# Chern Numbers of Matroids

Eline Mannino

###### Abstract.

We define Chern numbers of a matroid. These numbers are obtained when
intersecting appropriate matroid Chern-Schwartz-MacPherson cycles defined by
López de Medrano, Rincón, and Shaw. We prove that when a matroid arises from a
complex hyperplane arrangement the Chern numbers of the matroid correspond to
the Chern numbers of the log cotangent bundle.

A matroid $M$ of rank 3 has two Chern numbers. We prove that they are positive
and that their ratio is bounded by 3, which is analogous to the Bogomolov-
Miyaoka-Yau inequality. If the matroid is orientable, we generalize a result
of Eterović, Figuera, and Urzúa to prove that the ratio is bounded above by
5/2. Finally, we give a formula for the Chern numbers of the uniform matroid
of any rank.

##  1\. Introduction

We introduce new matroid invariants, namely their Chern numbers. These are
numbers obtained when intersecting the Chern Schwartz MacPherson cycles of a
matroid (CSM cycles) introduced by López de Medrano, Rincón, and Shaw in
[LdMRS20].

More specifically, to a matroid $M$ of rank $d+1$ Ardila and Klivans assign a
rational polyhedral fan $\Sigma_{M}$ called the Bergman fan of $M$ (Definition
2.4) having a cone for each chain of flats of $M$. Denote the Minkowski
weights supported on $\Sigma_{M}$ by $MW_{\ast}(\Sigma_{M})$, where the
grading is by dimension. Then $MW_{\ast}(\Sigma_{M})$ has the structure of a
graded ring where the product

$MW_{k}(\Sigma_{M})\times MW_{l}(\Sigma_{M})\to MW_{d-k-l}(\Sigma_{M})$

can be defined in several ways, see for example [Sha13], [FR13], and [AHK18].
Moreover, there exists an isomorphism
$w:MW_{0}(\Sigma_{M})\rightarrow\mathbb{Z}$ naturally defined by the weight
assigned to the vertex of the fan.

The CSM cycles of a matroid are collections of Minkowski weights
$\text{csm}_{k}(M)$ in $MW_{k}(M)$ for all $k=0,\dots,d$. The weights are
determined by the beta invariants of matroid minors. Our definition in this
text is the following.

######  Definition 1.1.

Let $M$ be a rank $d+1$ matroid. The Chern numbers of a matroid
$\bar{c}_{1}^{k_{1}}\bar{c}_{2}^{k_{2}}\cdots\bar{c}_{d}^{k_{d}}(M)$ are

$\bar{c}_{1}^{k_{1}}\bar{c}_{2}^{k_{2}}\cdots\bar{c}_{d}^{k_{d}}(M)=\text{w}(\textnormal{csm}_{d-1}^{k_{1}}(M)\textnormal{csm}_{d-2}^{k_{2}}(M)\cdots\textnormal{csm}_{0}^{k_{d}}(M)),$

where $\sum_{i=1}^{d}ik_{i}=d$, and the map w is the map returning the
multiplicity associated to the vertex.

In algebraic geometry, the Chern-Schwartz-MacPherson classes generalize the
Chern classes of the tangent bundle for varieties that are not smooth and
complete see [Mac74] and [BS81]. In [LdMRS20], the authors prove that the CSM
cycles of a matroid representable over the complex numbers are related to CSM
classes in algebraic geometry. Namely, when the matroid $M_{\mathcal{A}}$
arises from an arrangement $\mathcal{A}$ defined over $\mathbb{C}$, the
authors prove that

$\text{CSM}(1_{C(\mathcal{A})})=\sum_{k=0}^{d}\text{csm}_{k}(M_{\mathcal{A}})\in
MW_{*}(\Sigma_{\mathcal{A}}),$

where $\text{CSM}(1_{C(\mathcal{A})})$ is the CSM class of the group of
constructible function on the complement of the arrangement $C(\mathcal{A})$.
Moreover, Aluffi proves that $\text{CSM}(1_{C(\mathcal{A})})$ equals the Chern
class of the sheaf of the log cotangent bundle, see [Alu12]. This allows us to
to relate the Chern numbers of matroids representable over $\mathbb{C}$ to
Chern numbers in algebraic geometry.

######  Theorem 1.2.

Let $\mathcal{A}\subset\mathbb{P}^{d}_{\mathbb{C}}$ be a hyperplane
arrangement and let $M_{\mathcal{A}}$ be the corresponding matroid. Let
$W_{\mathcal{A}}$ be the maximal wonderful compactification of its complement
$C(\mathcal{A})$ and let $D=W_{\mathcal{A}}\setminus C(\mathcal{A})$. Let
$\Omega^{1}_{W_{\mathcal{A}}}(\log D)$ denote the bundle of log differentials
on $W_{\mathcal{A}}$ with poles along $D$ and
$\Omega^{1}_{W_{\mathcal{A}}}(\log D)^{*}$ denote its dual. Then

$c_{1}(\Omega^{1}_{W_{\mathcal{A}}}(\log D)^{*})^{k_{1}}\cdots
c_{d}(\Omega^{1}_{W_{\mathcal{A}}}(\log
D)^{*})^{k_{d}}=\bar{c}_{1}^{k_{1}}\cdots\bar{c}_{d}^{k_{d}}(M_{\mathcal{A}}).$

Next, for a few particular matroids we find an explicit formula for the Chern
numbers. For example, in Proposition 2.9, we give a formula for computing the
Chern numbers of the uniform matroid. In Proposition 3.1 we give a formula for
the Chern numbers of simple matroids of rank 3 in terms of the flats. Note
that when a matroid is representable over some field $k$, it arises from a
hyperplane arrangement over $k$. In fact the formula given in Proposition 3.1
corresponds to Eterović, Figuera, and Urzúa definition for the Chern numbers
of line arrangements given in terms of the incidence structure [EFU22].

Inspired by the study of the geography of manifolds, see for example [Hun89],
we then launch a study of the possible Chern numbers of matroids. Focusing
specifically on rank $3$, we consider the following question.

######  Question 1.3 (Geography of Chern numbers in rank 3).

For which pairs $(a,b)\in\mathbb{Z}_{\geq 0}\times\mathbb{Z}_{\geq 0}$ does
there exist a rank $3$ matroid such that

$(\overline{c}_{1}^{2}(M),\overline{c}_{2}(M))=(a,b)?$

We then establish some first results in the geography problem for matroids.

######  Proposition 1.4.

Let $M$ be a simple matroid of rank $3$ on a ground set $E$ of size $n$. The
Chern numbers $\overline{c}_{1}^{2}(M)$ and $\overline{c}_{2}(M)$ are bounded
by

$\displaystyle 0\leq\overline{c}_{1}^{2}(M)$
$\displaystyle\leq\overline{c}_{1}^{2}(U_{3,n}),\ \textup{and}$ $\displaystyle
0\leq\overline{c}_{2}(M)$ $\displaystyle\leq\overline{c}_{2}(U_{3,n}).$

We also prove upper and lower bounds on the quotient of the two Chern numbers
of rank $3$ matroids. The next theorem is a generalization of Proposition 3.4
in [EFU22] which proves the same bounds for the quotient of Chern numbers of
line arrangements.

######  Theorem 1.5.

Let $M$ be a simple matroid of rank $3$ on a ground set $E$ of size n with no
coloops. Then,

$\displaystyle\frac{2n-6}{n-2}\leq\frac{\overline{c}_{1}^{2}(M)}{\overline{c}_{2}(M)}\leq
3.$

Left equality holds if and only if M is the uniform matroid $U_{3,n}$, and
right equality holds if and only if M is the matroid of a finite projective
plane.

Curiously, the right inequality is the same inequality for the quotient of the
Chern numbers of complex algebraic surfaces given some restrictions, which is
known as the Bogomolov-Miyaoka-Yau inequality. Note that, also for the case of
representable matroids, the inequality in 1.5 is a different inequality than
the Bogomolov-Miyaoka-Yau inequality. While we are calculating Chern classes
of bundles of log differentials (see 1.2), they are calculating Chern classes
of the complex tangent bundles. Later on, Miyaoka stated an analogy of the
Bogomolov-Miyaoka-Yau inequality in a more general context that can be
restricted to our case for $X=W_{\mathcal{A}}$, see Corollary 1.2 in [Miy84].

Line arrangements appear in Hirzebruch’s study of the Bogomolov-Miyaoka-Yau
inequality. To a line arrangement $\mathcal{A}\in\mathbb{P}^{2}_{\mathbb{C}}$
and a $n\geq 2$, Hirzebruch associates a smooth algebraic surface $Y$ by
desingularizing a covering of $\mathbb{P}^{2}_{\mathbb{C}}$ of degree
depending on $n$, see [Hir83] for details. Hirzebruch proves that the right
inequality holds for the quotient of the Chern numbers of $Y$, see the theorem
in Section 2.5 of [Hir83]. Moreover, Sommes proves that the left inequality
holds for surfaces constructed in the same manner as Hirzebruch, see Theorem
5.1 in [Som84].

While Hirzebruch gives, for a few specific line arrangements over
$\mathbb{C}$, a surface covering the line arrangement having Chern numbers
quotient obtaining 3, we prove that the upper bound of $3$ in Theorem 1.5 is
obtained only for matroids arising from finite projective planes.

Before introducing the next theorem, we need some background. A simple closed
curve $L$ in $\mathbb{P}^{2}_{\mathbb{R}}$ is called a pseudoline if
$\mathbb{P}^{2}\setminus L$ has one connected component. Moreover, Folkman and
Lawrence have proved that there is a one to one correspondence between
arrangements of pseudolines and oriented matroids [FL78]. The next theorem
about the quotient of the Chern numbers is a generalization of Theorem 3.5 in
[EFU22] which is stated in terms of real line arrangements.

######  Theorem 1.6.

If $M$ is a simple oriented matroid of rank 3 with no coloops, then

$\displaystyle\frac{\overline{c}_{1}^{2}(M)}{\overline{c}_{2}(M)}\leq\frac{5}{2}.$

Equality is achieved if and only if the pseudoline arrangement of M is
simplicial.

In the following question we further relate the study of Chern numbers of
matroids to geometry.

######  Question 1.7 (Representability of Chern numbers in rank 3).

Given any simple matroid $M$ of rank 3 with Chern numbers
$(\overline{c}_{1}^{2}(M),\overline{c}_{2}(M))$, does there exist a
representable matroid $M_{\mathcal{A}}$ over some field $k$ of rank $3$ such
that

$(\overline{c}_{1}^{2}(M_{\mathcal{A}}),\overline{c}_{2}(M_{\mathcal{A}}))=(\overline{c}_{1}^{2}(M),\overline{c}_{2}(M))?$

Finally, we want to point out some possible implications of answers to this
question. In 1.5 we will see that for a simple matroid of rank 3 the quotient
$\overline{c}_{1}^{2}(M)/\overline{c}_{2}(M)=3$ if and only if $M$ is the
matroid of a finite projective plane. Moreover, in Example 3.3 we show that
the finite projective plane matroid $PG(2,q)$ of order $q$ has Chern numbers

$\displaystyle\overline{c}_{1}^{2}(PG(2,q))=$ $\displaystyle
3(q^{3}-q^{2}-q+1),$ $\displaystyle\overline{c}_{2}(PG(2,q))=$ $\displaystyle
q^{3}-q^{2}-q+1.$

Hence, finding out for which values of $q$ there exists a matroid $M$ having
Chern numbers
$(\overline{c}_{1}^{2}(M),\overline{c}_{2}(M))=(3(q^{3}-q^{2}-q+1),q^{3}-q^{2}-q+1)$,
and if so if the matroid is representable over a field, would in fact solve
the prime power conjecture for finite projective planes ([VB06]).

## Acknowledgements

This paper is based on my master thesis, see [Man22]. Firstly, I want to thank
my supervisor Kris Shaw who inspired and encouraged me. I also want to thank
Felipe Rincón and John Christian Ottem for carefully reading and commenting on
my thesis. I want to thank Edvard Aksnes and Jon Pål Hamre for insightful
discussions. I want to thank Ragni Piene for noticing a significant mistake in
the bibliography. Finally, I want to thank Matt Wendell Larson for pointing me
to the more general case of the the Bogomolov-Miyaoka-Yau inequality and Piotr
Pokora for pointing me to previous results regarding pseudoline arrangements.

I acknowledge the support of the Centre for Advanced Study (CAS) in Oslo,
Norway, which funded and hosted the Young CAS research project Real Structures
in Discrete, Algebraic, Symplectic, and Tropical Geometries during the
2021/2022 and 2022/2023 academic years. This research was also supported by
the Trond Mohn Foundation project “Algebraic and Topological Cycles in Complex
and Tropical Geometries”.

##  2\. Chern numbers of matroids

We begin by introducing the necessary background.

######  Definition 2.1.

A matroid on a finite set $E$ of rank $d+1$ is a function

$r:2^{E}\rightarrow\mathbb{Z}$

satisfying

  1. 1.

$0\leq r(S)\leq|S|$,

  2. 2.

$S\subseteq U$ implies $r(S)\leq r(U),$

  3. 3.

$r(S\cup U)+r(S\cap U)\leq r(S)+r(U)$ and

  4. 4.

$r(E)=d+1.$

######  Example 2.2.

Let $L\subseteq\mathbb{P}^{n}_{k}$ be a linear subspace over a field $k$. And
let

$\mathcal{A}=\\{H_{i}=L\cap V(x_{i})\ |\ 0\leq i\leq n-1\\}\subseteq L$

be the hyperplane arrangement given by intersecting $L$ with the coordinate
axis $V(x_{i})\subseteq\mathbb{P}^{n}_{k}$. We can define a function $r$ on
the power set of $\mathcal{A}$: given a subset $S\subseteq\mathcal{A}$ let

$r(S)=\text{codimension}\Big{(}\bigcap_{i\in S}H_{i}\Big{)}$

in $L$. The function $r$ is a matroid. ∎

Given a hyperplane arrangement $\mathcal{A}$ we denote by $M_{\mathcal{A}}$
the corresponding matroid.

A subset $S\subseteq E$ such that for any element $j$ of $E$ not in $S$ the
rank

$r(S\cup\\{j\\})>r(S)$

is called a flat. The flats of the matroid form a lattice under inclusion
called the lattice of flats of the matroid. A loop of a matroid is an element
$i\in E$ such that $r(i)=0$. A pair of parallel points consists of points
$i,j\in E$ such that $r(i,j)=1$. In most of our results, we assume that the
matroid is simple, i.e., having neither loops nor parallel points.

######  Example 2.3.

The _uniform matroid_ $U_{d+1,n}$ of rank $d+1$ on the ground set $E$ of size
$n$ is defined to be the rank function $r$ such that for $S\subseteq E$:

$r(S):=\begin{cases}|S|\quad\text{if }|S|<d+1\\\
d+1\quad\text{otherwise}.\end{cases}$ (2.1)

Hence, for the uniform matroid any subset $S$ of size $|S|\leq d$ is a flat. ∎

The lattice of flats of a matroid $M$ can be represented by a polyhedral fan
called its Bergman fan, introduced in [AK06], spanned by rays corresponding to
the flats of the matroid. More precisely, let the vectors $e_{i}$ for $i\in E$
be the standard basis of $\mathbb{Z}^{E}$, and let $N$ be the quotient space
$\mathbb{Z}^{E}/\mathbb{Z}\mathbf{1}$ where the vector $\mathbf{1}=\sum_{i\in
E}e_{i}\in\mathbb{Z}^{E}$. Moreover, let $u_{i}$ be the image of $e_{i}$ in
$N$, and let the vector $u_{S}=\sum_{i\in S}u_{i}$ for a subset $S\subset E$.

######  Definition 2.4 ([AK06, Section 3]).

Let $M$ be a loopless matroid of rank $r=d+1$ on a ground set $E$. The Bergman
fan $\sum_{M}$ is the pure $d$-dimensional polyhedral fan in
$N_{\mathbb{R}}:=N\otimes\mathbb{R}$ consisting of the cones

$\sigma_{\mathcal{F}}:=\text{Cone}(u_{F_{1}},u_{F_{2}},\cdots,u_{F_{k}})\subset
N_{\mathbb{R}}$

for each chain of flats $\mathcal{F}:\emptyset\subsetneq
F_{1}\subsetneq\cdots\subsetneq F_{k}\subsetneq E$ in the lattice of flats of
$M$.

######  Example 2.5.

The Bergman fan of the uniform matroid $U_{3,4}$ is a polyhedral fan of
dimension 2 living in $\mathbb{Z}^{4}/\mathbb{Z}\mathbf{1}\otimes\mathbb{R}$,
see Figure 1. The fan consists of 12 cones, corresponding to the flags
$\emptyset\subsetneq\\{i\\}\subsetneq\\{i,j\\}\subsetneq E$, for each pair
$i\neq j\in E$.

Figure 1. The Bergman fan of $U_{3,4}$

∎

Given a subset $S\subseteq E$, the deletion matroid $M\setminus S$ is the
matroid with ground set $E\setminus S$ and rank function $r_{M\setminus S}$
given by for a subset $U\subseteq E\setminus S$:

$r_{M\setminus S}(U)=r(U\cup S)-r(S).$

For a subset $T\subset E$ the restriction matroid $M|T$ is given by
$M\setminus(E\setminus T)$. Finally, for a subset $X\subseteq E$, the
_contraction_ $E/X$ is defined to be the matroid on the ground set $E\setminus
X$ and rank function $r_{M/X}$ given by for a subset $S\subseteq E\setminus
X$:

$r_{M/X}(S)=r(S\cup X)-r(X).$

Let $\beta(M)$ be the beta invariant of $M$, see [Cra67].

######  Definition 2.6 ([LdMRS20, Definition 4]).

Let $M$ be a loopless matroid, the $k$-dimensional CSM cycle
$\text{csm}_{k}(M)$ of $M$ is a $k$-dimensional Minkowski weight with weigths
given by

$\omega(\sigma_{\mathcal{F}}):=(-1)^{d-k}\prod^{k}_{i=0}\beta(M|F_{i+1}/F_{i}),$

for a flag $\mathcal{F}:=\\{\emptyset=F_{0}\subsetneq
F_{1}\subsetneq\cdots\subsetneq F_{k}\subsetneq F_{k+1}=\\{1,...,n\\}\\}$. The
minor $M|F_{i+1}/F_{i}$ is obtained by restricting to $F_{i+1}$ and
contracting to $F_{i}$.

######  Example 2.7.

We compute $\text{csm}_{k}(U_{n,d+1})$. For a cone $\sigma_{\mathcal{F}}$
corresponding to a flag $\mathcal{F}:=\emptyset=F_{0}\subseteq
F_{1}\subseteq\cdots\subseteq F_{k}\subseteq F_{k+1}=\\{1,...,n\\}$ such that
$r(F_{i+1})-r(F_{i})=1$ for all flats $F_{i}$ in the flag, the weight is given
by

$\displaystyle\omega(\sigma_{\mathcal{F}})$
$\displaystyle=(-1)^{d-k}\binom{n-k-2}{d-k}.$

Otherwise, the weight of the cone is zero. For example, for a top dimensional
cone $\sigma_{\mathcal{F}}$ the weight is given by:

$\displaystyle\omega(\sigma_{\mathcal{F}})=1.$

See Example 4. of [LdMRS20] for the details. ∎

We recall our main definition. See 1.1

######  Example 2.8.

Let $M$ be a rank $d+1$ matroid. The Chern number

$\overline{c}_{d}(M)=(-1)^{d}\beta(M),$

which is simply the weight of the vertex of the Bergman fan of the matroid
$M$. ∎

######  Proposition 2.9.

The Chern numbers of the uniform matroid $U_{d+1,n}$ are given by

$\displaystyle\bar{c}_{1}^{k_{1}}\cdots\bar{c}_{d}^{k_{d}}(U_{d+1,n})=(-1)^{d}\prod_{i=1}^{d}\binom{n-(d-i)-2}{i}^{k_{i}}.$

###### Proof.

Let the fan $C\in MW_{*}(\Sigma_{U_{d,n}})$ be the Bergman fan of $U_{d,n}$
with weight 1 on all its top dimensional cones. Then, by Example 15 in
[LdMRS20] and Example 2.8, the fan $C^{k}\in MW_{*}(\Sigma_{U_{d-k+1,n}})$ is
the Bergman fan of $U_{d-k+1,n}$ with weight 1 on all its top dimensional
cones. Hence, the Chern numbers of $U_{d+1,n}$ are given by

$\displaystyle\bar{c}_{1}^{k_{1}}\cdots\bar{c}_{d}^{k_{d}}(U_{d+1,n})=$
$\displaystyle\text{wt}_{0}\Bigg{(}\prod_{i=1}^{d}(-1)^{k_{i}(d-(d-i))}\binom{n-(d-i)-2}{d-(d-i)}^{k_{i}}C^{ik_{i}}\Bigg{)}$
$\displaystyle=$
$\displaystyle(-1)^{d}\prod_{i=1}^{d}\binom{n-(d-i)-2}{i}^{k_{i}},$

where the last equality follows from the fact that the fan $C^{d}$ is just the
vertex of the fan with weight 1, and from the fact that
$\sum_{i=1}^{d}ik_{i}=d$. ∎

When the matroid is representable over $\mathbb{C}$ the Chern numbers of the
matroid have in fact a geometric meaning. See 1.2

###### Proof.

Feichtner and Yuzvinsky have proven that for a representable matroid we have
the follwoing isomorphism $A^{*}(M_{\mathcal{A}})\simeq
A^{*}(W_{\mathcal{A}})$ [FY04]. Moreover, by Poincaré duality for the
wonderful compactification, the following map is an isomorphism of rings:

$[W_{\mathcal{A}}]\cap\colon A^{\ast}(W_{\mathcal{A}})\to
A_{d-\ast}(W_{\mathcal{A}}),$

and it respects the graded structures as indicated. This isomorphism is
compatible with the degree maps $A^{d}(W_{\mathcal{A}})\to\mathbb{Z}$ and
$A_{0}(W_{\mathcal{A}})\to\mathbb{Z}$, see Definition 5.9 in [AHK18] for a
construction of the degree map. inline]You could cite AHK here but since this
is the alg geo setting it is more standard.

By Theorem 1.1 of [Alu12]

$[W_{\mathcal{A}}]\cap\text{c}(\Omega^{1}_{W_{\mathcal{A}}}(\text{log}D)^{*})=\text{CSM}(1_{C(\mathcal{A})}).$

Therefore,

$[W_{\mathcal{A}}]\cap\text{c}_{k}(\Omega^{1}_{W_{\mathcal{A}}}(\text{log}D)^{*})=\text{CSM}_{d-k}(1_{C(\mathcal{A})})\in
A_{d-k}(W_{\mathcal{A}}).$

Furthermore, the isomorphism $A_{\ast}(W_{\mathcal{A}})\simeq
MW_{\ast}(\Sigma_{\mathcal{A}})$ follows by the Poincaré duality for the
wonderful compactification and the linear duality

$MW_{k}(\Sigma_{\mathcal{A}})\simeq
Hom_{\mathbb{Z}}(A^{k}(W_{\mathcal{A}}),\mathbb{Z})$

proven in Proposition 5.6 in [AHK18]. Moreover, by Theorem 3.1 of [LdMRS20],
we have

$\text{CSM}(1_{C(\mathcal{A}}))=\sum^{d}_{k=0}\text{csm}_{k}(M_{\mathcal{A}})\in
A_{*}(W_{\mathcal{A}})\simeq MW_{*}(\Sigma_{\mathcal{A}}).$

Finally, combining these two results we have that

$[W_{\mathcal{A}}]\cap\text{c}_{k}(\Omega^{1}_{W_{\mathcal{A}}}(\text{log}D)^{*})=\text{csm}_{d-k}(M_{\mathcal{A}}).$

Since the cap product with $[W_{\mathcal{A}}]$ induces a ring isomorphism
compatible with the degree maps we obtain the equality in the statement of the
theorem. ∎

##  3\. Chern numbers of matroids of rank 3

In this section we investigate properties of Chern numbers of matroids of rank
3. We first give a formula for the Chern numbers in terms of the number of
flats. This formula generalizes the formula of Chern numbers of line
arrangements defined and studied in [EFU22].

######  Proposition 3.1.

Let $M$ be a simple matroid of rank $3$ on a ground set $E$ of size $n$. Let
$t_{m}$ be the number of flats $F$ of rank $2$ of size $m$, then the Chern
numbers of $M$ are

$\displaystyle\overline{c}_{1}^{2}(M)=9-5n+\sum_{m\geq 2}(3m-4)t_{m},$
$\displaystyle\overline{c}_{2}(M)=3-2n+\sum_{m\geq 2}(m-1)t_{m}.$

###### Proof.

For the first statement, we find that the weight associated to the self
intersection of the 1-dimensional csm cycle of the Bergman fan of $M$ is

$\overline{c}_{1}^{2}(M)=(3-n)^{2}-\sum_{m\geq 2}(2-m)^{2}t_{m},$ (3.1)

by Definition 2.16 and Equation 3.2 in [Sha15]. Moreover, we prove that
Equation 3.1 equals the first statement of Proposition 3.1 by proving that

$n^{2}-n=\sum_{m\geq 2}(m^{2}-m)t_{m}$ (3.2)

by induction on $n$. For $n=3$ the matroid consists of 3 flats of rank 2 of
size 2, hence Equation 3.2 holds. Assume now that Equation 3.2 holds for any
matroid $M$ of size $|E|=n$. Then, adding an element to the matroid $M$
increases the left hand side by $2n$. For the right hand side, we first denote
by $M^{\prime}$ the matroid $M$ with one element $i$ added. Note that the rank
2 flats of $M^{\prime}$ consist of those not containing $i$, which are also
rank 2 flats of $M$, and those containing $i$. Now, let $\mathcal{F}_{i}$ be
the set of rank 2 flats of $M^{\prime}$ containing $i$. The set
$\mathcal{F}_{i}$ equals the disjoint union
$\mathcal{F}_{i}=\mathcal{F}_{1}\cup\mathcal{F}_{2}$, where $\mathcal{F}_{1}$
consists of the flats $F\in\mathcal{F}_{i}$ such that $F\setminus\\{i\\}$ is a
rank 2 flat of $M$, and $\mathcal{F}_{2}$ consists of the flats
$F\in\mathcal{F}_{i}$ such that $F\setminus\\{i\\}$ is a flat of rank 1 in
$M$, hence flats of size 2. Remark that, by the covering axiom of flats, the
size $|\mathcal{F}_{2}|=n-\sum_{F\in\mathcal{F}_{1}}(|F|-1)$. Then, adding an
element $i$ to the matroid $M$ changes the right hand side of Equation 3.2 by

$\displaystyle\sum_{F\in\mathcal{F}_{1}}(|F|^{2}-|F|)-\sum_{F\in\mathcal{F}_{1}}((|F|-1)^{2}-(|F|-1))$
$\displaystyle+2(n-\sum_{F\in\mathcal{F}_{1}}(|F|-1))$ $\displaystyle=2n.$

This proves Equation 3.2, hence the first statement in Proposition 3.1.

Recall that by Example 2.8 the Chern number

$\overline{c}_{d}(M)=(-1)^{d}\beta(M).$

Hence, for proving the second statement it suffices to compute $\beta(M)$. The
characteristic polynomial is given by

$\displaystyle\chi_{M}(\lambda)=\lambda^{3}-n\lambda^{2}+\sum_{m\geq
2}(m-1)t_{m}\lambda-(1-n+\sum_{m\geq 2}(m-1)t_{m}),$

where we have used that for flats of rank 2 we have that

$\sum_{r(F)=2}\mu(\emptyset,F)=\sum_{m\geq 2}(m-1)t_{m}.$

Then, the beta invariant is given by

$\displaystyle\overline{\chi_{M}}(1)=3-2n+\sum_{m\geq 2}(m-1)t_{m},$

which equals the Chern number $\overline{c}_{2}(M)$. ∎

######  Remark 3.2.

Recall that from a line arrangement, which is uniquely defined by a surface
$Y$, we obtain a matroid via Example 2.2. Recall that in [EFU22] the authors
give a combinatorial definition of Chern numbers of line arrangements in terms
of the incidence structure. Their definition is inspired from the work of
Hirzebruch and they claim that these numbers are the Chern numbers of the
corresponding log surface, see their Remark 3.2. Therefore, in the case of
representable matroids Proposition 3.1 follows from 1.2 and Remark 3.2 of
[EFU22].

Matroids arising from finite projective planes $PG(2,q)$ have particularly
nice properties.

######  Example 3.3.

The finite projective plane matroid $PG(2,q)$ is a matroid on a ground set $E$
of size $q^{2}+q+1$ having $q^{2}+q+1$ flats of rank 2 of size $q+1$. Hence,
its Chern numbers are given by

$\displaystyle\overline{c}_{1}^{2}(PG(2,q))$
$\displaystyle=3(q^{3}-q^{2}-q+1),$ $\displaystyle\overline{c}_{2}(PG(2,q))$
$\displaystyle=q^{3}-q^{2}-q+1.$

For example, the Fano plane $PG(2,2)$ has Chern numbers

$\displaystyle\overline{c}_{1}^{2}(PG(2,2))$ $\displaystyle=9,$
$\displaystyle\overline{c}_{2}(PG(2,2))$ $\displaystyle=3.$

∎

######  Example 3.4.

The non-Pappus matroid is obtained from the pseudoline arrangement in Figure
3. We compute the Chern numbers of the non-Pappus matroid:
$(\overline{c}_{1}^{2}(M),\overline{c}_{2}(M))=(28,13)$. Note, that the non-
Pappus matroid violates Pappus’s Theorem, hence it is not representable over
any field. However, there exists a representable matroid having the same Chern
numbers which is given by the line arrangement in Figure 3.

Figure 2.

Figure 3.

We now present results on the bounds of Chern numbers of simple matroids of
rank 3.

######  Proposition 3.5.

Let M be a simple matroid of rank $3$ on the ground set $E=\\{1,\ldots,n\\}$.
The Chern numbers are bounded by

$\displaystyle 0\leq$ $\displaystyle\overline{c}_{1}^{2}(M)$ $\displaystyle
0\leq$ $\displaystyle\overline{c}_{2}(M).$

Left inequality holds if and only if $M$ has a coloop.

###### Proof.

Recall that the Chern number $\overline{c}_{2}(M)=\beta(M)$. Moreover, the
beta invariant $\beta({M})\geq 0$ and $\beta({M})=0$ if and only if $M$ is
disconnected or a loop, see [Cra67]. Finally, note that $M$ disconnected
implies that $M$ has a coloop.

Now, we prove by induction on $n$, that $\overline{c}_{1}^{2}(M)$ is positive
if $M$ has no coloops. Assume that $n=4$ and recall that $t_{m}$ denotes the
number of flats of rank $2$ of size $m$. Then $t_{n}=t_{n-1}=0$ implies
$t_{2}=6$; so the Chern number of the matroid is $\overline{c}_{1}^{2}(M)=1$.
Assume now that $M$ is a matroid on a ground set $|E|=n\geq 5$, and let $i\in
E$ be an element contained in at least three flats of rank 2. Note that such
an element must exist by the assumption of no coloops.

Let $\mathcal{F}_{i}=\\{F_{1}\ldots F_{t}\\}$ be the rank 2 flats of $M$
containing $i$, and denote by $M\setminus i$ the deletion matroid. We
partition $\mathcal{F}_{i}$ as before, i.e., we let the set
$\mathcal{F}_{i}=\mathcal{F}_{1}\cup\mathcal{F}_{2}$, where $\mathcal{F}_{1}$
consists of flats $F$ such that $F\setminus i$ is a flat of rank 2 of
$M\setminus i$, and $\mathcal{F}_{2}$ consists of flats $F$ of rank 2 such
that $F\setminus i$ is a flat of rank 1 of $M\setminus i$. Recall also that
the size of $\mathcal{F}_{2}$ is
$|\mathcal{F}_{2}|=n-\sum_{F\in\mathcal{F}_{1}}(|F|-1)$. Then, the Chern
number $\overline{c}_{1}^{2}(M)$ is given by

$\displaystyle\overline{c}_{1}^{2}(M)=\ $
$\displaystyle\overline{c}_{1}^{2}(M\setminus
i)-5-\sum_{F\in\mathcal{F}_{1}}(3(|F|-1)-4)+\sum_{F\in\mathcal{F}_{1}}(3|F|-4)$
$\displaystyle+2(n-\sum_{F\in\mathcal{F}_{1}}(|F|-1))$ $\displaystyle=\ $
$\displaystyle\overline{c}_{1}^{2}(M\setminus
i)-5+\sum_{F\in\mathcal{F}_{1}}1+2(\sum_{F\in\mathcal{F}_{1}}1+n-\sum_{F\in\mathcal{F}_{1}}(|F|-1))$
$\displaystyle\geq\ $ $\displaystyle\overline{c}_{1}^{2}(M\setminus i)-5+2t$
$\displaystyle\geq\ $ $\displaystyle\overline{c}_{1}^{2}(M\setminus i)+1$
$\displaystyle>\ $ $\displaystyle 1.$

The last inequality follows from the induction hypothesis, i.e., that the
Chern number $\overline{c}_{1}^{2}(M\setminus i)\geq 0$.

Finally, if $M$ has a coloop, then $M$ has $t_{2}=n-1$ and $t_{n-1}=1$,
whereas $t_{k}=0$ for all other $k$’s. By inserting these values for
$c_{1}^{2}(M)$, we get that $c_{1}^{2}(M)=0.$ ∎

We also find an upper bound for the Chern numbers of simple matroids of rank
3. See 1.4

###### Proof.

Left inequality is proven in Proposition 3.5. For the right inequality we
begin with the first statement. Recall, by Equation 3.1, that the Chern number
$\overline{c}_{1}^{2}(M)$ of an arbitrary matroid is given by

$\displaystyle\overline{c}_{1}^{2}(M)=(3-n)^{2}-\sum_{m\geq 2}(2-m)^{2}t_{m},$

whereas the Chern number $\overline{c}_{1}^{2}(U_{3,n})$ of the uniform
matroid $U_{3,n}$ is given by

$\displaystyle\overline{c}_{1}^{2}(U_{3,n})=(3-n)^{2}.$

Then, since both the Chern number $\overline{c}_{1}^{2}(M)$ and the sum
$\sum_{m\geq 2}(2-m)^{2}t_{m}$ of an arbitrary matroid are positive, we get
the following inequality

$\overline{c}_{1}^{2}(M)\leq(3-n)^{2}=\overline{c}_{1}^{2}(U_{3,n}).$

The second statement follows from the inequality $\beta(M)\leq\beta(U_{d,n})$
when $M$ is of rank $\text{r}(M)=d$ on a ground set $|E|=n$. This is due to
Brylawski, see Corollary 7.14 in [Bry72]. ∎

We conjecture that a generalization of Proposition 1.4 holds for a matroid of
any rank.

######  Conjecture 3.6.

Let $M$ be a simple matroid of rank $d+1$ on a ground set $E$ of size $n$.
Then the Chern numbers of $M$ are bounded by

$|\bar{c}_{1}^{k_{1}}\cdots\bar{c}_{d}^{k_{d}}(M)|\leq|\bar{c}_{1}^{k_{1}}\cdots\bar{c}_{d}^{k_{d}}(U_{d+1,n})|$

with equality if and only if $M=U_{3,n}$.

######  Remark 3.7.

In algebraic geometry, the volume polynomial measures the top self-
intersection number of divisors on a smooth projective variety. Eur introduces
an analogous volume polynomial for matroids that agrees with the classical
volume polynomial of the wonderful compactification when the matroid is
representable [Eur20]. Moreover, he introduces a new invariant called the
shifted rank-volume of a matroid $\text{shRVol}(M)$ as a particular
specialization of its volume polynomial and proves that for representabe
matroids $\text{shRVol}(M)\leq\text{shRVol}(U_{d,n})$ with equality if and
only if $M=U_{d,n}$. Recall, that
$\overline{c}_{1}^{d}(M)=w(\text{csm}_{d-1}(M))$ and that by the following
isomorphism $MW_{*}(\Sigma_{M})\simeq A^{*}(M)$ the CSM-cycle
$\text{csm}_{d-1}(M)$ corresponds to a divisor $ch_{1}\in A^{1}(M)$. Recently,
Rincón and Fife have proven a formula for the CSM-cycles of a matroid in the
Chow ring of the matroid [FR22]. We can use their formula to show that
$ch_{1}(M)$ and Eur’s Shifted rank divisor $D_{M}$, which is used to compute
$\text{shRVol}(M)$, only differ by a term, see Proposition 5.4.8 in [Man22].
Moreover, for a simple matroid $M$ of rank 3, we also prove that
$\text{shRVol}(M)$ and $c_{1}^{2}(M)$ only differ by a function of $n$, see
Lemma 6.4.2. in [Man22].

We continue with our results on the quotient of the Chern numbers. The next
result, generalizes Proposition 3.4 in [EFU22] which gives bounds for the
quotient of Chern numbers of line arrangements. Moreover, these bounds are the
matroid analogue to the bound for complex projective surfaces. See 1.5

###### Proof.

Proving the left inequality is equivalent to showing the inequality below

$\displaystyle 0$
$\displaystyle\leq(n-2)\overline{c}_{1}^{2}(M)-(2n-6)\overline{c}_{2}(M).$
(3.3)

Recall from the proof of Proposition 3.1 that the equality

$\displaystyle-n^{2}+n=\sum_{m\geq 2}(m-m^{2})t_{m}$

holds. Then, by inserting for $\overline{c}_{1}^{2}(M)$, and
$\overline{c}_{2}(M)$ in the right hand side of Inequality 3.3 we get the
following equality

$\displaystyle(n-2)\overline{c}_{1}^{2}(M)-(2n-6)\overline{c}_{2}(M)$
$\displaystyle=-n^{2}+n+\sum_{m\geq 2}(mn-2n+2)t_{m}$
$\displaystyle=\sum_{m\geq 2}(-m^{2}+m)t_{m}+\sum_{m\geq 2}(mn-2n+2)t_{m}$
$\displaystyle=\sum_{m\geq 2}(-m^{2}+m(1+n)+(2-2n))t_{m}.$

Moreover, the inequality $-m^{2}+m(1+n)+(2-2n)\geq 0$ holds for the values
$2\leq m\leq n-1$ and the inequality $-m^{2}+m(1+n)+(2-2n))>0$ holds for all
$3\leq m\leq n-2$. And since $m$ is less then $n-1$ by our assumption,
Inequality 3.3 holds. The left equality in the theorem holds if and only if
$M$ has $n(n-1)/2$ flats of rank 2 of size 2, hence if and only if $M$ is the
uniform matroid $U_{3,n}$.

Proving the right inequality is equivalent to showing the following inequality

$\overline{c}_{1}^{2}(M)-3\overline{c}_{2}(M)=n-\sum_{m\geq 2}t_{m}\leq 0.$

The above inequality holds by Theorem 1 in [DW75] which states that for a
finite geometric lattice of rank 3 the following inequality $W_{1}\leq W_{2}$
holds, where $W_{1}$ is the number of lattice elements of rank 1 and $W_{2}$
is the number of lattice elements of rank 2. Moreover, right equality holds if
and only if the matroid arises from a finite projective plane $PG(2,q)$, see
Theorem 1 in [dBE48]. ∎

The next theorem about the quotient of the Chern numbers is a generalization
of Theorem 3.5 in [EFU22] which is stated in terms of Chern numbers of real
line arrangements. See 1.6

###### Proof.

Note first that

$\displaystyle 5\overline{c}_{2}(M)-2\overline{c}_{1}^{2}(M)=-3-\sum_{m\geq
2}(m-3)t_{m}.$

A pseudoline arrangement partitions $\mathbb{P}^{2}_{\mathbb{R}}$ into
polygons and let $p_{m}$ be the number of $m$-gons. As in the proof of Theorem
3.5 in [EFU22], we refer to p.115 in [Hir83] for the following statement. By
the Euler characteristic formula, and by using that the Euler characteristic
of $\mathbb{P}^{2}_{\mathbb{R}}$ is 1 one attains the following equality

$\sum_{m\geq 2}(m-3)p_{m}=-3-\sum_{m\geq 2}(m-3)t_{m}.$

Since every two pseudolines intersect in exactly one point, and all the
pseudolines do not intersect in the same point, the multiplicity $p_{2}=0$.
Hence, the sum $\sum_{m\geq 2}(m-3)p_{m}\geq 0$ and the original inequality
holds. Moreover, if the pseudoline arrangement $\mathcal{A}$ is simplicial,
the multiplicity $p_{k}=0$ for all $k\neq 3$. Hence, equality is achieved if
and only if the matroid $M$ arises from a simplicial pseudoline arrangement.

In fact, the inequality $-3-\sum_{m\geq 2}(m-3)t_{m}\geq 0$ for pseudoline
arrangements was first proven by Melchior in 1940 and published in the
unfavorable journal Deutsche Matematik [AMS], which we choose not to cite
here. ∎

In the table of Figure 4, we have listed the values of the Chern numbers
$\overline{c}_{1}^{2}(M)$, $\overline{c}_{2}(M)$ for some specific matroids
and their corresponding quotient. Note that we have also included the non-Fano
matroid, which arises from relaxing one of the flats of rank two of the Fano
matroid, and the matroid arising from the Braid arrangement. In the graph of
Figure 4, we have plotted the Chern number pairs
$(\overline{c}_{2}(M),\overline{c}_{1}^{2}(M))$. The slope of the lines
represents the bound of the quotient of the Chern numbers given in 1.5 and
1.6, which are respectively 3 and $\frac{5}{2}$.

$M$ |  $\overline{c}_{1}^{2}(M)$ |  $\overline{c}_{2}(M)$ |  $\overline{c}_{1}^{2}/\overline{c}_{2}(M)$ |
---|---|---|---|---
$U_{3,3}$ |  0 |  0 |  - |
$U_{3,4}$ |  1 |  1 |  1 |  $A$
$U_{3,5}$ |  4 |  3 |  $\approx 1.33$ |  $B$
$U_{3,7}$ |  16 |  10 |  1.6 |  $C$
$U_{3,9}$ |  36 |  21 |  $\approx 1.71$ |
$PG(2,2)$ |  9 |  3 |  3 |  $D$
$PG(2,4)$ |  135 |  45 |  3 |
$PG(2,8)$ |  1323 |  441 |  3 |
$PG(2,9)$ |  1920 |  640 |  3 |
non-Fano |  10 |  4 |  2.5 |  $E$
Pappus |  27 |  12 |  2.25 |  $F$
non-Pappus |  28 |  13 |  $\approx 2.15$ |  $G$
Braid |  5 |  2 |  2.5 |  H

$c_{1}^{2}(M)$$c_{2}(M)$$A$$B$$C$$D$$E$$F$$G$$H$

Figure 4. Chern numbers of matroids of rank 3.

## References

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Eline Mannino
Department of Mathematics
University of Oslo
0316 Oslo
Norway
<EMAIL_ADDRESS>


# Developing a Scenario-Based Video Game Generation Framework: Preliminary
Results

Elif Surer (1), Mustafa Erkayaoğlu (2), Zeynep Nur Öztürk (3), Furkan Yücel
(4), Emin Alp Bıyık (5), Burak Altan (6), Büşra Şenderin (7), Zeliha Oğuz (8),
Servet Gürer (9), H. Şebnem Düzgün (10)
(1) Department of Modeling and Simulation, Graduate School of Informatics,
Middle East Technical University, Ankara, Turkey
(2) Department of Mining Engineering, Middle East Technical University,
Ankara, Turkey
(3) Department of Computer Engineering, Bilkent University, Ankara, Turkey
(4) Department of Modeling and Simulation, Graduate School of Informatics,
Middle East Technical University, Ankara, Turkey
(5) Department of Modeling and Simulation, Graduate School of Informatics,
Middle East Technical University, Ankara, Turkey
(6) Department of Modeling and Simulation, Graduate School of Informatics,
Middle East Technical University, Ankara, Turkey
(7) Department of Modeling and Simulation, Graduate School of Informatics,
Middle East Technical University, Ankara, Turkey
(8) Department of Psychology, Bilkent University, Ankara, Turkey
(9) Department of Mining Engineering, Middle East Technical University,
Ankara, Turkey
(10) Colorado School of Mines, Brown Hall 268, CO 80401, USA
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Emergency training and planning provide structured curricula, rule-based
action items, and interdisciplinary collaborative entities to imitate and
teach real-life tasks. This rule-based structure enables the curricula to be
transferred into other systematic learning platforms such as serious games
—games that have additional purposes rather than only entertainment. Serious
games aim to educate, cure, and plan several real-life tasks and circumstances
in an interactive, efficient, and user-friendly way. Although emergency
training includes these highly structured and repetitive action responses, a
general framework to map the training scenarios’ actions, roles, and
collaborative structures to game mechanics and game dialogues, is still not
available. To address this issue, in this study, a scenario-based game
generator, which maps domain-oriented tasks to game rules and game mechanics,
was developed. Also, two serious games (i.e., Hospital game and BioGarden
game) addressing the training mechanisms of Chemical, Biological,
Radiological, Nuclear, and Explosives (CBRNe) domain, were developed by both
the game developers and the scenario-based game generator for comparative
analysis. The results show that although the game generator uses higher CPU
time, memory usage, and rendering time, it highly outperforms the game
development pipeline performance of the developers. Thus, this study is an
initial attempt of a game generator which bridges the CBRNe practitioners and
game developers to transform real-life training scenarios into video games
efficiently and quickly.

###### Index Terms:

Serious Games, Game Scenarios, Video Game Generator, CBRNe, Emergency
Training.

##  I Introduction

Serious gaming [1][2], the umbrella term describing the video and board games
having additional goals rather than only entertainment, is widely used in
several domains such as health [3], defense [4][5], and education [6][7].
CBRNe is an acronym for Chemical, Biological, Radiological, Nuclear, and
Explosives, and recent research on this domain focuses on personnel training,
emergency planning and organizing of field, tabletop, simulation, and serious
gaming exercises for preparedness [8].

One of the use cases of serious gaming in emergency planning is on firefighter
training. In a study by Heldal [9], firefighter training was examined by using
serious games and tools. To do so, qualitative questionnaires and observations
on two use cases (i.e., ship evacuation in Baltic Sea and railway accident
with cyanide leakage) were used to analyze the impacts of serious gaming on
non-users. Results showed that serious games would be useful in emergency
training situations, and in-depth training scenarios and evaluation methods
were necessary.

In another study, Lukosch et al. [10] performed the steps of the traditional
design process with the contributions of the end-users. The primary purpose of
the study was to check if the simulations could be used to train situational
awareness skills, and the end-users’ participation simulation demonstrated the
positive impact of using simulations. However, the main limitation of the
study was not having a game-scenario based approach, and future research would
focus on this aspect while creating virtual agents.

The use of virtual reality simulation was also a common topic in the
literature. Ingrassia et al. [11] focused on testing and comparing
performances of 56 medical students during mass casualty triage in real-world
and virtual reality (VR). The results showed that VR and live simulation were
both useful in improving the accomplishments of the medical students.
Ragazzoni et al. [12] also focused on VR training’s medical aspect, where the
objective was to increase the staff safety in life-or-death risks. Hybrid
simulation for infection control and Ebola treatment were also successfully
performed virtually, and the results demonstrated that awareness of the health
personnel increased.

Serious gaming in CBRNe has been a recent topic, and there are some
misinterpretations on the definitions and core concepts, such as the misuse of
the words ‘game’ or ‘simulation’. To overcome these misinterpretations, a pre-
development survey was developed [13] to be used before implementing the
serious games of the European Network Of CBRN TraIning CEnters (eNOTICE)
project [14]. In the pre-development survey, 24 questions were asked to the
practitioners and experts of CBRNe under the following subgroups: 1)
Participant’s video gaming background, 2) Participant’s knowledge on serious
games, and 3) Participant’s expectations on eNOTICE serious games. Results
from 14 CBRNe professionals showed that the majority of the participants were
highly positive on using serious games in CBRNe and provided open concepts,
suggestions, and guidelines to develop serious games for CBRNe domain [15].

In this study, a scenario-based game generator, which maps linear real-life
scenarios to serious games with training objectives, was developed. The main
focus of this study is the CBRNe domain, where the roles, tasks, and goals of
the participants are clearly defined. The main objectives of the games are as
follows: 1) Providing a tool for additional training, 2) Synergy building, and
3) Transporting a different domain and a new concept to the CBRNe community.
Two of the games that were developed by both the scenario-based game generator
and by the game developers were based on the real exercises that were
performed during eNOTICE project’s joint activities in Nimes (France) and
Brussels (Belgium). The comparative analysis regarding CPU usage, rendering,
memory usage, and game development pipeline on the generator-based and
developer-based games was also performed.

##  II Materials and Methods

In this section, details of the scenario-based game generator (Figures 1-3),
two serious games that were developed by both the game designers and the game
generator, and their evaluation are explained in detail.

###  II-1 Scenario and Task Definitions

The theme of the scenario, active players, location, and interaction
mechanisms were collected in advance from the practitioners. Workflow and
state diagrams were used to create a detailed scenario where different roles
and entities communicate with each other. In the beginning, two different
CBRNe scenarios that were based on real practices were mapped to the workflow
and state diagram structures. One of the scenarios is based on the subset of
the BioGarden exercise (i.e., linear version of the scenario in which the
players do not change the flow of the events), which was held in June 2018 in
Belgium as part of eNOTICE joint activities. The other scenario is based on
the Nimes exercise, which was held in France in January 2018 again as part of
eNOTICE joint activities.

Defined scenarios and interaction mechanisms were mapped to game ideas, linear
game stories, and interaction mechanisms. Interaction mechanisms were
converted into concrete tasks and user roles. A generic system, built on top
of the initial scenario definitions, was conceptualized and implemented. Then,
the generated system was fine-tuned with goals, feedback measurements, and
score adaptations.

The scenarios of the exercises were designed by different institutions such as
fire departments, research centers, and hospitals. Thus, breaking the
scenarios down into actions and events was a crucial step so that the game
mechanics, reward mechanisms, and scoring could be systematized. Also,
different roles in the scenarios were assigned to different player types so
that the active role of the player and the role of non-player characters
(NPCs) were defined.

Figure 1: Initial interaction mechanism including Entry and Exit States, a
Stationary Object interacting (i.e., Move, Push, Pull, etc.) and using
stationary objects. Figure 2: In an attack scenario, chasing, attacking and
interacting use cases are modeled. Figure 3: Fire fighting scenario includes
prioritizing the steps, extinguishing the fire and rescuing the affected
people.

Before starting to implement the Hospital game, a detailed survey, which was
briefly mentioned above, was conducted on 14 professionals in CBRNe field —7
of them being game players [15]. The scope and the purposes of the study were
as follows: 1) Learning the user’s gamer profile, 2) Understanding the user’s
perspective on serious games, 3) Retrieving the expectations of the user, 4)
Clarifying the differences between the video/serious games and simulations,
and 5) Asking for suggestions. The initial results of the survey were positive
and encouraging. The participants’ gamer profiles involved playing strategy
games and multiplayer games to learn new skills and to relieve stress. After
the detailed analysis of the results, a tutorial mode was added to the initial
game prototype.

###  II-2 Scenario-Based Game Generator

Scenario-based game generator is developed in combination with Unity
software’s Animator Controller tool and is composed of four different
components: 1) Main Code, 2) Control Code, 3) Transition Code, and 4) User
Interface. Main Code is where the state definitions and structures —the
definitions of the final consequences of the actions— are initialized. Control
Code works as a mechanism to form and map action methods and their related
states. Transition Code is the link where the game generator works with
Unity’s Animator Controller. Finally, additional user interface mechanisms
such as feedback, scores, and health points are added and grouped under the
User Interface component (Figure 4).

While developing the scenario-based game generator tool, the following steps
are executed: 1) Creating an environment, 2) Defining state diagrams, 3)
Creating animations, 4) Adding basic artificial intelligence (AI) to states,
5) Resolving player and NPC interactions and 6) Adding basic AI for
interactions.

In this scenario-based game generator, only linear scenarios, where the
decision making of players do not modify the outcomes of the actions, are
implemented. The scenario-based game generator is used to generate the
duplicates of Hospital and BioGarden games. First, the game scenarios are
tested using simple 3D game objects such as cubes, spheres, and capsules
(Figures 5 and 6). Then, after checking that the scenario works correctly,
initial 3D game objects are automatically replaced with game assets using tag
information of the assets.

Figure 4: Structure of the scenario-based game generator.

###  II-3 Hospital Game

Hospital Game is based on the Nimes scenario, which was performed during the
eNOTICE joint activity in January 2018. The main purpose of this scenario is
training medical staff for CBRNe circumstances. In this game, the player
learns taking security measures such as using gloves, using masks, blocking
the entrance of the hospital, and applying decontamination procedures. Players
can play different roles, such as a doctor, nurse, and secretary. It is based
on a linear scenario, and when the players make wrong choices, they lose game
points (Figures 7 and 8).

This game was developed by a second-year Middle East Technical University
(METU) Multimedia Informatics program student and the same game scenario was
also given to the scenario-based game generator. The initial results of both
environments were compared. In both versions of the games, Quadart’s Hospital
Lowply pack [16], which provides several realistic modular assets, was used.

###  II-4 BioGarden Game

BioGarden game is based on the eNOTICE joint activity, which was played in
Belgium in June 2018. Although it had a nonlinear scenario, only linear parts
of the scenario were implemented so that a comparison with the scenario-based
game generator would be possible. In the scenario, there were different
laboratories with different structures and responsibilities. The role-playing
part was composed of decontamination, role assignment, and evaluation (Figures
9 and 10).

As in the case of the Hospital game, BioGarden game was developed by a second-
year METU Multimedia Informatics program student, and the same game scenario
was also given to the scenario-based game generator, and the initial results
were compared. In both versions of the games, 3LB Games’ Low Poly laboratory
pack [17], which provides several realistic models, textures, and diffuse
maps, was used.

##  III Results

In this section, performance outcomes of the scenario-based game generator and
the two serious games that were developed by the game developers were compared
in terms of CPU usage, rendering time, memory usage, and game development
pipeline. All the tests were performed on a laptop having Intel Core i7 9750HQ
CPU, 16GB RAM, and NVIDIA GeForce GTX 1660TI graphics.

Table I and Table II present the comparison on CPU usage (i.e. game
generator’s output vs. developer-based game), Table III and Table IV on memory
usage, and finally, Table V and Table VI present a comparison using rendering
parameters. The rendering profile used the SetPass Calls, Draw Calls, Total
Batches, Triangles and Vertices as parameters. SetPass parameter is defined as
“the number of rendering passes” [18], a Draw Call as a “call to the graphics
API to draw objects” [18] and Batch as a “package with data that will be sent
to the GPU” [18] on the Unity’s Renderer Profiler page [18].

Figure 5: Initial tests of the scenario-based game generator were performed on simple game objects. Figure 6: Initial tests of the scenario-based game generator were performed on simple game objects such as cubes and capsules. Figure 7: Screenshots from the Hospital game and a demo of Dialogue menu. Figure 8: Screenshots from the Hospital game and a demo of the interaction mechanism. Figure 9: The interior design of the Candestine lab from the BioGarden game and menu interactions. Figure 10: The interior design of the Clandestine lab from the BioGarden game. TABLE I: CPU Usage of the Hospital Game Generated by the Scenario-Based Video Game Generator vs. Hospital Game CPU Usage | Hospital (Generator) | Hospital Game
---|---|---
CPU | 9 ms | 4 ms
TABLE II: CPU Usage of the BioGarden Game Generated by the Scenario-Based Video Game Generator vs. BioGarden Game Memory | BioGarden (Generator) | BioGarden Game
---|---|---
CPU | 22.6 ms | 8.5 ms
TABLE III: Memory Usage of the Hospital Game Generated by the Scenario-Based Video Game Generator vs. Hospital Game Memory | Hospital (Generator) | Hospital Game
---|---|---
Used Total | 1.03 GB | 0.45 GB
Reserved Total | 1.32 GB | 0.63 GB
System Memory Usage | 1.96 GB | 1.38 GB
TABLE IV: Memory Usage of the BioGarden Game Generated by the Scenario-Based Video Game Generator vs. BioGarden Game Memory | BioGarden (Generator) | BioGarden Game
---|---|---
Used Total | 0.62 GB | 0.28 GB
Reserved Total | 0.92 GB | 0.49 GB
System Memory Usage | 1.65 GB | 1.27 GB
TABLE V: Rendering Results of the Hospital Game Generated by the Scenario-Based Video Game Generator vs. Hospital Game Rendering | Hospital (Generator) | Hospital Game
---|---|---
SetPass Calls | 106 | 136
Draw Calls | 252 | 298
Total Batches | 217 | 243
Triangles | 463.9K | 504.9K
Vertices | 319.5K | 361.1K
TABLE VI: Rendering Results of the BioGarden Game Generated by the Scenario-Based Video Game Generator vs. BioGarden Game Rendering | BioGarden (Generator) | BioGarden Game
---|---|---
SetPass Calls | 1826 | 2200
Draw Calls | 2584 | 3007
Total Batches | 2584 | 3007
Triangles | 3.1M | 3.2M
Vertices | 2.3M | 2.4M

It took three weeks to develop and implement the scenario-based game
generator. The most time-consuming part was the state transitions and handling
the outcomes of the actions. After the game generator was built, it took three
and a half hours to generate the Hospital game and four hours to generate the
BioGarden game using the game generator.

The development of the original BioGarden game took 25 days in total: one week
for the scenario clarification and role assignment; one week for the text-
based decision mechanisms, nine days to finish the user interface and menus
and two days to add the assets to the game.

The original Hospital game was first refactored from the initial prototype,
which took one week. Then, it took another two weeks adapting the new
scenario, merging different game modes, and dialogue generation.

##  IV Discussion

In this study, real-life exercise scenarios of two eNOTICE joint activities
were developed by game developers as well as a scenario-based game generator
—developed during this study. All the frameworks used the same scenario-to-
game mechanics mapping. All four versions of the games (i.e., developed by the
game developer vs. generated by the game generator) were compared in terms of
CPU usage, memory usage, rendering, and game development timeline
perspectives. Scenario-based game generator’s CPU usage, memory usage, and
rendering time were higher when compared with the developer-based games. The
reason for the game generator’s higher resource usage was the complex
structure of the scenario-based game generator, tag search, and communication
with the Unity’s Animator Controller.

The rendering performance results of both versions of the games were very
similar because the working principle of the game generator was not dependent
on the visual contents of the games. Although the game generator used higher
memory, CPU, and rendering time, its game development timeline efficiency
highly outperformed the game developers (i.e., four hours vs. three weeks).
This is a highly promising outcome that will enable further exercise scenarios
to be mapped into games in a short period of time. This outcome can benefit
the practitioners in two ways: 1) Visualizing the action-state diagrams of the
exercise so that they can see the flaws or unassigned roles of their
exercises, and 2) Having a rapid game prototype which becomes a fast,
interactive testbed and training tool.

The proposed game generator framework will be extended using state machines so
that nonlinear scenarios can also be generated quickly. Also, use case
scenarios will be adapted to VR environments [19]. The players will interact
with their surroundings in the VR environment, achieve their goals, interact
with other users and receive feedback regarding the success of their outcomes,
which will enable us to build a detailed training environment where training
scenarios can easily be modified and played in two different settings: on
computers and using VR headsets. All the game versions (i.e., developer-based
and game generator-based) will be played by the users, and a comparative
analysis will be performed. Finally, the performance, technology acceptance,
immersion, and usability outcomes of the proposed system will be tested on
participants and practitioners. Besides collecting game-related parameters
such as interaction time and score, standard questionnaires on usability [20]
and technology acceptance model [21] will also be applied to users. This tool
will also be used to develop new prototype games for the future joint
activities of the eNOTICE project till 2022.

##  V Conclusion

In this study, a scenario-based video game generator, which targets the
scenarios in CBRNe domain, was developed. This initial version of the game
generator used linear scenarios that were based on the joint activities of the
eNOTICE project. The effectiveness of the game generator was tested in
comparison with two serious games, which were developed by the game
developers. Even though the performance of the game generator lacked on the
rendering, memory usage, and CPU usage aspects, it highly outperformed the
game development pipeline of the game developers. This is a promising result
that will enable the practitioners to visualize their scenarios while also
generating prototype games rapidly so that the training of CBRNe personnel
will be enriched. This current version of the game generator will be improved
with usability tests, adaptation to VR and feedback of the CBRNe personnel so
that an end-to-end and easy-to-use serious game generator for the CBRNe field
will be provided.

## Acknowledgments

This study was developed during the 15th Summer Workshop on Multimodal
Interfaces (eNTERFACE’19) which was held at Bilkent University, Ankara, Turkey
between the dates of July 8 and August 2, 2019. This framework is fully
supported by European Network Of CBRN TraIning Centers (eNOTICE) project
funded under EU H2020 (Project ID: 740521). The authors would like to thank
Dr. Olga Vybornova (Center for Applied Molecular Technologies, Université
catholique de Louvain, Brussels, Belgium) and Prof. Gilles Dusserre (IMT Mines
Alès, Alès, Languedoc Roussillon, France) for their help in scenarios of
BioGarden and Hospital games, respectively. The authors would also like to
thank Oğuzcan Ergün (Multimedia Informatics, Middle East Technical University,
Ankara, Turkey) for his help and feedback on VR instrumentation.

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|  Elif Surer Elif Surer received her Ph.D in Bioengineering in 2011 from the
University of Bologna. She received her M.Sc. and B.Sc. degrees in Computer
Engineering from Boğaziçi University in 2007 and 2005, respectively. She is
currently working as an Assistant Professor at the METU Graduate School of
Informatics’ Multimedia Informatics program. She is funded by the H2020
project eNOTICE as METU local coordinator as of September 2017 and also
collaborates as a researcher in several interdisciplinary national and EU-
funded projects. Her research interests are serious games, virtual/mixed
reality, reinforcement learning and human and canine movement analysis.
---|---
|  Mustafa Erkayaoğlu Asst. Prof. Erkayaoğlu, holding a PhD degree in mining
engineering and a minor degree in Systems and Industrial Engineering earned
from the University of Arizona, is an expert in business intelligence in the
mining industry with a broad academic and practical experience. He has a
proven track of teaching experience on mining engineering courses and his
expertise also extends to project management, continuous improvement, and
performance measurement skills acquired in the projects he worked as a
Business Intelligence (BI) expert.
---|---
|  Zeynep Nur Öztürk Zeynep Nur Öztürk is studying on her Bachelor’s degree in
Computer Science at Bilkent University. During her Bachelor’s degree, she
studied basics of artificial intelligence and programming games. Her focus is
on game development with Unity, Unreal Engine and Java.
---|---
|  Furkan Yücel Furkan Yücel received his B.Arch. degree in Architecture from
Bilkent University in 2019, where he studied generative algorithms and
computational design in architecture. He is a first-year Master’s student at
METU Multimedia Informatics program.
---|---
|  Emin Alp Bıyık Emin Alp Bıyık received his B.Arch degree in Architecture
from Middle East Technical University (METU) in 2018, where he studied
creative coding and generative design in architecture. He is a first-year
Master’s student METU Multimedia Informatics program. His research interests
are game development, virtual/mixed reality, generative systems and multimedia
arts.
---|---
|  Burak Altan Burak Altan currently studies at METU Graduate School of
Informatics’ Multimedia Informatics program. He received his B.Sc. degree in
Computer Engineering from Başkent University in 2017. He currently works at
Ekin Teknoloji as a software engineer where his work includes front-end web
development, Android development and database management. His research
interests are serious games, virtual/mixed reality, and AAA game development.
---|---
|  Büşra Şenderin Büşra Şenderin is a second-year Master’s student at METU
Multimedia Infromatics program. Her current research focuses on serious games
and virtual reality. She obtained her BS.c degree in Computer Engineering from
Yıldırım Beyazıt University in 2018.
---|---
|  Zeliha Oğuz Zeliha Oğuz is a second-year psychology student at Bilkent
University. She has been working at Bilkent University’s National Magnetic
Resonance Research Center (UMRAM) as a research assistant. The research is
about understanding gene-environment interactions by using neuroimaging,
genetic and psychological analysis and multimedia arts.
---|---
|  Servet Gürer Servet Gürer received his BS.c degree in METU Mining
Engineering Department and continues his Master’s degree in the same
department. After receiving his degree, he worked as a mining engineer in
different companies and then became an occupational safety specialist. Since
2011, he has been working as a labor inspector at the Ministry of Family,
Labor and Social Services. He is a member of TMMOB Chamber of Mining Engineers
and Labor Inspectors Association. He served as the 44th board member of the
Chamber of Mining Engineers. He works in the fields of 3D modeling, animation,
game development, virtual reality.
---|---
|  H. Şebnem Düzgün Dr. H. Şebnem Düzgün is Professor and Fred Banfield
Distinguished Endowed Chair in Mining Engineering at Colorado School of Mines,
Golden, USA. She also holds a joint appointment in the Department of Computer
Science at Mines. Her recent research areas involve quantitative risk and
resilience assessment for mining hazards and geohazards, big data analytics,
Earth observation in geosciences, virtual/augmented/mixed reality (VR/AR/MR)
and serious gaming for technical training and collaborative decision making.
---|---