertex $V$ is $n$-valent, the rest of them are 3-valent, and for each
each edge is contained in the unique path to a unique marking. Moreover, the
weight of all these polyhedra is $1$.

###### Proof.

Let $[h:\mathsf{C}\to\mathbb{R}^{2}]$ be such a curve. Since for any
$i=4,\ldots,n$ we have $\mathrm{ft}_{i}([h])=0$, for each $i$ there must be a
unique vertex $V_{i}$ in $\mathsf{C}$ and $4$ different edges
$e_{1},e_{2},e_{3},e_{i}$ attached to $V_{i}$ that are part of the unique path
between $V_{i}$ and $p_{1}$, $p_{2}$, $p_{3}$ and $p_{i}$, respectively. In
particular, $V_{i}$ must be contained in the unique path joining $p_{1}$,
$p_{2}$, and in the unique path containing $p_{1}$, $p_{3}$. Therefore, there
exists a vertex $V$ such that $V_{i}=V$ for all $i$. By [Gol21, Lemma 3.11],
for any vertex $W$ of $\mathsf{C}$

$\mathrm{val}(W)=3+|\\{i\mid V_{i}=W\\}|,$

And this implies the first assertion of the lemma.
The multiplicities of such polyhedra are also computed in the same lemma
[Gol21, Lemma 3.11], and, in the case we are concerned with, they are all $1$.
This amounts to say that there is only one (see Figure 4 below) way of
resolving the cross ratios $(p_{1},p_{2};p_{3},p_{i})$ for $i=4,\ldots,n$. ∎

$p_{1}$$p_{2}$$p_{3}$$p_{4}$$p_{5}$$\ddots$$p_{n}$resolve$(p_{1},p_{2};p_{3},p_{4})$$p_{1}$$p_{2}$$p_{5}$$\ddots$$p_{n}$$p_{4}$$p_{3}$resolve$(p_{1},p_{2};p_{3},p_{5})$$p_{1}$$p_{2}$$\ddots$$p_{n}$$p_{5}$$p_{4}$$p_{3}$$\ldots$$p_{1}$$p_{2}$$p_{n}$$\vdots$$p_{5}$$p_{4}$$p_{3}$
Figure 4: Resolving the cross-ratios in the proof of Lemma 21.

######  Definition 22.

In the situation of Lemma 21, we will refer to the $n$-valent vertex $V$ as
the central vertex of $\mathsf{C}$. Also, we will call the connected
components of $\mathsf{C}\smallsetminus\\{V\\}$ leaves of $\mathsf{C}$.

In light of the previous lemma, each leaf contains a unique marked point and
all of its vertices are trivalent.

######  Corollary 23.

The contribution of a curve $[h:\mathsf{C}\to\mathbb{R}^{2}]$ to

$\prod_{i=1}^{n}\mathrm{ev}_{i}^{*}(x_{i})\cdot Z,$

where the points $p_{i}$ are in general position, is equal to the product of
its local $\mathrm{ev}$-multiplicities at each vertex, as defined in [Gol21,
Definition 3.18].

###### Proof.

By the previous lemma the weights of $Z$ are all $1$, thus this immediately
follows from [Gol21, Proposition 3.16] and [Gol21, Lemma 3.19]. ∎

######  Corollary 24.

Let $[h:\mathsf{C}\to\mathbb{R}^{2}]$ be a curve that contributes to

$\prod_{i=1}^{n}\mathrm{ev}_{i}^{*}(x_{i})\cdot Z,$

where the points are in general position. Then $\mathsf{C}$ has one of the two
shapes in Proposition 10.

###### Proof.

Denote by $V$ the central vertex of $\mathsf{C}$ and let $\mathsf{L}$ be one
of the leaves, which contains the marked point $p_{i}$. Then:

  * (1)

if $\mathsf{L}$ has no vertices, it has to be itself the marked point;

  * (2)

if $\mathsf{L}$ has one vertex, it must consist of one bounded edge, one end
and the marked point $p_{i}$;

  * (3)

if $\mathsf{L}$ has two vertices, it must consist of two bounded edges, one
marking and two ends. Moreover, the vertex with no marking attached has to be
between $V$ and $p_{i}$. Otherwise we could find a string (i.e. an embedding
$\mathbb{R}\to\mathsf{C}$ disjoint from any marked point), so we could deform
the curve without changing its ev-multiplicity, and so it cannot contribute;

  * (4)

$\mathsf{L}$ cannot have more than 2 vertices, or there would be again be a
string in $\mathsf{L}$ and the contribution would be $0$.

$\bullet$$V$$p_{i}$$\bullet$$V$$p_{i}$$p_{i}$$\bullet$$V$$\bullet$$V$$p_{i}$
Figure 5: The pictures for the situations (1), (2), (3) and (4) in the proof.

If $L_{i}$ is the number of leaves with $i$ vertices, counting the number of
markings we obtain $L_{0}+L_{1}+L_{2}=n$, whereas counting the number of ends,
$L_{1}+2L_{2}=|\Delta|=2(n-1)$. In particular, $2L_{0}+L_{1}=2$, which leaves
only two possibilities, that correspond to the description of A and B. ∎

###  3.2 Proof of Theorem 7

We now assume $X=\mathcal{H}_{a}$ and $\Sigma$ is the fan in 6. Corollary 24
motivates the following definition.

######  Definition 25.

For a curve $[h:\mathsf{C}\to\mathbb{R}^{2}]$ contributing to (6), let $V$ be
its central vertex and define:

$\displaystyle\alpha([h])=\text{ number of leaves from }V\text{ having ends
whose primitive vectors are }n_{1}\text{ and }n_{2},$
$\displaystyle\beta([h])=\text{ number of leaves from }V\text{ having ends
whose primitive vectors are }n_{1}\text{ and }n_{3},$
$\displaystyle\gamma([h])=\text{ number of leaves from }V\text{ having ends
whose primitive vectors are }n_{2}\text{ and }n_{3},$
$\displaystyle\delta([h])=\text{ number of leaves from }V\text{ having ends
whose primitive vectors are }n_{3}\text{ and }n_{4},$
$\displaystyle\chi([h])=\text{ number of leaves from }V\text{ having ends
whose primitive vectors are }n_{4}\text{ and }n_{1},$
$\displaystyle\epsilon_{1}([h])=\text{ number of leaves from }V\text{ having
one end whose primitive vector is }n_{1},$
$\displaystyle\epsilon_{2}([h])=\text{ number of leaves from }V\text{ having
one end whose primitive vector is }n_{2},$
$\displaystyle\epsilon_{3}([h])=\text{ number of leaves from }V\text{ having
one end whose primitive vector is }n_{3},$
$\displaystyle\epsilon_{4}([h])=\text{ number of leaves from }V\text{ having
one end whose primitive vector is }n_{4}.$

######  Remark 26.

Note that these are the possibilities that can occur. More precisely, let

$\displaystyle\sigma_{1}=\\{x<0,ax+y>0\\},$
$\displaystyle\sigma_{2}=\\{x>0,y>0\\},$
$\displaystyle\sigma_{3}=\\{x>0,y<0\\},$
$\displaystyle\sigma_{4}=\\{x<0,ax+y<0\\}$

be the interiors of the maximal cones of the fan $\Sigma$ in Notation 6. Then
leaves counted in $\alpha([h])$ corresponds to points $x_{i}$ in $\sigma_{1}$,
leaves in $\beta([h])$ corresponds to points $x_{i}$ in $\sigma_{1}$ or
$\sigma_{2}$, leaves in $\gamma([h])$ corresponds to points in $\sigma_{2}$,
leaves in $\delta([h])$ corresponds to points in $\sigma_{3}$ and leaves in
$\chi([h])$ corresponds to points in $\sigma_{4}$.

######  Lemma 27.

For a curve $[h:\mathsf{C}\to\mathbb{R}^{2}]$ contributing to (6), the vector

$(\alpha,\beta,\gamma,\delta,\chi,\epsilon_{1},\epsilon_{2},\epsilon_{3},\epsilon_{4})=(\alpha([h]),\beta([h]),\gamma([h]),\delta([h]),\chi([h]),\epsilon_{1}([h]),\epsilon_{2}([h]),\epsilon_{3}([h]),\epsilon_{4}([h])).$

satisfies the following system

$\left\\{\begin{array}[]{ll}\alpha+\beta+\chi&=|\mu_{1}|-\epsilon_{1}\\\
\alpha+\gamma&=|\mu_{2}|-\epsilon_{2}\\\
\beta+\gamma+\delta&=|\mu_{3}|-\epsilon_{3}\\\
\chi+\delta&=|\mu_{4}|-\epsilon_{4}\end{array}\right.$ (7)

###### Proof.

The proof is clear. ∎

In particular

$\alpha+\beta+\gamma+\delta+\chi=n-1-\delta_{B}^{\text{type of }[h]}$

where $\delta_{B}^{\text{type of }[h]}$ is $1$ when the type of $[h]$ is $B$
and $0$ when it is $A$. Solving the system in $\alpha$, we obtain

$\left\\{\begin{array}[]{ll}\beta&=n-1-\delta_{B}^{\text{type of
}[h]}-|\mu_{4}|-|\mu_{2}|+\epsilon_{2}+\epsilon_{4}\\\
\gamma&=|\mu_{2}|-\epsilon_{2}-\alpha\\\
\delta&=|\mu_{3}|+|\mu_{4}|-(n-1-\delta_{B}^{\text{type of
}[h]})-\epsilon_{3}-\epsilon_{4}+\alpha\\\ \chi&=n-1-\delta_{B}^{\text{type of
}[h]}-|\mu_{3}|+\epsilon_{3}-\alpha\end{array}\right.$ (8)

######  Remark 28.

If system (8) does not have an admissible solution, then
$\mathsf{tropTev}^{\mathcal{H}_{a}}_{\mathsf{\Gamma}}=0$. In particular, if

$|\mu_{2}|+|\mu_{4}|>n-1$

then $\beta<0$ (note that $\epsilon_{2}([h])$ and $\epsilon_{4}([h])$ cannot
both be $1$ for $[h]$ contributing to (6) being the $x_{i}$ in general
position) and $\mathsf{tropTev}^{\mathcal{H}_{a}}_{\mathsf{\Gamma}}=0$.
Similarly, if

$|\mu_{3}|>n-1$

then $\mathsf{tropTev}^{\mathcal{H}_{a}}_{\mathsf{\Gamma}}=0$. Finally, there
is an isomorphism of $\Sigma$ preserving $n_{2}$ and $n_{4}$ and switching
$n_{1}$ and $n_{3}$, so also

$|\mu_{1}|>n-1$

imlpies $\mathsf{tropTev}^{\mathcal{H}_{a}}_{\mathsf{\Gamma}}=0$

The next lemma computes the multiplicities of any curve to 6 for Hirzebruch
surfaces.

######  Lemma 29.

The contribution of a curve $[h:\mathsf{C}\to\mathbb{R}^{2}]$ to 6 is

$\left(\prod_{i=1}^{4}\prod_{j=1}^{|\mu_{ij}|}\mu_{i,j}\right)a^{n-1-|\mu_{2}|-|\mu_{4}|}$

###### Proof.

By Corollary 23, we need to compute the product of the
$\mathrm{ev}$-multiplicity at each vertex.
For a leaf $\mathsf{L}_{i}$ with two vertices, marking $x_{i}$, and weighted
vectors associated to its end $\lambda_{i}\cdot n_{j_{i}}$, $\eta_{i}\cdot
n_{k_{i}}$, the multiplicity at the vertex of $\mathsf{L}_{i}$ that is not
adjacent to $x_{i}$ is precisely
$\lambda_{i}\eta_{i}|\det(n_{j_{i}},n_{k_{i}})|$. Note that for $j<k$,

$\det(n_{j},n_{k})=\left\\{\begin{array}[]{ll}a&\text{ if }j=1,k=3\\\ 0&\text{
if }j=2,k=4\\\ 1&\text{ otherwise}\end{array}\right.$

  * •

If the curve is of type $A$, $V$ will have local multiplicity $1$ by [Gol21,
definition 3.18], and as $i$ goes from $2$ to $n$, we obtain a factor of $a$
for each leaf counted in $\beta([h])=n-1-|\mu_{2}|-|\mu_{4}|$ (see Equation
8), and the numbers $\lambda_{i}$ and $\eta_{i}$ go through all the elements
of each partition, so we obtain the number in the statement of the lemma.

  * •

If the curve is of type $B$, we can assume without lost of generality that the
leaves with one end are $\mathsf{L}_{1}$, $\mathsf{L}_{2}$, with markings
$x_{1}$, $x_{2}$ and that the end of $\mathsf{L}_{i}$ has weight $\lambda_{i}$
and primitive vector $n_{j_{i}}$, for $i=1,2$.
Then the leaves $\mathsf{L}_{1}$, $\mathsf{L}_{2}$ are precisely the fixed
components of $V$, in the language of [Gol21, Definition 3.18]. We divide into
cases:

    * –

If $\epsilon_{1}([h])+\epsilon_{3}([h])=1$, the $\mathrm{ev}$-multiplicity of
$V$ is $\lambda_{1}\lambda_{2}$ and the local multiplicity for the rest of the
vertices from the leaves $\mathsf{L}_{3},\ldots,\mathsf{L}_{n}$ are counted as
in the case $A$, so the multiplicity of the curve would be

$\left(\prod_{i=1}^{4}\prod_{j=1}^{|\mu_{ij}|}\mu_{i,j}\right)a^{\beta([h])}.$

Note that, by 8, in this case $\beta([h])=n-1-|\mu_{4}|-|\mu 2|$ because
$\epsilon_{2}([h])+\epsilon_{4}([h])=1$.

    * –

If $\epsilon_{1}([h])+\epsilon_{3}([h])=2$, the $\mathrm{ev}$-multiplicity of
$V$ is $a\lambda_{1}\lambda_{2}$ and the rest of local multiplicities are
computed in the same way giving the number

$\left(\prod_{i=1}^{4}\prod_{j=1}^{|\mu_{ij}|}\mu_{i,j}\right)a^{\beta([h])+1}.$

In this case, by looking at 8, $\beta([h])=n-2-|\mu_{2}|-|\mu_{4}|$.

∎

Next, we explain which curves appear in 6. We distinguish two cases.

####  3.2.1 Case $|\mu_{3}|+|\mu_{4}|\geq n-1$

In this case, the following

$\left\\{\begin{array}[]{ll}\bar{\alpha}&=0,\\\ \bar{\delta}_{B}^{\text{type
of }h}&=0\\\ \bar{\epsilon}_{i}&=0\text{ for }i=1,\ldots,4,\\\
\bar{\beta}&=n-1-|\mu_{2}|-|\mu_{4}|\\\ \bar{\gamma}&=|\mu_{2}|\\\
\bar{\delta}&=|\mu_{3}|+|\mu_{4}|-(n-1)\\\
\bar{\chi}&=n-1-|\mu_{3}|\end{array}\right.$ (9)

is a solution of the system (8).

We put the points $x_{1},\ldots,x_{n}$ in the plane $\mathbb{R}^{2}$ in
general position and in such a way that: $x_{1}$ at $(0,0)$, and there are
exactly $\bar{\alpha}$ points in $\sigma_{1}$, $\bar{\beta}+\bar{\gamma}$ in
$\sigma_{2}$, $\bar{\delta}$ in $\\{ax+y>0,y<0\\}\subseteq\sigma_{3}$ and
$\bar{\chi}$ in $\\{x<0,y<0\\}\subseteq\sigma_{4}$.

It is then clear that we can form

$\prod_{i=1}^{4}|\mu_{i}|!\binom{\bar{\beta}+\bar{\gamma}}{\bar{\beta}}=\prod_{i=1}^{4}|\mu_{i}|!\binom{n-1-|\mu_{4}|}{|\mu_{2}|}$

curves $[h:\mathsf{C}\to\mathbb{R}^{2}]$ contributing to the intersection (6)
with domain $\mathsf{C}$ of type $A$ and
$(\alpha([h]),\ldots\chi([h]))=(\bar{\alpha},\ldots,\bar{\chi})$ as in Remark
26. Here the factorial terms are counting the different ways of labelling the
ends of $\mathsf{C}$. We will prove in §3.2.3 that these are all the
contributing curves in this case.

####  3.2.2 Case $|\mu_{3}|+|\mu_{4}|<n-1$

This case is more complicated. In this case, the following

$\left\\{\begin{array}[]{ll}\bar{\alpha}&=n-1-|\mu_{3}|-|\mu_{4}|,\\\
\bar{\delta}_{B}^{\text{type of }h}&=0\\\ \bar{\epsilon}_{i}&=0\text{ for
}i=1,\ldots,4,\\\ \bar{\beta}&=n-1-|\mu_{4}|-|\mu_{2}|\\\
\bar{\gamma}&=n-1-|\mu_{1}|\\\ \bar{\delta}&=0\\\
\bar{\chi}&=|\mu_{4}|\end{array}\right.$ (10)

is a solution of (8).

As before, we put the points $x_{1},\ldots,x_{n}$ in the plane
$\mathbb{R}^{2}$ in general position and in such a way that: $x_{1}$ is at
$(0,0)$, and there are exactly $\bar{\alpha}$ points in $\sigma_{1}$,
$\bar{\beta}+\bar{\gamma}$ in $\sigma_{2}$, $\bar{\delta}$ in
$\\{ax+y>0,y<0\\}\subseteq\sigma_{3}$ and $\bar{\chi}$ in
$\\{x<0,y<0\\}\subseteq\sigma_{4}$.

We can form

$\prod_{i=1}^{4}|\mu_{i}|!\binom{\bar{\beta}+\bar{\gamma}}{\bar{\beta}}=\prod_{i=1}^{4}|\mu_{i}|!\binom{|\mu_{3}|}{n-1-|\mu_{4}|-|\mu_{2}|}$
(11)

curves $[h:\mathsf{C}\to\mathbb{R}^{2}]$ contributing to the intersection (6)
with domain $\mathsf{C}$ of type $A$ as in the previous case. However, there
also are some curves $[h:\mathsf{C}\to\mathbb{R}^{2}]$ of type $B$ which we
now describe. The vertex $V$ is mapped to

$h(V)\in\\{x>0,ax+y=0\\},$

and

$\displaystyle(\epsilon_{1}([h]),\epsilon_{2}([h]),\epsilon_{3}([h]),\epsilon_{4}([h]),\alpha([h]),\beta([h]),\gamma([h]),\delta([h]),\chi([h]))=(1,1,0,0,\bar{\alpha}-1,\bar{\beta},\bar{\gamma},\bar{\delta},\bar{\chi}).$

Note that the marking $x_{1}$ is reached from $V$ by a single bounded edge in
the direction $n_{1}$. There are

$\displaystyle\begin{split}&\prod_{i=1}^{4}|\mu_{i}|!\sum_{k=1}^{\bar{\beta}}\binom{\bar{\alpha}+k-1}{\bar{\alpha}-1}\binom{\bar{\beta}+\bar{\gamma}-k}{\bar{\beta}-k}\\\
=&\prod_{i=1}^{4}|\mu_{i}|!\sum_{k=1}^{n-1-|\mu_{2}|-|\mu_{4}|}\binom{n-2-|\mu_{3}|-|\mu_{4}|+k}{n-2-|\mu_{3}|-|\mu_{4}|}\binom{|\mu_{3}|-k}{n-1-|\mu_{2}|-|\mu_{4}|-k}\end{split}$
(12)

of such curves.

######  Lemma 30.

The curves in (11) and (12) are a total of

$\prod_{i=1}^{4}|\mu_{i}|!\binom{n-1-|\mu_{4}|}{|\mu_{2}|}$

curves.

###### Proof.

We will use the following two well-known combinatorial identities

$\binom{x}{y}=(-1)^{y}\binom{y-x-1}{y}\ $ (13)

and

$\binom{x}{y}=(-1)^{x-y}\binom{-y-1}{x-y}$ (14)

valid for $x\in\mathbb{Z}_{>0}$ and $y\in\mathbb{Z}_{\geq 0}$, and Vandermonde
identity

$\sum_{k=0}^{N}\binom{x}{k}\binom{y}{N-k}=\binom{x+y}{N}$ (15)

valid for $N\in\mathbb{Z}_{\geq 0}$ and $x,y\in\mathbb{C}$.

We start with noticing that the $k=0$ term in the sum (12) is exactly the
binomial coefficient in (11). Therefore, the sum of the two contributions is
(up to taking the product with $\prod_{i=1}^{4}|\mu_{i}|!$)

$\displaystyle\sum_{k=0}^{n-1-|\mu_{2}|-|\mu_{4}|}\binom{n-2-|\mu_{3}|-|\mu_{4}|+k}{n-2-|\mu_{3}|-|\mu_{4}|}\binom{|\mu_{3}|-k}{n-1-|\mu_{1}|}$
$\displaystyle=$
$\displaystyle(-1)^{n-1-|\mu_{3}|-|\mu_{4}|}\sum_{k=0}^{n-1-|\mu_{2}|-|\mu_{4}|}\binom{|\mu_{3}|+|\mu_{4}|-(n-1)}{k}\binom{|\mu_{1}|-n}{|\mu_{1}|+|\mu_{3}|-(n-1)-k}$
$\displaystyle=$
$\displaystyle(-1)^{n-1-|\mu_{3}|-|\mu_{4}|}\sum_{k=-\infty}^{\infty}\binom{|\mu_{3}|+|\mu_{4}|-(n-1)}{k}\binom{|\mu_{1}|-n}{|\mu_{1}|+|\mu_{3}|-(n-1)-k}$
$\displaystyle=$
$\displaystyle(-1)^{n-1-|\mu_{3}|-|\mu_{4}|}\binom{-|\mu_{2}|-1}{|\mu_{1}|+|\mu_{3}|-(n-1)}$
$\displaystyle=$ $\displaystyle\binom{n-1-|\mu_{4}|}{n-1-|\mu_{4}|-|\mu_{2}|}$

where in the first equality we used (13), in the second equality the fact that
$|\mu_{1}|+|\mu_{3}|-(n-1)-k<0$ for $k>n-1-|\mu_{2}|-|\mu_{4}|$, in the third
equality (15) and in the last equality (14). This concludes the proof. ∎

We will prove in §3.2.3 that these are all the contributing curves in this
case.

####  3.2.3 Exclusion of further contributions

We have to prove that the curves listed above are all the curves contributing
to the intersection 6.

We start with adopting a unifying perspective.

######  Remark 31.

The solutions (9) and (10) are determined as the unique solution of the system
(8) where we set $\delta_{B}^{\text{type of }[h]}=0$, $\epsilon_{i}=0$ and ask
for $\alpha$ to be the minimum possible.

Call $(\bar{\alpha},\bar{\beta},\bar{\gamma},\bar{\delta},\bar{\chi})$ such
solution (so either as in (9) or as in (10))

######  Remark 32.

Suppose $(\alpha,\beta,\gamma,\delta,\chi)$ is a solution of (8), with
$\delta_{B}^{\text{type of }[h]}=1$ and $\epsilon_{i}=1$ for exactly two
indices $i$ and $\epsilon_{i}=0$ otherwise. Then $\alpha\geq\bar{\alpha}$
unless

$(\epsilon_{1},\epsilon_{2},\epsilon_{3},\epsilon_{4},\alpha,\beta,\gamma,\delta,\chi)=(1,1,0,0,\bar{\alpha}+1,\bar{\beta},\bar{\gamma},\bar{\delta},\bar{\chi}).$

###### Proof.

Suppose for example $\epsilon_{1}=\epsilon_{2}=0$ and
$\epsilon_{3}=\epsilon_{4}=1$. Then $(\alpha,\beta,\gamma+1,\delta,\chi)$ is a
solution of the system (8) with $\delta_{B}^{\text{type of }[h]}=0$ and
$\epsilon_{i}=0$ for all $i$. Therefore $\alpha\geq\bar{\alpha}$. The other
cases are treated similarly. The case $\epsilon_{2}=\epsilon_{4}=1$ and
$\epsilon_{1}=\epsilon_{3}=0$ is not possible because the points $x_{i}$ are
in general positions and in particular there are no two of them in the same
line $x=\mathrm{constant}$. ∎

We first exclude other contributions $[h:\mathsf{C}\to\mathbb{R}^{2}]$ with
$\mathsf{C}$ of type $A$. For such $[h]$ we would have:

  1. $\bullet$

if $h(V)\in\sigma_{1}$, then $\alpha([h])\leq\bar{\alpha}$ and
$\delta([h])>\bar{\delta}$,

  2. $\bullet$

if $h(V)\in\sigma_{2}$, then $\gamma([h])>\bar{\gamma}$,

  3. $\bullet$

if $h(V)\in\sigma_{3}\cap\\{ax+y>0\\}$, then $\gamma([h])>\bar{\gamma}$,

  4. $\bullet$

if $h(V)\in\\{ax+y<0\\}$, then
$\alpha([h])+\beta([h])+\gamma([h])>\bar{\alpha}+\bar{\beta}+\bar{\gamma}$,

each of which is in contradiction with system (8) and the choice of
$\bar{\alpha}$.

Assume now we are not in the exceptional situation of Remark 32. Under this
assumption, we now prove that there are no more contributions of type B. For
such a $[h:\mathsf{C}\to\mathbb{R}^{2}]$ we would have one of the following
contradictions:

  1. $\bullet$

if $h(V)\in\\{y>0\\}\cup\\{x>0,y=0\\}$, then
$\alpha([h])+\beta([h])+\gamma([h])\leq\bar{\alpha}+\bar{\beta}+\bar{\gamma}-1$
and therefore by (8) it must be $\epsilon_{4}([h])=0$, which would then imply
$\alpha([h])+\beta([h])+\gamma([h])=\bar{\alpha}+\bar{\beta}+\bar{\gamma}-2$
in contradiction with (8),

  2. $\bullet$

if $h(V)\in\\{x>0,ax+y>0\\}$, then $\chi([h])>\bar{\chi}$ in contradiction
with (8) and our assumption on $\bar{\alpha}$,

  3. $\bullet$

if $h(V)\in\\{x>0,ax+y=0\\}$, then $\epsilon_{1}([h])=1$ and
$\epsilon_{4}([h])=0$ for the choice of the points $x_{i}$. If
$\epsilon_{3}=0$, then system (8) prescribes $\chi([h])<\bar{\chi}$ but
$\chi([h])\geq\bar{\chi}$,

  4. $\bullet$

if $h(V)\in\\{x\geq 0,ax+y<0\\}$ we have
$\alpha([h])+\beta([h])+\gamma([h])\geq\bar{\alpha}+\bar{\beta}+\bar{\gamma}$
and so, by (8), this is an equality and $\epsilon_{4}([h])=1$. However,
$\epsilon_{4}([h])=0$ by the position of the points $x_{i}$,

  5. $\bullet$

if $h(V)\in\\{x<0,y<0\\}$ then $\delta([h])+\chi([h])<\bar{\delta}+\bar{\chi}$
and so by (8), $\epsilon_{4}([h])=1$. It follows that $\epsilon_{2}([h])=0$
and so $\delta([h])+\chi([h])\leq\bar{\delta}+\bar{\chi}-2$, which is in
contradiction with (8),

  6. $\bullet$

if $h(V)\in\\{x<0,y=0\\}$ then $\epsilon_{3}([h])=1$. If
$\epsilon_{4}([h])=0$, then $\alpha([h])<\bar{\alpha}$ which is not possible
for the choice of $\bar{\alpha}$ and our assumptions; if instead
$\epsilon_{4}([h])=1$, then $\chi([h])<\bar{\chi}$ and so, by (8),
$\alpha([h])>\bar{\alpha}$ which is in contradiction with the fact that there
are no points $x_{i}$ which are reachable from $h(V)$ via leaves counted in
$\delta([h])$ and are not reachable with the same type of leaves from $(0,0)$.

Finally, we deal with curves of type B in the exceptional case of Remark 32.
In this case:

  1. $\bullet$

$h(V)$ cannot be on $\\{ax+y\leq 0\ ,x\geq 0\\}\smallsetminus\\{(0,0)\\}$
otherwise $\epsilon_{1}([h])=0$,

  2. $\bullet$

$h(V)\in\\{x<0,ax+y>0\\}$, then $\alpha([h])\leq\bar{\alpha}-2$ which is not
the case,

  3. $\bullet$

$h(V)$ cannot be on $\\{y>0,x=0\\}$ otherwise $\epsilon_{2}([h])=0$,

  4. $\bullet$

if $h(V)\in\\{x>0,ax+y>0\\}$, then $\chi([h])>\bar{\chi}$ which is also not
the case,

  5. $\bullet$

if $h(V)\in\\{x>0,ax+y=0\\}$ then the curves of type B in §3.2.2 appear,

  6. $\bullet$

if $h(V)\in\\{x>0,ax+y<0\\}$ then
$\alpha([h])+\beta([h])+\gamma([h])\geq\bar{\alpha}+\bar{\beta}+\bar{\gamma}$
which is not happening.

This proves that the curves listed in §3.2.1 and §3.2.2 are all the curves
contributing to the intersection (6) and concludes the proof of Theorem 7.

##  4 Absence of rational curves interpolating $n$ points on Hirzebruch
surfaces

Let $a\geq 2$ and assume that $\mu_{i,j}=1$ for all $i,j$. Let also $\beta\in
H_{2}(X,\mathbb{Z})$ be the unique curve class with $\beta.D_{i}=|\mu_{i}|$
for $i=1,2,3,4$. As in Remark 8, we have

$\mathsf{vTev}^{\mathcal{H}_{a}}_{\mathsf{\Gamma}}=0.$

By Lemma 17, this is equivalent to say that the map (2) is not dominant. We
aim to now explain this phenomenon geometrically.

By the proof of Theorem 5, it is enough to show that the restriction

$\overset{\circ}{\tau}:\mathcal{M}_{\mathsf{\Gamma}}(\mathcal{H}_{a})\to\mathcal{M}_{g,n}\times\mathcal{H}_{a}$

of (2) is not dominant. By contradiction, suppose that $\overset{\circ}{\tau}$
is dominant.

Let
$\rho:\mathcal{H}_{a}=\mathbb{P}(\mathcal{O}\oplus\mathcal{O}(a))\to\mathbb{P}^{1}$
be the projective bundle map and let

$0\to\mathcal{O}_{\mathcal{H}_{a}}(-1)\to\rho^{*}(\mathcal{O}_{\mathbb{P}^{1}}\oplus\mathcal{O}_{\mathbb{P}^{1}}(a))\to
Q\to 0$ (16)

be the associated universal exact sequence.

######  Lemma 33.

We have $2|\mu_{1}|\geq n-1$.

###### Proof.

Composing maps in $\mathcal{M}_{\mathsf{\Gamma}}(\mathcal{H}_{a})$ with
$\rho$, we obtain maps $\mathbb{P}^{1}\to\mathbb{P}^{1}$ of degree
$|\mu_{1}|$. Let $\mathcal{M}_{\mathsf{\Gamma}^{\prime}}(\mathbb{P}^{1})$ be
the moduli space of logarithmic stable maps
$(\mathbb{P}^{1},p_{1},...,p_{n},q_{1},...,q_{2d})\to\mathbb{P}^{1}$ with
prescribed intersection multiplicities with $\partial\mathbb{P}^{1}$ all $1$.
By Lemma 17, this space is irreducible. It follows that the natural map
$\mathcal{M}_{\mathsf{\Gamma}^{\prime}}(\mathbb{P}^{1})\to\mathcal{M}_{0,n}\times(\mathbb{P}^{1})^{n}$
is also dominant and in particular that

$2|\mu_{1}|+n-2=\mathrm{dim}(\mathcal{M}_{\mathsf{\Gamma}^{\prime}}(\mathbb{P}^{1}))\geq\mathrm{dim}(\mathcal{M}_{0,n}\times(\mathbb{P}^{1})^{n})=2n-3$

The conclusion follows. ∎

Given a map $f:\mathbb{P}^{1}\to\mathcal{H}_{a}$, we can can pull back to
$\mathbb{P}^{1}$ the exact sequence (16), obtaining

$\mathcal{O}_{\mathbb{P}^{1}}\oplus\mathcal{O}_{\mathbb{P}^{1}}(a|\mu_{1}|)\xrightarrow{v}\mathcal{O}_{\mathbb{P}^{1}}(|\mu_{2}|+a|\mu_{1}|)\to
0$

such that

$f(p)=\mathrm{ker}(v).$

We will think of $v$ as the data of two sections $s_{1}\in
H^{0}(\mathbb{P}^{1},\mathcal{O}_{\mathbb{P}^{1}}(|\mu_{2}|))$ and $v_{2}\in
H^{0}(\mathbb{P}^{1},\mathcal{O}_{\mathbb{P}^{1}}(|\mu_{2}|+a|\mu_{1}|)$.

The fact that $\overset{\circ}{\tau}$ is dominant amounts to say that for
general points

$\mathsf{L}_{i}\in\mathbb{P}(\mathcal{O}_{\mathbb{P}^{1}}(|\mu_{2}|)\oplus\mathcal{O}_{\mathbb{P}^{1}}(|\mu_{2}|+a|\mu_{1}|))\cong\mathcal{H}_{a}\text{
and }p_{1},\ldots,p_{n}\in\mathbb{P}^{1}$

there are sections $v_{1}\in
H^{0}(\mathbb{P}^{1},\mathcal{O}_{\mathbb{P}^{1}}(|\mu_{2}|))$ and $v_{2}\in
H^{0}(\mathbb{P}^{1},\mathcal{O}_{\mathbb{P}^{1}}(|\mu_{2}|+a|\mu_{1}|)$ such
that

$\langle v_{1}(p_{i}),v_{2}(p_{i})\rangle=\mathsf{L}_{i}$

for each $i=1,\ldots,n$. Since the codimension of

$\oplus_{i=1}^{n}\mathsf{L}_{i}\subseteq\oplus_{i=1}^{n}\mathcal{O}_{\mathbb{P}^{1}}(|\mu_{2}|)|_{p_{i}}\oplus\mathcal{O}_{\mathbb{P}^{1}}(|\mu_{2}|+a|\mu_{1}|)|_{p_{i}}$

is $n$, the rank of the evaluation map

$\Psi:H^{0}(\mathbb{P}^{1},\mathcal{O}_{\mathbb{P}^{1}}(|\mu_{2}|))\oplus
H^{0}(\mathbb{P}^{1},\mathcal{O}_{\mathbb{P}^{1}}(|\mu_{2}|+a|\mu_{1}|))\to\bigoplus_{i=1}^{n}\mathcal{O}_{\mathbb{P}^{1}}(|\mu_{2}|)|_{p_{i}}\oplus\mathcal{O}_{\mathbb{P}^{1}}(|\mu_{2}|+a|\mu_{1}|)|_{p_{i}}$

must be at least $n+1$. Also, by Lemma 33, it is at most

$2|\mu_{2}|+a|\mu_{1}|+2=2(n-1)-2|\mu_{1}|+2\leq n+1.$

Therefore, $2|\mu_{1}|=n-1$ and

$n-1=|\mu_{2}|+|\mu_{4}|\geq|\mu_{4}|=a|\mu_{1}|+|\mu_{2}|>n-1$

for $a\geq 2$. This yields a contradiction.

##  5 Proof of Theorem 13

It is well-known that for any $a\in\mathbb{N}$ the Hirzebruch surfaces
$\mathcal{H}_{a}$ and $\mathcal{H}_{a+2}$ are deformation equivalent. We will
use this fact to compute certain virtual Tevelev degrees of $\mathcal{H}_{a}$
by reducing to $\mathcal{H}_{0}=\mathbb{P}^{1}\times\mathbb{P}^{1}$ or
$\mathcal{H}_{1}=\mathrm{Bl}_{p}\mathbb{P}^{2}$. Theorem 13 will then follow
by comparing the result with Theorem 7 above.

Assume $a\geq 2$.

######  Lemma 34.

Let $0<2j\leq a$. Then there exists a smooth family
$\pi:\mathcal{U}\to\mathbb{A}^{1}$ such that:

  1. 1.

$\pi^{-1}(0)$ is isomorphic to $\mathcal{H}_{a}$;

  2. 2.

$\mathcal{U}\smallsetminus\pi^{-1}(0)$ is isomorphic to
$\mathcal{H}_{(a-2j)}\times(\mathbb{A}^{1}\smallsetminus\\{0\\})$ over
$\mathbb{A}^{1}\smallsetminus\\{0\\}$;

  3. 3.

$\pi$ admits a section.

###### Proof.

This is probably a well-known construction. Let

$\pi:\mathcal{U}=\\{([x_{0},x_{1}],[y_{0},y_{1},y_{2}],s)\ |\
x_{0}^{a}y_{1}-x_{1}^{a}y_{0}+sx_{0}^{j}x_{1}^{j+1}y_{2}=0\\}\to\mathbb{A}^{1}$

where
$\mathcal{U}\subseteq\mathbb{P}^{1}\times\mathbb{P}^{2}\times\mathbb{A}^{1}$
and $\pi$ is the projection onto $\mathbb{A}^{1}$. We claim that $\pi$ is as
stated in the lemma. In order to provide the required isomorphisms we will use
the following description [Cox95, Man04]:

$\mathcal{H}_{a}=(\mathbb{C}^{2}\smallsetminus\\{0\\})\times(\mathbb{C}^{2}\smallsetminus\\{0\\})/\sim$

where the equivalence relation $\sim$ is given by the
$(\mathbb{C}^{*})^{2}$-action is given by

$(\lambda,\eta).(l_{0},l_{1},t_{0},t_{1})=(\lambda l_{0},\lambda
l_{1},\lambda^{a}\eta t_{0},\eta t_{1}).$

Then

$\displaystyle\mathcal{H}_{a}$ $\displaystyle\to\pi^{-1}(0)\hskip 71.13188pt$
$\displaystyle[l_{0},l_{1},t_{0},t_{1}]$
$\displaystyle\mapsto([l_{0},l_{1}],[t_{1}l_{0}^{a},t_{1}l_{1}^{a},t_{0}])$

and

$\displaystyle\mathcal{H}_{(a-2m)}\times(\mathbb{A}^{1}\smallsetminus\\{0\\})$
$\displaystyle\to\mathcal{U}\smallsetminus\pi^{-1}(0)$
$\displaystyle([l_{0},l_{1},t_{0},t_{1}],s)$
$\displaystyle\mapsto([l_{0},l_{1}],[sl_{0}^{j}t_{0},sl_{1}^{j+1}t_{1},l_{1}^{a-j-1}t_{0}-l_{0}^{a-j}t_{1}],s)$

are isomorphisms and

$\displaystyle s:\mathbb{A}^{1}$ $\displaystyle\to\mathcal{U}$ $\displaystyle
s\mapsto$ $\displaystyle([0,1],[0,s,1])$

is a section. ∎

Write $a=2j$ or $a=2j+1$ for $j\in\mathbb{Z}_{\geq 1}$ depending on the parity
of $a$ and let

$\pi:\mathcal{U}\to\mathbb{A}^{1}$

be a smooth family as in Lemma 34.

######  Remark 35.

Given a line bundle $L$ on $\mathcal{U}\smallsetminus\pi^{-1}(0)$, we can
always extend $L$ to a line bundle on the all $\mathcal{U}$ (which is possible
being $\mathcal{U}$ smooth). Even if there are many extensions of $L$, for
each $s\in\mathbb{A}^{1}$ the restriction $L_{s}$ of $L$ to $\pi^{-1}(s)$ is
independent of the extension.

We treat the two case in Theorem 13 separately.

####  5.0.1 Case $a=2j$

Let $L$ be the pullback of $\mathcal{O}(1)\boxtimes\mathcal{O}(1)$ from
$\mathcal{H}_{0}\cong\mathbb{P}^{1}\times\mathbb{P}^{1}$ on
$\mathcal{U}\smallsetminus\pi^{-1}(0)\cong\mathcal{H}_{0}\times(\mathbb{A}^{1}\smallsetminus\\{0\\})$.

######  Lemma 36.

We have

$c_{1}(L_{0})=(1+j)D_{1}+D_{2}\in A_{*}(\mathcal{H}_{a})$

where $D_{1}$ and $D_{2}$ are the toric divisors of $\mathcal{H}_{a}$ as in
Notation 6

###### Proof.

Clearly

$c_{1}(L_{0})^{2}=c_{1}(L_{1})^{2}=2.$ (17)

Also since the normal bundle to each fiber of $\pi$ is trivial, we have

$c_{1}(L_{0}).c_{1}(\mathcal{T}_{\pi^{-1}(0)})=c_{1}(L_{1}).c_{1}(\mathcal{T}_{\pi^{-1}(1)})=4$
(18)

These two equations completely determine $c_{1}(L_{0})$. Namely, $D_{1}$ and
$D_{2}$ form a basis of $A_{1}(\mathcal{H}_{a})$ and writing
$c_{1}(L_{1})=xD_{1}+yD_{2}$, Equations 17 and 18 yields

$\begin{cases}2=2xy-y^{2}a,\\\ 4=2x+(2+a)y-2ya\,\end{cases}$

whose integral solution is $(x,y)=(1+j,1)$. ∎

Fix $d>0$ such that $2d=n-1$ and let $\beta=d[(1+j)D_{1}+D_{2}]$. Since
Gromov-Witten invariants are deformation invariant, we get

$1=\mathsf{vTev}^{\mathbb{P}^{1}\times\mathbb{P}^{1}}_{0,n,(d,d)}=\mathsf{vTev}^{\mathcal{H}_{a}}_{0,n,\beta}$

The first equality follows from [BP21, Example 2.2 and Proposition 2.3]. Note
that the existence of a section of $\pi$ guarantee that on each fiber the
point class can be realized as the restriction of a class from $\mathcal{U}$.

To conclude, if $\mathsf{\Gamma}$ and $\alpha$ are as in Theorem 13, we have

$\mathsf{vTev}_{\mathsf{\Gamma}}^{\mathcal{H}_{a}}=0\neq
1=\mathsf{vTev}^{\mathcal{H}_{a}}_{0,n,\beta}$

and thus
$\alpha_{*}[\overline{\mathcal{M}}_{\mathsf{\Gamma}}(X)]^{\mathrm{vir}}\neq[\overline{\mathcal{M}}_{0,n}(X,\beta)]^{\mathrm{vir}}$
as desired.

####  5.0.2 Case $a=2j+1$

This case is very similar, but uses [CL23b, Theorem 14] instead of [BP21].

Let $L_{\mathsf{H}}$ (resp. $L_{\mathsf{E}}$) be the pullback of
$\mathcal{O}(\mathsf{H})$ (resp $\mathcal{O}(\mathsf{E})$) from
$\mathcal{H}_{1}\cong\mathrm{Bl}_{p}\mathbb{P}^{2}$ on
$\mathcal{U}\smallsetminus\pi^{-1}(0)\cong\mathcal{H}_{1}\times(\mathbb{A}^{1}\smallsetminus\\{0\\})$.
Here $\mathsf{H}$ (resp. $\mathsf{E}$) is the hyperplane class (resp. the
class of the exceptional divisor) on $\mathrm{Bl}_{p}\mathbb{P}^{2}$.

######  Lemma 37.

We have

$c_{1}((L_{\mathsf{H}})_{0})=(1+j)D_{1}+D_{2}$

and

$c_{1}((L_{\mathsf{E}})_{0})=jD_{1}+D_{2}$

###### Proof.

Proceed as in Lemma 36 ∎

Fix $d$ and $k$ integers such that $0\leq k\leq d$, $0\leq k\leq n-1-d$ and
$3d-k=2(n-1)$. Call $\beta=[j(d-k)+d]D_{1}+(d-k)D_{2}$. Then

$\binom{n-1-d}{k}=\mathsf{vTev}^{\mathrm{Bl}_{p}\mathbb{P}^{2}}_{0,n,d\mathsf{H}-k\mathsf{E}}=\mathsf{vTev}^{\mathcal{H}_{a}}_{0,n,\beta}$

where the first equality follows from [CL23b, Theorem 14]. If
$\mathsf{\Gamma}$ and $\alpha$ are as in Theorem 13, we have

$\mathsf{vTev}_{\mathsf{\Gamma}}^{\mathcal{H}_{a}}=0\neq\binom{n-1-d}{k}=\mathsf{vTev}^{\mathcal{H}_{a}}_{0,n,\beta}$

and thus
$\alpha_{*}[\overline{\mathcal{M}}_{\mathsf{\Gamma}}(X)]^{\mathrm{vir}}\neq[\overline{\mathcal{M}}_{0,n}(X,\beta)]^{\mathrm{vir}}$.

This concludes the proof of Theorem 13.

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# Learn Class Hierarchy using Convolutional Neural Networks

Riccardo La Grassa  Ignazio Gallo  Nicola Landro  University of Insubria,
Department of Theoretical and Applied Science, Varese, Italy

###### Abstract

A large amount of research on Convolutional Neural Networks has focused on
flat Classification in the multi-class domain. In the real world, many
problems are naturally expressed as problems of hierarchical classification,
in which the classes to be predicted are organized in a hierarchy of classes.
In this paper, we propose a new architecture for hierarchical classification
of images, introducing a stack of deep linear layers with cross-entropy loss
functions and center loss combined. The proposed architecture can extend any
neural network model and simultaneously optimizes loss functions to discover
local hierarchical class relationships and a loss function to discover global
information from the whole class hierarchy while penalizing class hierarchy
violations. We experimentally show that our hierarchical classifier presents
advantages to the traditional classification approaches finding application in
computer vision tasks.

t1t1footnotetext: This paper is under review to Applied Intelligence journal

##  1 Introduction

In recent years researchers have become increasingly interested in the multi-
label and hierarchical learning approaches, finding many application to
several domain, including classification Wehrmann et al. (2018); Cesa-Bianchi
et al. (2006), image annotation Dimitrovski et al. (2011), bioinformatics
Valentini (2009); Yan et al. (2019) Chen et al. (2018) Abacha et al. (2019).
Nowadays, machine learning is commonly used to resolve complex problem into
pattern recognition where an object is classified assigning a label in
according with the model’s rule used. However, classes are not always disjoint
from others and objects within them can be related to others as a hierarchical
structure Silla and Freitas (2011). Human beings perceive the world with
different types of granularity and can translate information from coarse-
grained to fine-grained and on the contrary, perceiving different levels of
abstraction of the information acquired Hobbs (1990); McCalla et al. (1992).
This concept is reflected in the taxonomy of the multi-label general
approaches under the idea of structured output prediction Su et al. (2015).

In terms of neural models, the main difference between the prediction of
structured output and flat multi-label classification lies in the level of
neurons that contains the label prediction. In fact, in the presence of a
structured output, the information is based on a different level of
abstraction, while with the multi-label flat approach it is based on a single
level.

Hierarchical multi-label classification (HMC) is a variant of the
classification task where instances may belong to multiple classes at the same
time and classes are organized in a hierarchy. In HMC approaches a
relationship among classes and can be formalized by a tree or directed acyclic
graph (DAG). Our approach to HMC exploits the annotation hierarchy by building
a single neural network that can simultaneously predict all categorization of
an input source exploiting multiple layers of a neural model. For example,
considering the class label prediction for an image containing a tiger, the
proposed system can simultaneously predict that a ”tiger” has been found but
at the same time the same object is also a ”feline” and a ”mammal”.

In literature exists two main approaches to HMC problem, known as local and
global Costa et al. (2007); Xu et al. (2019); Silla and Freitas (2011). In the
global approach, the output of the final layer predicts the test instance in
which only one classifier sees information globally without having local
information. In the local approach, there is a set of trained classifiers that
follows a top-down strategy, in particular, the training process is
independently for each base classifier.

Different local approaches have been proposed in the literature, like Local
classifier per Node (LCN) Valentini (2009), Local classifier per parent node
(LCPN), Local classifier per level (LCL) Cerri et al. (2011). LCN strategy
trains a local classifier for each node of a graph providing a local decision
to make predictions. LCPN uses a multi-class classifier for each internal
class to recognize classes from its sub-classes and LCL methods train a multi-
class classifier per hierarchical level. In contrast with local (LCN, LCL,
LCPN) and global approaches, we use a single trained model and a single back-
propagation error with many different layers fully connected, responsible to
synchronize with a concept linked to a given hierarchical structure.

A recent work Wehrmann et al. (2018) describes a novel method to solve HMC
problem, that preserves local and global information simultaneously to
discover the local hierarchical relationship among classes. Unlike this work,
our architecture exploits recent neural network potentialities and facilitates
the multi-class prediction for each deep layers to capture local context
following the hierarchical structure of the information. In our approach, we
have a cascade of fully connected linear layers each one with softmax plus
cross-entropy, where the output of a layer $l-1$ is the input of layer $l$;
instead, in Wehrmann et al. (2018) the model has ReLu activation functions on
two different layers fully connected with softmax and binary cross-entropy per
block. Another difference with Wehrmann et al. (2018) is that the input of
each layer $l$ fuse with the input, instead, in our approach the input per
layer is the output of the previous layer. The last difference is that our
model uses local classification as final prediction in according to
hierarchical multi-label classification task, instead of in HMCN-F the final
layer is used as flat layer plus another layer that uses jointly local and
global output information to obtain the final prediction.

Our work can be summarized in the following key contributions:

  * 1.

We propose a new hierarchical deep loss approach (HDL) as an extension of
convolutional neural networks to assign hierarchical multi-labels to images.
Our extension can be adapted to a generic Convolutional Neural Network as
final step.

  * 2.

To prove the effectiveness of our hierarchical classification approach we
conduct empirical studies on three different datasets. First, we created
Animals_Taxonomy8 dataset based on real animal images from Flickr on three
groups of taxonomy (Class, Family, Species) with their relative label
annotations. Second, we used a well-known biomedical dataset (VQA-Med 2019)
contains radiology real images on different levels of hierarchy and third, we
created Geometry_shapes_annotations that contains thousands of shapes images
on three depth hierarchy levels. Further, all datasets have a different number
of instances (2.8k,8k,40k) useful to prove the robustness of our approach.

##  2 The Proposed Approach

As mentioned above, our solution is an architectural extension that can be
adapted to a generic neural network. In this paper, we used a standard
Convolutional Neural Network, the ResNet18, as a base model to which we added
our solution to solve a hierarchical images classification problem. As
graphically represented in Figure 1, what we do is to extend the output layer
with some fully connected layers equal to the number of layers available in
the classes hierarchy tree of the problem to be solved, and to associate a
loss function to each of these new layers added. In practice, we construct a
mapping between the layers of a class hierarchy and the new layers of the
neural network ($linear_{1},\dots,linear_{N}$ in Figure 1) so that the network
can learn to discriminate between all class labels belonging to a given layer
of the hierarchy. To minimize the intra-class variance and at the same time to
keep the features among different classes separated we compute the Center Loss
Wen et al. (2016) on each training mini-batch and update all class centers
after each training epoch. More formally we compute the center loss
$\mathcal{L}_{C}$ as follow:

$\mathcal{L}_{C}=\sum_{i=1}^{m}{\lVert x_{i}-c_{y_{i}}\rVert}_{2}^{2}$ (1)

where $c_{y_{i}}\in\mathbb{R}^{d}$ denotes the center for the class
$y{{}_{i}}$ in the features space of the deep model. In our experiments, we
chose a Resnet-18 as a general model and apply Center Loss after the adaptive
pooling layer. Finally, let $l_{1}$ be the linear layer at first level with
dimension equal to the number of classes at first level of hierarchy, more
formally:

$l_{G}^{1}=\phi(W_{G}^{1}x+b_{G}^{1})$ (2)

where $W_{G}^{1}\in\mathbb{R}^{|l_{G}^{1}|\times|d|}$ ,
$b_{G}^{1}\in\mathbb{R}^{|l_{G}^{1}|\times 1}$ is the bias vector with $\phi$
linear activation function and $d$ be the number of features. Then, we add a
linear layer $l$ for each hierarchical level in a generic dataset and we
perform the cross-entropy loss to maximize the inter-class variance.
Precisely, we apply softmax function from logits of layer $l_{G}^{l}$ and use
cross-entropy loss as Eq.3

$\mathcal{L}_{l_{G}^{l}}=-\sum_{i=1}^{m}log\frac{e^{W_{y_{i}}^{T}x_{i}+b_{yi}}}{\sum_{j=1}^{n}e^{W_{j}^{T}x_{i}+b_{j}}}$
(3)

Where, $l$ is the layer l-th, m and n are the mini-batch size and number of
classes respectively, $x_{i}\in\mathbb{R}^{d}$ denotes the ith deep feature,
belonging to the $y_{i}$th class and b is the bias.

Fig. 1: The proposed architecture of HDL

Finally, our total loss is:

$\displaystyle\mathcal{L}=\lambda_{0}\cdot\mathcal{L}_{C}+\lambda_{1}\cdot\mathcal{L}_{1}+\dots+\lambda_{N}\cdot\mathcal{L}_{N}$
(4)

Where $\lambda_{0,1,...,N}=1$, $\mathcal{L}_{C}$ it the centers loss value and
$\mathcal{L}_{0,1,..,N}$ is the cross-entropy loss value of the layer
$\\{1,...,N\\}$. The general formulation with $N$ layer is defined as Eq.5

$\displaystyle\mathcal{L}=\lambda_{0}\cdot\mathcal{L}_{C}+\sum_{l=1}^{N}{\lambda_{l}\cdot\mathcal{L}_{l}}$
(5)

##  3 Datasets

To evaluate the proposed method, we created our own datasets as there is no a
standard benchmarked dataset on hierarchical multi-label images
classification, available in the literature.

The medical Visual Question Answering task (VQA-Med 2019) Abacha et al. (2019)
is focused on radiology images (example in Fig. 4) grouped in four main
classes: Modality, Plane, Organ system, Abnormality. The original challenge is
to classify an image from a question linked to it, indeed for each image in
the training we have a paired question. Our focus is on the hierarchical
multi-label classification of images, therefore, we will exclude our
experiment from text classification task. We use all train size and use the
validation set as a test set (because the test set is not labelled with all
labels), respectively 2816/340 objects. In total, we consider three levels of
hierarchy (Modality Class, Plane Class, Organ Class) with their relative
different type of concepts. These classes have a size of 44, 15, 10
respectively per classes. In these experiments, our goal is to prove
experimentally the effectiveness and robustness of our model to discriminate
different concepts also in the case we have a few examples per classes in the
train.

We have created a synthetic geometric shapes dataset which contains 2
different shapes (Triangle, Square, some image sample into fig. 2) at the
first level of our hierarchy. Each shape has 6 different full colours and
other 6 different colours for out-fill, the last two represents the second and
third level of the hierarchy. The possible configuration is 72 so, we generate
train/test with 20000 and 6000 objects respectively. The dimension of the
images is 128x128x3. In these experiments, we want to answer the question
”Which kind of shape is this? What is the fill colour? and the out fill
colour?.

(a) Green square with orange border

(b) Gray square with red border

(c) Green triangle with red border

(d) Gray triangle with red border

Fig. 2: Some sample of geometric dataset.

This data set is created from Flickr animals images, the hierarchy represents
a small taxonomy with class, family and species as in Fig. 6. The selected
class is mammalia and reptilia. The second level of hierarchy is the family,
in particular felidae and ursidae for mammalia and crocodyle, iguanidae,
emydidae and pythonidae for reptilia. The last hierarchy level represent the
species( example of images in fig. 5) as malaysia tiger, felis catus known as
cat, ailuropoda melanoleuca known as giant panda, ursus maritimus known as
polar bear, python molurus known as green python, trachemys scripta as small
turtle, iguana iguana and crocodylus niloticus well known as nilus crocodile.
A whole representation of the dataset is in Fig. 3.

Fig. 3: Hierarchy of categories used in Animals_Taxonomy8

(a) t1, sagittal, skull and contents

(b) xr-plain film, lateral, lung-mediastinum-pleura

(c) xr-plain film, ap, musculoskeletal

(d) ct w/contrast (iv), axial, skull and contents

Fig. 4: Four images extracted from VQA-Med 2019 Abacha et al. (2019) dataset
(synpic371, synpic10103, synpic16486, synpic48315). The labels separated by
comma belong to the sub-categories of the three main classes following the
order: Modality, Plain, Organ.

(a) Felis catus

(b) Malaysia tiger

(c) Ursus maritimus

(d) Ailuropoda malanoleuca

(e) Python molurus

(f) Trachemys scripta

(g) Green iguana

(h) Crocodylus niloticus

Fig. 5: Example of images extracted from Animals_Taxonomy8

Fig. 6: Animals_Taxonomy8 dataset classification hierarchy.

##  4 Experiments

To evaluate the proposed method we develop four empirical studies.

  1. 1.

In the first one, we use a well-known dataset (VQA-Med 2019) to test our
approach with biomedical real images, also in the case we have few data
available.

  2. 2.

In the second we test the capability of abstraction of our approach on a
synthetic dataset created in the context we have thousand of instances
available.

  3. 3.

In the third, we extract hierarchical structure on the real-images dataset
contains images of three types of animal taxonomy levels (Class-Family-
Species) and prove the robustness of our HDL in the case which images are hard
to recognize and they contain noise.

  4. 4.

In the four experiment, we compare our HDL with a ResNet18 proving the
effectiveness of our approach.

First experiment In this experiment, we test our model in the case we have few
instances and with a high complexity of images. Our hypothesis is that the
performance in terms of accuracy in a layer is higher when the number of
different concepts to distinguish is inferior to a layer with many concepts to
recognize. As we show in 1 at the first row (VQA-Med 2019), we have accuracies
of 38.05, 74.04, 66.66 for the size of layers 44, 15, 10 respectively. We can
observe that the accuracy of the first layer is lower of 1.94 times than the
second layer and to 1.75 times than the third layer, this proves that our
model offers better scalability when we have few concepts per layer to learn.
Similar results can be found in Animals_Taxonomy8, where the higher accuracy
of the third layer at the third row of Table 1 than others, is due to the fact
we have only two concept (mammals or reptiles) to distinguish than the second
layer (8 concepts) Figs. 7 and 8.

Second experiment In a second experiment, we use a synthetic dataset with
simple geometric shapes and several instances 7.10 times greater than VQA-Med
2019. Our intuition is that attribute more samples per classes can improve the
training of our model and subsequently, to obtain better performance in terms
of accuracy than the first experiments. To prove this conjecture, we train
with 20K instances our HDL and test it with 6k instances. The results in
Table. 1 at the second row per tables, confirm our expectations. The higher
number of instances jointly with the simplicity of images allows the model to
reach high accuracy starting from the first ten epochs. Furthermore, we
conduct three different runs with a learning rate of 0.005, 0.001, 0.01 using
batch-size of 64.

Third experiment In these experiments, we test our model using more instances
than the first experiment and with images of animal (Animals_Taxonomy8) that
contains noise. In particular, our model offers good performance also in the
case the images are not simple as in the second experiments and when they
contain noise or offers little comprehensibility, indeed many images are not
clear, like for example a snake completely hidden by forest or a bar sign with
a panda logo. However, as we show in 1, the accuracy of the third layer,
responsible to recognize mammals or reptile is very high. We conclude that
considering the poor understanding of images, noise and hard images to
recognize, experimental results prove the robustness of our model.

Fourth experiment HDL is designed to maximize the learning capacity and to
extract the hierarchical structure from the labelled data. Our intuition is
that our model, lead to different losses at any level, with the power to
reduce intra-variance and to maximize inter-variance, can obtain better
accuracy than a classical convolutional neural network. To prove this, we
conduct six different experiments using a classic ResNet18 and our HDL on
Animals_Taxonomy8 using two learning rate and a batch size of 64. The results
in Fig. 2 and 9,10 clearly confirm our expectations. In all cases, the
accuracy is higher than a classical ResNet18, this experiment proves the
effectiveness of our proposed model.

###  4.1 Experiments settings

We build our hierarchical multi-label classifier model as an extension on a
Resnet-18, but is it possible to apply to any Convolutional Neural networks.
We implement our extension in Python using Pytorch framework. Fig.1 shows the
architecture used for experiments. The size of the input images is re-scaled
to 64x64x3 for Geometry dataset and 256x256x3 for VQA-Med 2019 and
Animals_Taxonomy8 datasets. We do not apply any preprocessing of images as
data augmentation, rotation or normalization. The kernel size of the first
convolutional layers is 7x7 with a stride of 2 pixels, followed by a
normalization of layer and a non-linear layer with ReLu activation function. A
max-pooling operation over 3x3 regions and a stride of 2 pixels. Then, we have
four blocks of Convolution, with 64, 128, 256, 512 numbers of plans
respectively and apply an adaptive average pooling over 1x1 region. Finally,
we add three fully connected linear layer, where each layer corresponding to
the total number of concepts in our hierarchical dataset. In the forward
process, we take the output after the adaptive average pooling and apply
Center loss function and for each linear layers we apply softmax function and
then cross-entropy loss. The total loss will be the sum of the local loss per
layers. Our network was trained with Adam optimizer Kingma and Ba (2014). The
batch-size used, learning rate, epochs are described jointly with the results
for each dataset.

Fig. 7: Training losses Animals_Taxonomy8 with $lr=0.01$

Fig. 8: Training losses Animals_Taxonomy8 with $lr=0.005$. We emphasize the
different descent of losses. This is due to the number of concepts to
distinguish from each layer. Each line represents the loss for each layer. For
this dataset, we design our model with a shape of 6,8,2 to distinguish Family,
Species and Classes respectively. As we show also in 1, the line yellow that
represents linear layer 3 with 2 concepts (Mammals or Reptiles) has more
descent power, indicating that our model quickly learns a few concepts rather
than many as red line or green.

Fig. 9: HDL vs original Resnet18 with $lr=0.005$

Fig. 10: HDL vs original Resnet18 with $lr=0.01$

##  5 Results and Discussion

This study is placed in the sub-category of multi-label classification called
Structure output learning. In according with experimental results at Tables 1,
2, we achieved good results on three different datasets finding the way to
exploit the dependency among classes and make accurate predictions, reducing
the misclassification than a classic ResNet18. The main reason we have created
these datasets is to prove our proposal in the field of computer vision and
with more than 2 levels of depth, indeed CIFAR100 contains only two levels of
depth (Super Class, Classes) and other datasets with many depths find
applicability only in text classification or in bioinformatics, where the
inputs are not images.

Table 1: Accuracies comparison using three different datasets on different learning rate. We use a batch size of 32 on VQA-MED and 64 on the other datasets. | lr=0.005
---|---
Datasets | 1l | 2l | 3l
VQA-Med | 38.05% | 74.04% | 66.66%
Shapes | 100% | 100% | 100%
Animals_Taxonomy8 | 71.98% | 69.07% | 92.82%
| lr=0.001
VQA-Med | 35.98% | 70.20% | 67.84%
Shapes | 100% | 100% | 100%
Animals_Taxonomy8 | 72% | 69.12% | 92.89%
| lr=0.01
VQA-Med | 34.8% | 71.97% | 69.61%
Shapes | 100% | 100% | 100%
Animals_Taxonomy8 | 69.2% | 66.53% | 91.32%
Table 2: Accuracies comparison of our model with a original ResNet on different layers. The batch size used for these experiments is 64. | lr=0.005 |  |
---|---|---|---
Our Model | 1l | 2l | 3l | ResNet18
| 71.98% | - | - | 71.19%
Animals_Taxonomy8 | - | 69.07% | - | 68.58%
| - | - | 92.82% | 90.86%
| lr=0.01 |  |
Our Model | 1l | 2l | 3l | ResNet18
| 69.2% | - | - | 68.34%
Animals_Taxonomy8 | - | 66.53% | - | 65.36%
| - | - | 91.32% | 90.98%

##  6 Conclusion

In literature, multi-label classification is an important field in machine
learning and it is strongly related to many real-world applications for
example, in biomedical images annotation, document categorization and whatever
problem which the instances inside the classes are not disjoint but they keep
a hierarchical structure. In this work, we have conducted four empirical
studies on different datasets to prove by experimental results the
effectiveness and robustness of our proposed model, that can be applied as an
extension to any Convolutional Neural Network.

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