# A Note on the Strict Transformation of an Effective Cartier Divisor

Yu Zhao

###### Abstract.

Let $Y$ be an effective Cartier divisor of a smooth variety $Z$. Let $X_{i}$,
$i\in\\{1,\cdots,n\\}$ be a set of pairwise disjoint smooth subvarieties in
$Y$ such that their union contains the singular locus of $Y$. In this paper,
we give a sufficient condition such that $Bl_{X}Y$ is smooth, where $X$ is the
disjoint union of all $X_{i}$. Moreover, we prove that assuming a dimensional
condition, there is an admissible subcategory of $D^{b}(Bl_{X}Y)$ which is a
weak categorical crepant resolution of $Y$, in the sense of Kuznetsov [3].

##  1\. Introduction

###  1.1. The main result of this paper

In this paper, we work over an algebraically closed field $\mathbb{F}$ of any
characteristic.

Let $Z$ be an irreducible smooth variety and $\\{X_{i}\\}_{i=1,\cdots n}$ be a
set of pairwise disjoint smooth closed subvarieties of $Z$. We denote
$X:=\cup_{i=1}^{n}X_{i}$. Let $Y$ be an effective Cartier divisor that
contains $X$. The purpose of this paper is to understand the following
questions:

######  Question 1.1.

Let $\tilde{Y}$ be the strict transform of $Y$ in $Bl_{X}Z$. When is
$\tilde{Y}$ a smooth divisor of $Bl_{X}Z$?

Let $\pi:Bl_{X}Z\to Z$ be the projection morphism. Let $C_{i}$ be the conormal
bundle of $X_{i}$ in $Z$. Then the exceptional divisor
$E_{X_{i}}Z\cong\mathbb{P}_{X_{i}}(C_{i})$ and we denote
$\pi_{i}:E_{X_{i}}Z\cong\mathbb{P}_{X_{i}}(C_{i})\to X_{i}$ as the projection
morphisms. If $\tilde{Y}$ is smooth, we have

$\mathcal{O}(\tilde{Y})\cong\pi^{*}\mathcal{O}(Y)-\sum_{i=1}^{n}k_{i}E_{X_{i}}Z$

for some $k_{i}\in\mathbb{Z}$. Moreover, if $Y$ is normal, by the adjunction
formula, we have

$K_{\tilde{Y}}\cong\pi_{Y}^{*}K_{Y}+\sum_{i=1}(d_{i}-k_{i}-1)E_{X_{i}}Z|_{\tilde{Y}},$

where $d_{i}$ is the codimension of $X_{i}$ in $Z$, and $K_{\tilde{Y}}$ and
$K_{Y}$ are the canonical divisors of $\tilde{Y}$ and $Y$ respectively.

######  Question 1.2.

How do we compute those coefficients $k_{i}$?

Let $\mathcal{I}_{i}$ be the ideal sheaf $X_{i}$ in $Z$, and $k_{i}$ be the
maximal integer such that $\mathcal{O}(-Y)\subset\mathcal{I}_{i}^{k_{i}}$. By
restricting to $X$, it induces canonical morphisms
$\mathcal{O}(-Y)|_{X_{i}}\to Sym^{k_{i}}(C_{i})$ and thus induce global
sections

$\Phi_{i}\in\Gamma(X_{i},Sym^{k_{i}}(C_{i})\otimes\mathcal{O}(Y_{i})|_{X_{i}}).$

Over $\mathbb{P}_{X_{i}}(C_{i})$, we have canonical morphisms
$\pi_{i}^{*}Sym^{k}C_{i}\to\mathcal{O}_{\mathbb{P}_{X}(C_{i})}(k_{i})$ and
thus $\Phi_{i}$ induce global sections

$s_{i}\in\Gamma(\mathbb{P}_{X_{i}}(C_{i}),\mathcal{O}_{\mathbb{P}_{X_{i}}(C_{i})}(k_{i})\otimes\pi_{i}^{*}\mathcal{O}(Y)|_{X_{i}})$

Let $E_{X_{i}}Y$ be the zero locus of $s_{i}$. Let $pr:\tilde{Y}\to Y$ and
$pr_{i}:E_{X_{i}}Y\to X_{i}$ be the respective projection morphisms. In this
paper, we will prove that

######  Theorem 1.3.

If $Y-X$ is smooth and $E_{X_{i}}Y$ is a smooth effective divisor of
$E_{X_{i}}Z$ for each $i$, then $\tilde{Y}$ is smooth and $E_{X_{i}}Y$ is the
exceptional divisor of $X_{i}$. Moreover, we have

$\mathcal{O}(\tilde{Y})\cong\pi^{*}\mathcal{O}(Y)-\sum_{i=1}^{n}k_{i}E_{X_{i}}Z.$

We denote $B_{i}$ as the zero locus of $\Phi_{i}$. Our second theorem is that

######  Theorem 1.4.

Assume that all $k_{i}$ is $1$. If $Y-X$ is smooth and for each $i$, $B_{i}$
is smooth and $dimB_{i}=2dimX_{i}-dimZ$, then $\tilde{Y}$ is smooth and

$\mathcal{O}(\tilde{Y})\cong\pi^{*}\mathcal{O}(Y)-\sum_{i=1}^{n}E_{X_{i}}Z.$

Moreover, for any $i$ that $d_{i}=2$, we have the exceptional divisor

$E_{X_{i}}Y\cong Bl_{B_{i}}X_{i}.$

Under this isomorphism, we have

$\displaystyle E_{B_{i}}X_{i}\cong
pr_{i}^{*}det(C_{i})\otimes\mathcal{O}(\tilde{Y}+2E_{X_{i}}Z)|_{E_{X_{i}}Y},$

where $E_{B_{i}}X_{i}$ is the exceptional divisor of $B_{i}$ in $X_{i}$.

###  1.2. Lefschetz decomposition of the exceptional divisor and weakly
categorical crepant resolutions

In [2], Kuznetsov introduced the notion of Lefschetz decomposition:

######  Definition 1.5 (Definition 4.1 of [2]).

Let $X$ be an algebraic variety with a line bundle $\mathcal{L}$ on $X$. A
Lefschetz decomposition of the derived category $D^{b}(X)$ is a semi-
orthogonal decomposition of $D^{b}(X)$ of the form

$\displaystyle
D^{b}(X)=<A_{0},A_{1}\otimes\mathcal{L},\cdots,\mathcal{A}_{m-1}\otimes\mathcal{L}^{m-1}>$
$\displaystyle 0\subset A_{m-1}\subset A_{m-2}\subset\cdots\subset
A_{1}\subset A_{0}\subset D^{b}(X)$

such that $0\subset A_{m-1}\subset A_{m-2}\subset\cdots\subset A_{1}\subset
A_{0}\subset D^{b}(X)$ is a chain of admissible sub-categories of $D^{b}(X)$.
Lefschetz decomposition is called rectangular if $A_{m-1}=\cdots=A_{1}=A_{0}$.

Similarly, a dual Lefschetz decomposition of $D^{b}(X)$ is a semi-orthogonal
decomposition of the form

$D^{b}(X)=<B_{m-1}\otimes\mathcal{L}^{1-m},\cdots,B_{1}\otimes\mathcal{L}^{-1},B_{0}>$

where

$0\subset B_{m-1}\subset\cdots\subset B_{1}\subset B_{0}\subset D^{b}(X).$

From now on we assume the assumptions in Theorem 1.3 hold.

######  Theorem 1.6.

If $k_{i}<d_{i}$, the functor

$Lpr_{i}^{*}(-)\otimes\mathcal{O}_{\mathbb{P}_{X_{i}(C_{i})}}(l):D^{b}(X_{i})\to
D^{b}(E_{X_{i}}Y)$

is fully faithful for any $l\in\mathbb{Z}$. Moreover, let

$A_{1}^{i}=A_{2}^{i}=\cdots=A_{d_{i}-k_{i}-1}^{i}:=Lpr_{i}^{*}(D^{b}(X_{i}))$

and $A_{0}^{i}$ be the left orthogonal of

$<A_{1}^{i}\otimes\mathcal{O}_{\mathbb{P}_{X_{i}(c_{i})}}(1),\cdots
A_{d_{i}-k_{i}-1}^{i}\otimes\mathcal{O}_{\mathbb{P}_{X_{i}(c_{i})}}(d_{i}-k_{i}-1)>$

in $D^{b}(E_{X_{i}}Y)$. Then
$<A_{l}^{i}\otimes\mathcal{O}_{\mathbb{P}_{X_{i}(C_{i})}}(i)>_{0\leq l\leq
d_{i}-k_{i}-1}$ is a Lefschetz decomposition of $D^{b}(E_{X_{i}}Y_{i})$.

Similarly, let

$B_{d_{i}-k_{i}-1}^{i}\cong\cdots\cong B_{1}^{i}:=Lpr_{i}^{*}(D^{b}(X_{i}))$

and $B_{0}$ be the right orthogonal of

$<B_{d_{i}-k_{i}-1}^{i}\otimes\mathcal{O}_{\mathbb{P}_{X_{i}(c_{i})}}(1+k_{i}-d_{i}),\cdots,B_{1}^{i}\otimes\mathcal{O}_{\mathbb{P}_{X_{i}(c_{i})}}(-1)>$

in $D_{b}(E_{X_{i}}Y_{i})$. Then we have the dual Lefschetz decomposition

$D^{b}(E_{X_{i}}Y)=<B_{d_{i}-k_{i}-1}^{i}\otimes\mathcal{O}_{\mathbb{P}_{X_{i}(c_{i})}}(1+k_{i}-d_{i}),\cdots,B_{0}>$

In [3], Kuznetsov introduced the notion of categorical resolutions of
singularities:

######  Definition 1.7 (Definition 3.1 of [3]).

A triangulated category $D$ is regular if it is equivalent to an admissible
subcategory of the bounded derived category of a smooth variety.

Given a triangulated category $D$, we denote $D^{perf}$ as the full
subcategory of perfect objects in $D$.

######  Definition 1.8 (Categorical resolution, Definition 3.2 of [3]).

A categorical resolution of a triangulated category $D$ is a regular
triangulated category $\tilde{D}$ and a pair of functors

$\pi_{*}:\tilde{D}\to D,\pi^{*}:D^{perf}\to\tilde{D},$

such that

  1. (1)

$\pi^{*}$ is left adjoint to $\pi_{*}$ on $D^{perf}$, that is

$Hom_{\tilde{D}}(\pi^{*}F,G)\cong Hom_{D}(F,\pi^{*}G)$

for any $F\in D^{perf}$, $G\in\tilde{D}$,

  2. (2)

the natural morphism of functors $id_{D^{perf}}\to\pi_{*}\pi^{*}$ is an
isomorphism.

######  Definition 1.9 (Definition 3.4 of [3]).

A categorical resolution $(\tilde{D},\pi_{*},\pi^{*})$ of $D$ is weakly
crepant if the functor $\pi^{*}$ is right adjoint to $\pi_{*}$ on $D^{perf}$:

$Hom_{\tilde{D}}(G,\pi^{*}F)\cong Hom_{D}(\pi_{*}G,F)$

for any $F\in D^{perf},G\in\tilde{D}$.

######  Remark 1.10.

In contrast to the weakly crepant categorical resolution, Kuznetsov [3] also
introduced the notion of strong crepant categorical resolution by requiring
the relative Serre functor as identity.

In [3], Kuznetsov revealed the relation between the Lefschetz decomposition of
the exceptional divisor and weakly (and strongly) crepant categorical
resolution of singularities. By applying his theorem (with a very mild
generalization), we prove that

######  Theorem 1.11.

Assuming the setting of Theorem 1.3 and $k_{i}<d_{i}$ for all $i$, let
$\tau_{i}:E_{X_{i}}Y\to\tilde{Y}$ be the closed embeddings of exceptional
divisors. Then we have a semi-orthogonal decomposition

$D^{b}(\tilde{Y})=<R\tau_{i*}(Lpr_{i}^{*}D^{b}(X_{i})\otimes\mathcal{O}_{\mathbb{P}_{X_{i}(C_{i})}}(l_{ij})),\tilde{D}>_{1\leq
i\leq n,k_{i}-d_{i}<l_{ij}<0},$

such that $\tilde{D}$ is a weak crepant categorical resolution of $D^{b}(Y)$.

###  1.3. Notations

All algebraic varieties are assumed to be of finite type over $\mathbb{F}$.
For any algebraic variety $X$, we denote $D^{b}(X)$ as the bounded derived
category of coherent sheaves on $X$. For a morphism $f:X\to Y$, we denote
$\mathbb{R}f_{*}$ as the derived push-forward functor and $Lf^{*}$ as the
derived pull-back functor.

###  1.4. Acknowledgements

This paper is inspired by the recent work of the author on the derived blow-
ups of quasi-smooth derived schemes [6] and Will Donovan on the homological
comparison of resolution and smoothing. Will Donovan suggested that it might
be interesting to rewrite some part of [6] in a more classical and elementary
language, and the author would acknowledge him for his encouragement.

The first version of this paper was written when the author was visiting
HKUST, Tsinghua YMSC, and CUHK, and the author would like to thank those
institutions and Wei-Ping Li, Changjian Su, Micheal Mcbreen, Bin Gui, Siqi He,
and Xinzhou Guo for their hospitality and many helpful discussions.

The author is supported by World Premier International Research Center
Initiative (WPI initiative), MEXT, Japan, and Grant-in-Aid for Scientific
Research grant (No. 22K13889) from JSPS Kakenhi, Japan.

##  2\. Smoothness of the Strict Transform

Before proving Theorem 1.3, we first prove the following lemma:

######  Lemma 2.1.

Let $X$ be a smooth variety with pairwise disjoint effective smooth divisors
$E_{i}$, $1\leq i\leq m$. Let $Y$ be an effective divisor of $X$. If
$Y-\cup_{i=1}^{m}E_{i}$ is smooth and $Y\cap E_{i}$ is a smooth effective
divisor of $E_{i}$ for each $i$, then $Y$ is also smooth.

###### Proof.

By induction, we can reduce to the case that $m=1$. As the smoothness property
is a local property, and preserved under the etale morphisms, we can assume
that $X$ is an open subvariety of the affine space $\mathbb{A}^{n}=Spec\
\mathbb{F}[x_{1},x_{2},\cdots,x_{n}]$, such that $E_{1}$ is the zero locus of
$x_{n}$ and $Y$ is the zero locus of

$f=x_{n}^{l}g_{n}+x_{n}^{l-1}g_{l-1}+\cdots+g_{0}$

where $g_{i}\in k[x_{1},\cdots,x_{n-1}]$. Then for any $p\in E_{1}$ such that
$f(p)=0$, we have

$\frac{\partial f}{\partial x_{i}}(p)=\frac{\partial g_{0}}{\partial
x_{i}}(p)$

for any $1\leq i<n$. Hence the singular locus of $f$ in $Y$ is inside the
singular locus of $g_{0}$ in $Y\cap E_{1}$. ∎

###### Proof of Theorem 1.3.

As $\mathcal{O}(-Y)\subset\mathcal{I}_{i}^{k_{i}}$, by [5],
$\pi^{*}\mathcal{O}(Y)-\sum_{i=1}^{n}k_{i}E_{X_{i}}Z$ is also an effective
divisor of $X$, which we denote as $\mathcal{O}(\tilde{Y})$. Moreover, we have
$E_{X_{i}}Z\cong\mathbb{P}_{X_{i}}(C_{i})$ and $E_{X_{i}}Z\cap\tilde{Y}$ is
the zero locus of $s_{i}$. If all the zero locus of $s_{i}$ are smooth
effective divisors of $E_{X_{i}}Z$ respectively, by Lemma 2.1 $\tilde{Y}$ is
smooth. Moreover, it is the schematic closure of $Y-X$ in $Bl_{X}Z$, and thus
is the strict transform of $Y$. The preimage of $X_{i}$ in $\tilde{Y}$ is the
zero locus of $s_{i}$, and hence is the exceptional divisor $E_{X_{i}}Y$. ∎

To prove Theorem 1.4, we first prove the following lemmas:

######  Lemma 2.2.

Let $X$ be a smooth variety. Let $V$ be a locally free sheaf over $X$ and $L$
be a line bundle over $X$. Let $\Phi$ be a global section of $V\otimes L$ and
let $S$ be the zero locus of $s$. Let $pr_{V}$ be the projection from
$\mathbb{P}_{X}(V)$ to $X$ and $t$ be the composition of morphisms:

$\mathcal{O}\to pr_{V}^{*}(V\otimes
L)\xrightarrow{taut}pr_{V}^{*}L\otimes\mathcal{O}_{\mathbb{P}_{X}(V)}(1)$

which is a global section of
$pr_{V}^{*}L\otimes\mathcal{O}_{\mathbb{P}_{X}(V)}(1)$. Let $T$ be the zero
locus of $t$. Then $S$ is a $dim(X)-rank(V)$ dimensional smooth subvariety of
$X$ iff $T$ is an effective smooth divisor of $\mathbb{P}_{X}(V)$.

###### Proof.

Actually, it follows from linear Koszul duality [4] or the singularity
category of projectivization of two-term complexes [1]. Here we give
elementary proof for the convenience of readers.

As the smoothness of $S$ and $T$ are both local properties, and stable under
the etale base change of $X$, we can assume that $X$ as an open subvariety of
$\mathbb{A}^{n}=Spec\ \mathbb{F}[x_{1},\cdots x_{n}]$, and
$L\cong\mathcal{O}_{X}$ and $V\cong\mathcal{O}_{X}^{m}$ for some positive
integer $m$. Then $\Phi$ is represented by

$f=(f_{1},f_{2},\cdots f_{m})$

where $f_{i}\in k[x_{1},\cdots x_{m}]$. We consider $\mathbb{A}^{n+m}=Spec\
\mathbb{F}[x_{1},\cdots,x_{n},y_{1},\cdots,y_{m}]$ and

$g:=\sum_{i=1}^{m}y_{i}f_{i}.$

Here is a $\mathbb{G}_{m}$ action on $\mathbb{A}^{n+m}$ by

$t\cdot(x_{1},\cdots,x_{n},y_{1},\cdots,y_{m})=(x_{1},\cdots,x_{n},ty_{1},\cdots,ty_{m})$

and we denote $T^{\prime}$ as the zero locus of $g$ on
$\mathbb{A}^{n+m}-\mathbb{A}^{n}\times\\{0\\}$. Then $T\cong
T^{\prime}/\mathbb{G}_{m}$.

Given $p\in\mathbb{A}^{n}$, we consider the deravative matrix with the $(i,j)$
entry

$Jac(f)_{p}=\begin{pmatrix}\frac{\partial f_{i}}{\partial
x_{j}}(p)\end{pmatrix}|_{1\leq i\leq m,1\leq j\leq n}$

and the critical locus of $f$, which we denote as $crif(f)$, is defined the
the locus that $Jac(f)(p)$ has rank less than $m$. Then the singular locus of
$S$ is $S\cap crit(f)$. The singular locus of $T^{\prime}$ is

$\\{(x_{1},\cdots,x_{n},y_{1},\cdots,y_{m})|x\in S,Jac(f)_{x}(y)=0\\}$

where $x:=(x_{1},\cdots,x_{n})$ and $y:=(y_{1},\cdots,y_{m})$. As
$y\neq\\{0\\}$, the singular locus of $T^{\prime}$ (and also $T$) is empty iff
the singular locus of $S$ is empty. ∎

######  Lemma 2.3.

We assume all the assumptions of Lemma 2.2. Moreover, we assume that
$rank(V)=2$ and $S$ is smooth of $dim(X)-2$. Then $T\cong Bl_{S}X$ and the
projection morphism of the blow-up factors through the projection morphism
$pr_{V}:\mathbb{P}_{X}(V)\to X$. Moreover, under the isomorphism, we have

(2.1) $\mathcal{O}_{\mathbb{P}_{X}(V)}(1)|_{T}\cong pr_{V}^{*}(det(V)\otimes
L)|_{T}\otimes(E_{S}X)^{-1}$

###### Proof.

We have the exact sequence

$0\to det(V\otimes L)^{-1}\to V^{\vee}\otimes
L^{-1}\xrightarrow{\Phi^{\vee}}\mathcal{O}_{X}\to\mathcal{O}_{S}\to 0.$

We notice that $V^{\vee}\otimes L^{-1}=det(V)^{-1}\otimes V\otimes L^{-1}$.
Thus we have the short exact sequence:

$0\to L^{-1}\xrightarrow{\Phi\otimes L^{-1}}V\to det(V)\otimes
L\otimes\mathcal{I}_{S}\to 0$

where $\mathcal{I}_{S}$ is the ideal sheaf of $S$. As

$Bl_{S}X\cong Proj_{X}\oplus_{i=0}^{\infty}Sym^{i}(\mathcal{I})\cong
Proj_{X}\oplus_{i=0}^{\infty}Sym^{i}(det(V)\otimes L\otimes\mathcal{I})$

we have $T\cong Bl_{S}X$ and 2.1 follows from the shift of line bundles. ∎

###### Proof of Theorem 1.4.

The smoothness of $\tilde{Y}$ follows from Theorem 1.3 and Lemma 2.2. When
$d_{i}=2$, we have

$E_{X_{i}}Z|_{E_{X_{i}}Y}\cong\mathcal{O}_{\mathbb{P}_{X_{i}}(C_{i})}(-1)|_{E_{X_{i}}Y}.$

On the other hand, by Lemma 2.3, we have

$\displaystyle E_{B_{i}}X_{i}$
$\displaystyle\cong(\pi_{i}^{*}(det(C_{i})\otimes\mathcal{O}(Y))\otimes\mathcal{O}_{\mathbb{P}_{X_{i}}(C_{i})}(-1))|_{E_{X_{i}}Y_{i}}$
$\displaystyle\cong(\pi_{i}^{*}det(C_{i})\otimes\mathcal{O}(\tilde{Y}+2E_{X_{i}}Z)|_{E_{X_{i}Z}})|_{E_{X_{i}}Y}$
$\displaystyle\cong
pr_{i}^{*}det(C_{i})\otimes\mathcal{O}(\tilde{Y}+2E_{X_{i}}Z)|_{E_{X_{i}}Y}.$

∎

##  3\. Lefschetz Decompositions and Weak Crepant Categorical Resolutions

In this section, we prove Theorem 1.6 and Theorem 1.11.

###### Proof of Theorem 1.6.

Given $F,G\in D^{b}(X_{i})$ and $0\leq l<d_{i}-k_{i}-1$, by the projection
formula we have

$Hom_{E_{X_{i}}Y}(Lpr_{i}^{*}F,Lpr_{i}^{*}G\otimes\mathcal{O}_{\mathbb{P}_{X_{i}}(c_{i})}(l))\cong
Hom_{X_{i}}(F\otimes Rpr_{i*}\mathcal{O}_{E_{X_{i}}Y}(-l),G).$

Thus we only need to prove that

(3.1) $Rpr_{i*}\mathcal{O}_{E_{X_{i}}Y}\cong\mathcal{O}_{X_{i}},\quad
Rpr_{i*}\mathcal{O}_{E_{X_{i}}Y}(-l)\cong 0$

if $0<l\leq d_{i}-k_{i}-1$. We notice that $E_{X_{i}}Y$ is the zero locus of
$s_{i}$ in $\mathbb{P}_{X_{i}}(C_{i})$. Thus regarding $E_{X_{i}}Y$ as a
closed subscheme of $\mathbb{P}_{X_{i}}(C_{i})$, we have the short exact
sequence:

$0\to\pi_{i}^{*}\mathcal{O}(-Y)\otimes\mathcal{O}_{\mathbb{P}_{X_{i}}(C_{i})}(-k_{i})\to\mathcal{O}_{\mathbb{P}_{X_{i}}(C_{i})}\to\mathcal{O}_{E_{X_{i}}Y}\to
0.$

Thus (3.1) follows from Serre’s theorem that

$R\pi_{*}\mathcal{O}_{\mathbb{P}_{X_{i}}(C_{i})}\cong\mathcal{O}_{X_{i}},\quad
R\pi_{*}\mathcal{O}_{\mathbb{P}_{X_{i}}(C_{i})}(m)\cong 0$

if $-d_{i}<m<0$. ∎

Before we prove Theorem 1.11, we summarize Kuznetsov’s theory in Section 4 of
[3] in the following theorem (notice that we formulate a mild generalization
such that there are several pairwise disjoint exceptional divisors):

######  Theorem 3.1 (Proposition 4.1, Theorem 4.4 and Proposition 4.5 of [3]).

Let $pr:\tilde{Y}\to Y$ be a resolution of singularities such that $Y$ is
Gorenstein. Let $E_{i}$ ($i\in\\{1,\cdots,n\\}$) be the exceptional divisors
of $pr$ which does not intersect pairwisely, $X_{i}:=pr(E_{i})$,
$pr_{i}=pr|_{E_{i}}:E_{i}\to X_{i}$ and $\iota_{i}:E_{i}\to\tilde{Y}$ are the
closed embeddings of exceptional divisors. Let $K_{\tilde{Y}}\cong
pr^{*}K_{Y}+\sum_{i=1}^{n}(m_{i}-1)E_{i}$, where all $m_{i}\in\mathbb{Z}$, and
$\mathcal{L}_{i}:=-E_{i}|_{E_{i}}$. If for each $i$, there exists a dual
Lefschetz decomposition

$D^{b}(E_{i})=<B_{m_{i}-1}^{i}\otimes L_{i}^{1-m},B_{m-1}^{i}\otimes
L_{i}^{2-m},\cdots,B_{1}^{i}\otimes L_{i}^{-1},B_{0}^{i}>$

such that $Lpr_{i}^{*}D^{perf}(X_{i})\subset B_{m_{i}-1}^{i}$, then there
exists a semi-orthogonal decomposition

$\displaystyle D^{b}(\tilde{Y})=$
$\displaystyle<R\iota_{1*}(B_{m_{1}-1}^{1}\otimes
L_{1}^{1-m_{1}}),\cdots,R\iota_{1*}(B_{1}^{1}\otimes L_{1}^{-1}),\cdots,$
$\displaystyle R\iota_{n*}(B_{m_{1}-1}^{1}\otimes
L_{n}^{1-m_{1}}),\cdots,R\iota_{n*}(B_{n}^{1}\otimes L_{n}^{-1}),\tilde{D}>$

such that $\tilde{D}$ is a weak categorical crepant resolution of $Y$.

######  Remark 3.2.

The order of $E_{i}$ in Theorem 3.1 does not matter. For any $\sigma$ being
permutation of $n$ elements, we can replace $E_{i}$ by $E_{\sigma_{i}}$ and
Theorem 3.1 still holds.

###### Proof of Theorem 1.11.

It follows from Theorem 3.1 and Theorem 1.3. ∎

##  4\. A Toy Example

Now we give a toy example where all the above theorems apply. Given a positive
integer $n$, let $Z_{n}:=\mathbb{A}^{2n}=Spec\
\mathbb{F}[X_{1},\cdots,x_{n},y_{1},\cdots,y_{n}]$ and $Y_{n}$ be the zero
locus of the following equation in $Z_{n}$:

$f_{n}=x_{1}y_{1}+x_{2}y_{2}+\cdots+x_{n}y_{n}.$

The singular locus of $Y_{n}$ is the origin
$O_{n}:=(0,\cdots,0)\in\mathbb{A}^{2n}$. In this section, we take two smooth
subvarieties which contain $O_{n}$:

  1. (1)

First we take

$X_{n}:=\mathbb{A}^{n}\times\\{0\\}$

which contains elements that $y_{1}=y_{2}=\cdots=y_{n}=0$. In this situation,
we apply Theorem 1.4 and the zero locus $B=\\{O_{n}\\}$ is smooth. Thus
$Bl_{X_{n}}Y_{n}$ is smooth.

  2. (2)

Then we take

$X_{n}^{\prime}=\\{O_{n}\\}.$

We apply Theorem 1.3 and in this situation, we have $k=2$. The zero locus of
$s$ is

$\\{[x_{1},\cdots,x_{n},y_{1},\cdots,y_{n}]\in\mathbb{P}^{2n-1}|f_{n}(x_{1},\cdots,x_{n},y_{1},\cdots,y_{n})=0\\}$

which is smooth. Thus $Bl_{X_{n}^{\prime}}Y_{n}$ is also smooth, too.

By Theorem 1.11, the above two resolutions induce two weakly categorical
crepant resolutions of $D^{b}(Y_{n})$, which we denote as $D_{n}$ and
$D_{n}^{\prime}$ respectively. Moreover, we have $D_{1}\cong D_{1}^{\prime}$
be the theory of Atiyah flops.

## References

  * [1] Qingyuan Jiang and Naichung Conan Leung.  Derived category of projectivization and flops.  arXiv preprint arXiv:1811.12525, 2018.
  * [2] Alexander Kuznetsov.  Homological projective duality.  Publ. Math. Inst. Hautes Études Sci., (105):157–220, 2007.
  * [3] Alexander Kuznetsov.  Lefschetz decompositions and categorical resolutions of singularities.  Selecta Math. (N.S.), 13(4):661–696, 2008.
  * [4] Ivan Mirković and Simon Riche.  Linear Koszul duality.  Compos. Math., 146(1):233–258, 2010.
  * [5] Jarosław Włodarczyk.  Simple Hironaka resolution in characteristic zero.  J. Amer. Math. Soc., 18(4):779–822, 2005.
  * [6] Yu Zhao.  Derived blow-ups and birational geometry of nested quiver varieties.  arXiv preprint arXiv:2303.01063, 2023.

Kavli Institute for the Physics and Mathematics of the Universe (WPI),
University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, 277-8583, Japan.

E-mail address<EMAIL_ADDRESS>


††thanks: These authors contributed equally to this work.††This document is
the Accepted Manuscript version of a Published Work that appeared in final
form in Nature Communications, after peer review and technical editing by the
publisher. To access the final edited and published work see
https://doi.org/10.1038/s41467-022-28392-8, Nat. Commun. 13, 981 (2022).
††thanks: These authors contributed equally to this work.

# Anionic Character of the Conduction Band of Sodium Chloride

Christopher C. Leon  Abhishek Grewal  Klaus Kuhnke<EMAIL_ADDRESS>Max-
Planck-Institut für Festkörperforschung, Heisenbergstraße 1, 70569 Stuttgart,
Germany  Klaus Kern  Max-Planck-Institut für Festkörperforschung,
Heisenbergstraße 1, 70569 Stuttgart, Germany  Institut de Physique, École
Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland  Olle
Gunnarsson<EMAIL_ADDRESS>Max-Planck-Institut für
Festkörperforschung, Heisenbergstraße 1, 70569 Stuttgart, Germany

###### Abstract

The alkali halides are ionic compounds. Each alkali atom donates an electron
to a halogen atom, leading to ions with full shells. The valence band is
mainly located on halogen atoms, while, in a traditional picture, the
conduction band is mainly located on alkali atoms. Scanning tunnelling
microscopy of NaCl at 4 K actually shows that the conduction band is located
on Cl- because the strong Madelung potential reverses the order of the Na+
$3s$ and Cl- $4s$ levels. We verify this reversal is true for both atomically
thin and bulk NaCl, and discuss implications for II-VI and I-VII compounds.

## Introduction

Insulating compounds provide a rich and very varied physics. One distinguishes
between ionic [1], Slater [2], Mott [3], and charge transfer [4, 5, 6, 7]
insulators, involving quite different mechanisms and having different
properties. Slater and Mott insulators primarily involve a partly filled band,
in which antiferromagnetic or Coulomb interactions open up a gap. The late
transition metal oxides are examples of charge transfer insulators, involving
the O $2p$-band and the, partly filled, transition metal $3d$ band. Coulomb
interactions are essential also in this case. Ionic insulators, e.g., alkali
halides, appear to be conceptually simpler. In a one-particle picture, all
bands are completely full or empty, and Coulomb interactions are less
essential. There is a large charge transfer from the alkali atoms to the
halogen atoms, which become positively and negatively charged ions. The
valence band is primarily of halogen $p$ character. This band is separated
from the conduction band by a large gap. The conduction band is assumed to
have mainly alkali character, involving an empty alkali $s$-level outside a
full shell. While this picture has been presented in many text books and
publications [1, 8, 9, 10, 11, 12, 13], this view has been occasionally be
questioned on theoretical grounds. Slater and Shockley [14] suggested a
different picture for NaCl, in which the conduction band also has a
substantial Cl $4s$ character. Clark [15, 16], de Boer and de Groot [17, 18],
as well as Olsson et al. [19], performed band structure calculations for NaCl
and concluded that the conduction band is actually mainly located on the Cl-
ions. These theoretical results, however, seem to have been largely overlooked
or ignored, possibly because of problems of uniquely assigning charges to
ions.

We present a heuristic calculation which suggests that a very large Madelung
potential [20] ($\approx 9$ eV) can reverse the order of the Cl- $4s$ (above
the vacuum level) and Na+ $3s$ (at $\approx-5$ eV) levels and imply a
conduction band of mainly Cl character. (The Madelung potential is the
potential at any ion position in an ionic crystal due to the combined
electrostatic potentials of the infinite number of ions in the crystal.)
However, such a calculation (and similar considerations) alone is not decisive
proof of this reality because of confounding factors such as the large spatial
extent of s orbitals and non-unique assignments of charges to ions. For
exactly this reason, we perform an experimental study of the conduction band
of NaCl using scanning tunnelling microscopy (STM) providing a real space
picture of states, which are centred on the Cl- ions across the entire band
gap.

Figure 1: STM of 2 ML NaCl(100) on Au(111) with topography and dI/dV
measurements. (a) Overview of the measurement area ($150\times 140$ Å2,
$I_{T}=8$ pA, $V_{S}=-50$ mV). $25\times 25$ Å2 region marked, investigated
for bias values between $-3.0$ V and $+2.5$ V. Inset: Fourier transform of (f)
centred over a 8 nm${}^{-1}\times$8 nm-1 region in reciprocal space. (b)
Differential conductance (d$I$/d$V$) measured atop 2 ML NaCl(100)/Au(111)
(marked by the black ring in (a)). The arrows point to onset of valence (VB)
and conduction band (CB). Inset. Surface state features of Au(111) and related
interface state of 2 ML NaCl(100)/Au(111) measured at the positions marked in
(a) by a gray ring and a black ring, respectively. Vertical scale: arbitrary
units. (c-h) Raw data showing grayscale topography images obtained for sample
bias between $-3.0$ V and $+2.5$ V as indicated. The red and blue dots in (c)
mark the positions of Na and Cl ions of the NaCl(100) lattice, respectively.
The corresponding tunnelling parameters are indicated next to each topography
image.

## Results

Despite the many STM studies addressing the structure and growth of NaCl films
[21, 22, 23, 24, 25], such a detailed study of the NaCl conduction band has
not yet been performed, possibly because it is very challenging to obtain
atomic resolution on NaCl at positive bias (e.g., $U>1$ V) [22]. These
difficulties preclude a straightforward spatial mapping of the NaCl conduction
band by STM and an identification of the relative local density of states at
Na and Cl positions. The instability of the tunnelling condition necessary to
achieve atomic resolution simply prohibits addressing the conduction band
directly.

We circumvent this problem by making the bias very positive to approach the
conduction band very closely without exceeding the band gap, and harnessing
our understanding of the tunnelling process under these conditions. We present
STM images of NaCl(100) on Au(111), varying the bias over a large range. They
show that tunnelling happens through the Cl- ions, even for energies just
below the conduction band, indicating that the conduction band also has mainly
Cl character.

To support these conclusion we present tight-binding (TB) calculations for two
models of a NaCl film on a Au surface, with the conduction band mainly on
either the Na+ or Cl- ions. These calculations show that features in STM
images for energies just below the conduction band indeed are located on the
same ions as the conduction band itself.

We compare NaCl on Au and bulk NaCl theoretically. We find that the order of
the Na $3s$ and the Cl $4s$ levels are reversed by similar amounts in both
cases, as discussed extensively in Supplementary Note 2. The conclusions
obtained for NaCl on Au should therefore also apply to bulk NaCl.

We furthermore extend the discussion of NaCl to II-VI and other I-VII
compounds, suggesting that the conduction band could have a substantial weight
on the anion also in these cases.

### STM topography of NaCl over the entire band gap

Because few experimental techniques can characterise the conduction band
character of the alkali halides with atomic resolution, we investigated the
NaCl(100) surface with STM. Since directly accessing the conduction band of
NaCl(100) involves high electric fields that destroy its structural integrity
[22], we do measurements at voltages as close as possible to the conduction
band edge while staying within the band gap of NaCl. Topographs are measured
at constant current whose colour scale represent tip-sample distance. By
repeating the measurements at certain voltages, the tip quality was monitored
and care was taken that the relatively high bias voltages did not modify the
tip character or shift the tip apex. Similarly, the tip drift over time,
typical for STM, was monitored and corrected in the presented data.

Figure 1a shows a region of Au(111) covered with a 2 ML thick (apparent
height: $309\pm 3$ Å) NaCl terrace, consistent with prior studies [26]. Atop
the NaCl terrace, the protrusions form a square lattice whose unit cell length
is 4 Å (Fig. 1a). The square lattice is reproduced in reciprocal space
(Fourier transform inset, Fig. 1a).

We survey the electronic structure of the system at selected positions marked
by gray and black rings in Fig. 1a. The pronounced surface state of Au(111)
shows an onset at $-500$ mV measured at the gray ring. This onset shifts to
$\approx-250$ mV when measured on Au-NaCl at the black ring. This shift
originates from the Pauli interaction experienced between NaCl and Au(111).
NaCl acts as a dielectric, polarises and weakens the image potential arising
from Au(111) alone [27, 28] (inset Fig. 1b) measured at the black ring. A
large range voltage scan of differential conductance indicating density of
states of NaCl is shown in Fig. 1b. Arrows point to the valence and conduction
band onsets at $-3.0$ V and $+3.6$ V.

The region marked by the white box, in the fcc region of herringbone
reconstruction of Au(111), in Fig. 1a is scanned with high stability and
signal to noise ratio (Fig. 1c-h) for applied sample voltages from $-3$ V to
$+2.5$ V. respectively. At applied bias of $-3.0$ V (Fig. 1c), a voltage close
to the valence band edge of NaCl (Fig. 1b), a square lattice of protrusions is
seen whose intensity is proportional to the $z$-corrugation. Each protrusion
is assigned to Cl. The assignment of protrusions at negative voltage to Cl is
in agreement with many earlier studies [22, 29, 19]. We do not observe any
change in position of $z$-corrugation at negative voltages (Fig. 1c-e). These
measurements further corroborate the valence band being mainly Cl in
character.

Figure 2: STM of 3 ML NaCl(100) on Au(111) at voltages close to both edges of
the band gap. (a-g) Raw (grayscale) and (h-n) Fourier filtered (coloured) at
the indicated voltages, $I_{T}=15$ pA. Scale: $24\times 24$ Å2. The white
dashed lines mark the position of Na ion rows of the NaCl(100) lattice.

We now consider the same region scanned at positive bias. Simple but
ultimately misleading electrostatic arguments suggest that the topographic
protrusions would become centered over Na+ ions. For instance, electron
extraction at negative bias is helped at the Cl- positions with the Cl- being
relatively electron-rich. Electron injection at positive bias is helped at the
Na+ positions with the Na+ being relatively electron-poor. Thus, one might
anticipate that inverting the applied bias polarity should result in an
inversion of the topographic contrast due to $z$-corrugation resulting from
Na+ ions instead of Cl- ions.

We perform this experiment and obtain evidence to the contrary. Figure 1f-h
show the result of probing the NaCl layer at positive bias up to $+2.5$ V. By
comparing these topographic measurements with those obtained at negative bias
(Fig. 1c-e), it is evident that the protrusions remain centred over Cl and do
not shift over to Na. If the conduction band were cationic in character, one
would have expected to see topographic intensity over the Na+ ions. Yet no
contrast inversion is observed when the bias polarity is inverted. In the band
gap of NaCl, the $z$-corrugation is correlated to the electronic states of
energetically nearest neighbouring bands. Thus, the experiment shows that the
electronic states of NaCl have substantial weight on Cl rather than Na,
irrespective of bias polarity. When one tunnels at energies within the band
gap of NaCl, electron transport in NaCl is dominated by the role of Cl- ions
at both negative and positive bias. This implies that both the valence and
conduction bands of NaCl are anionic in character, a point that is to be
emphasised.

The expected role of Na+ ions in the conduction band of NaCl asserted by many
textbooks and publications is simply absent. Rather, we see evidence for Cl
functioning as both an electron donor and acceptor in NaCl. We conceptualise
the electronic structure of NaCl as that of spectator Na+ cations holding
together Cl- anions that are mainly responsible for electron conduction.

To further evidence the markedly anionic (rather than cationic) character of
the NaCl conduction band, we expand the voltage range of the measurements to
$+3.5$ V in exchange for greatly reduced tip stability and signal to noise
ratio. However, we still keep the absolute piezo drift ($0.014$ nm min-1)
during data acquisition to be significantly smaller than the Cl-–Na+ ion
spacing ($\approx 0.3$ nm) and along a direction that avoids introducing
spurious contrast inversions due to drift. These efforts are shown in Fig. 2
in which a different sample area of 3 ML NaCl(100) ($483.5\pm 5.4$ pm
thickness) on Au(111) is scanned at select voltages and analysed with Fourier
filtered scans. As discussed by Lauwaet et al. [26], the interface state (see
inset Fig. 1b) wave function for NaCl(100) on Au(111) does not extend further
than two NaCl layers. In the discussion in Supplementary Note 2, we conclude
that electronically, 2 ML and 3 ML NaCl are identical. In contrast to Fig. 1,
the scans in Fig. 2 are taken across the well-defined fcc and hcp regions of
the Au(111) herringbone reconstruction

At bias $>+2$ V, the risk of tip changes and spontaneous defect creation in
the NaCl layer increase considerably. As seen in Fig. 2a-g, while tip
transients do prevent smooth scanning at high bias, they do not completely
destroy the important atomic resolution. The low and high frequencies are
Fourier filtered to mitigate noise and to clearly highlight the periodic
$z$-corrugation. The measurements in Fig. 2 attain atomic resolution on NaCl
even at high bias than the ones in Fig. 1. Once again, at $-3$ V,
$z$-corrugation is assigned to Cl positions and does not shift to Na positions
(lattice positions marked by white dashed line in Fig. 2h-n) at any of the
indicated applied voltages in Fig. 2. Repeating these measurements at
intermediate voltages between $-3.0$ V to $+3.5$ V reveals no evidence for
electron density maxima at Na. The results in Fig. 2 show that electron
density remains centred over Cl, even up to $+3.5$ V (Fig. 2g, n) which is
quite close to the empirically measured conduction band edge ($\approx+3.6$ V)
shown in Fig. 1b. The second data set thus demonstrates that the conclusions
about tunnelling being weighted on Cl in NaCl holds more generally and is
observed on different NaCl layer heights, on fcc as well as hcp areas and by
using a tip after different preparation.

### Tight binding calculations

We now perform tight-binding calculations for a model of a three-layer
NaCl(100) film on a Au(111) surface (see Supplementary Note 1). We use two
different sets of parameters for the NaCl film, resulting in a conduction band
mainly of either Na $3s$ or Cl $4s$ character. In the first set we use $3s$
and $3p$ basis states on the Cl atoms and in the second set the Cl $3s$ basis
states are replaced by $4s$ basis states. States in the energy gap of NaCl are
described as linear combinations of valence and conduction states.

In Fig. 3a, we use conventional parameters, including $3s$ and $3p$ states on
the Cl atoms [12] and neglect the Madelung potential. (Note, that in the
calculation with Cl $3s$ and $3p$ orbitals only, the conduction band is
repelled upwards by these states. Including higher states on Cl would tend to
have the opposite effect, requiring a higher Na $3s$ level to obtain the
correct band gap. This may then require a contribution from the Madelung
potential also in this case.) As we increase the energy through the gap, the
character of the state changes from mainly Cl character close to the valence
band to mainly Na character close to the conduction band, in contradiction to
the experimental finding.

Figure 3b shows results for the second set of parameters, including $3p$ and
$4s$ states on the Cl atoms. In this case, it is essential to include a
Madelung potential to obtain a conduction band of mainly Cl character and a
correct band gap. The result is that the gap states have mainly Cl character
throughout the whole gap, in agreement with experiment. This then provides
further evidence that not only the gap states but also the conduction band has
mainly Cl character. We emphasize again, the inclusion of Cl $4s$ states in
these calculations.

Tunnelling through an insulator band gap proceeds via the electronic states of
the energetically nearest neighbouring bands. If NaCl were composed of a Cl-
based valence band and a Na+ based conduction band, an atomically resolved STM
map of it would show voltage-dependent contrast that inverts at a specific
voltage in the band gap suggested in Fig. 3a. However, this inversion does not
happen in the experiment. In fact, the same type of ion appears bright at
energies near both the valence and conduction bands, corroborating the picture
that both bands have most of their weight on the same type of ion,
specifically Cl- in the case of NaCl (Fig. 3b).

## Discussion

Naively applying classical electrostatics, one would predict that the
negatively charged electrons avoid the negatively charged Cl- in favour of the
positively charged Na+. Moreover, because Cl- has a full shell electron
configuration [Ne]3s23p6, the extra tunnelling electron must be accommodated
in the higher energy Cl $4s$ orbital, which also appears to impose an
insurmountable energetic cost.

To understand why the Cl $4s$ states might, nevertheless, be so important, we
make a few very simple considerations. For fully ionised atoms (in the sense
of Cl- and Na+ having exact full shell electronic configurations), the
Madelung potential is 8.9 eV, leading to a strong upward shift of the Na $3s$
level, which for a neutral atom is just at $-5.1$ eV [30]. Even if the $3s$
orbital relaxes somewhat, it may then be pushed above the vacuum level by the
Madelung potential.

Figure 3: Character of the gap states of NaCl including Madelung potential.
Normalised relative weight of gap states on Na and Cl atoms in the outermost
NaCl layer as a function of energy, $E$, in or close to the gap (indicated by
the arrows). The parameters were adjusted so that the conduction states have
mainly (a) Na character (b) Cl character. The Figure illustrates that
electrons travelling through the gap at energies close to the conduction band,
have similar character as those in the conduction band.

Meanwhile, the $4s$ level of a free Cl- ion is above the vacuum level. The
issue is whether the Madelung potential can pull this level below the Na $3s$
level. To puzzle this out, we use a thought experiment to increase the nuclear
charge of Cl by two. This turns Cl- into K+, for which the $4s$ level is at
$-4.1$ eV [30], below the (raised) Na $3s$ level. The question is if the
Madelung potential has a similar attractive effect on the Cl $4s$ level. We
use Slater’s rules [31] to approximate the $4s$ state of K+ as $\phi_{4s}\sim
r^{2.7}\exp{(-2.2r/3.7)}$, where $2.2$ is the effective nuclear charge and
$n=3.7$ is the effective principal quantum number. We then calculate the
attractive potential on the $4s$ state due to increasing the nuclear charge of
Cl by $\Delta Z=2$. The magnitude of this potential is indeed comparable to
the Madelung potential of 8.9 eV. In reality, the atoms are not fully ionised
and so the Madelung potential is smaller. Simple estimates suggest, however,
that this effect and several other smaller effects do not change the picture
in an essential way (see Supplementary Note 3). The K $4s$ orbital is very
extended and complicates the interpretation of these results. Nevertheless,
these simple considerations make it plausible that the Madelung potential can
reverse the order of the Cl- $4s$ level and the Na+ $3s$ level, resulting in a
conduction band of mainly Cl character in NaCl. This is confirmed by the STM
measurements in Figs. 1 and 2 that show well-defined features centred on the
Cl ions. This interpretation is robust because the STM measurements are
performed in real space and do not depend on interpretations expressed in
terms of orbitals.

We now discuss differences between bulk NaCl and NaCl on Au, using tight-
binding calculations described in the Supplementary Note 2. For a NaCl film on
Au, we find an appreciable charge transfer to the Au substrate, consistent
with an observed substantial reduction of the work function. This raises the
potential of the outer layers of the film substantially, but the effect on the
potential difference between the Na and Cl sites in the outermost layer is
very small. The assumed smaller lattice parameter of the NaCl film on Au
increases the Madelung potential, while the finite thickness of the film
reduces it. Adding up all effects, we find that the tendency to put the
conduction band on the Cl atoms should be comparable (difference on the order
of $0.1$ eV, see Supplementary Table IV) for bulk NaCl and for NaCl on Au,
implying that the effects observed here should apply also to bulk NaCl.

## Conclusion

In nanoscience, a NaCl film is often used as a buffer between a substrate and
a molecule to be studied [32, 33]. Due to the large NaCl gap, the applied
voltage is typically such that the electrons tunnel through the NaCl gap. The
way the substrate couples to the molecule then depends on whether the
corresponding electrons mainly tunnel through the Cl- or Na+ ions. The results
above should then be important for interpreting such experiments.

In contrast to the discussion above, in nanoscience it is often assumed that
the electrons tunnel through a band gap via some state, not well specified,
being different from both the valence and conduction states [34]. To address
this issue, we could have included additional basis states, which would then
have to be orthogonal to the valence and conduction states of the NaCl film.
But such states would necessarily be higher in energy than the conduction
states already included. This implies that the additional basis states would
not be very important for describing gap states, which are located below the
conduction band. By contrast, the explicit use of just valence and conduction
states as basis states should already give a good qualitative picture of what
happens when tunnelling through the gap, and such calculations are relatively
easy to interpret.

In view of these results for Na+Cl- it is interesting to discuss Mg2+O2-. In
analogy to the discussion above, we do the thought experiment of increasing
the nuclear charge of O2- by three, obtaining Na+ with its 3s level at $-5.1$
eV. The attractive interaction of this additional nuclear charge with the $3s$
level is $20.0$ eV, comparable to the Madelung potential $23.9$ eV for the
ions Mg2+ and O2-. This suggests that the conduction band of MgO could be
located on oxygen, involving mainly O $3s$ levels. Indeed, this is what was
found in band structure calculations by de Boer and de Groot [35]. It would
then be very interesting to experimentally check the character of the
conduction bands both for the II-VI and for other I-VII ionic compounds.

Beyond these simple salts, it would indeed be fruitful to further experiment
with materials whose valence and conduction bands take on unusual combinations
of cationic and anionic character, e.g., that of the valence band being
cationic [36], or that of the conduction band being anionic [37]. There may
also be complementary mechanisms to probe that induce electronic structure
rearrangements like that of the Madelung potential.

To summarise, using STM we have shown that the conduction band of NaCl has
mainly Cl character, contrary to widespread belief. That the conduction band
can be anionic in character is related to the large Madelung potential, which
pushes the Na $3s$ level upwards and pulls the Cl $4s$ level downwards, which
for a free Cl- species is above the vacuum level. We provided a back of the
envelope calculation to make it plausible that the Madelung potential could
actually reverse the order of the Cl- $4s$ and the Na+ $3s$ levels, leading to
this result. This electronic structure may couple into other physical
processes. Incidentally, it is the negative ion vacancies that are important
to electrical conduction near the melting point of some alkali halides [38].

## Methods

Sample preparation and details of STM measurements – The experiments were
carried out with a home-built low-temperature STM operated at $T=4.3$ K, UHV
($<10^{-11}$ mbar) [39]. The Au(111) single-crystal ($>99.999\%$) sample was
cleaned by repeated cycles of Ar+ ion sputtering at $10^{-6}mbar$ range argon
pressure with 600 eV acceleration energy and subsequent annealing to 873 K.
The sample heating and cooling rate was about 1 K/s. NaCl was evaporated
thermally from a Knudsen cell held at $900$ K, with the Au(111) surface held
at $300$ K, to obtain defect-free, (100)-terminated NaCl islands. An
electrochemically etched gold wire [40] ($99.95\%$ purity) was used as tip in
the experiment. To ensure a metallic tip, the Au wire is further prepared by
controlled tip indentations ($1-3$ nm, $V=50-100$ mV) in Au(111) until atomic
resolution is obtained at tunnelling current set point: $I_{T}=10$ pA, $+1$ V.
The text always specifies bias voltages of the metal substrate with respect to
the grounded tip. Differential conductance (d$I$/d$V$) spectra were measured
using a standard lock-in technique with a bias modulation of $V_{rms}=4$ mV
(Fig. 1b inset) and $V_{rms}=10$ mV (Fig. 1b) at 629 Hz. Scanning tunnelling
microscopy/spectroscopy data were analysed using self-written MATLAB code.

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# On the Privacy of National Contact
Tracing COVID-19 Applications:
The Coronavírus-SUS Case

Jéferson Campos Nobre, Laura Rodrigues Soares, Briggette Olenka Roman
Huaytalla,
Elvandi da Silva Júnior, Lisandro Zambenedetti Granville
Federal University of Rio Grande do Sul, Brazil
Email: {jcnobre, lrsoares, borhuaytalla, elvandi.junior,
<EMAIL_ADDRESS>
###### Abstract

The 2019 Coronavirus disease (COVID-19) pandemic, caused by a quick
dissemination of the Severe Acute Respiratory Syndrome Coronavirus 2 (SARS-
CoV-2), has had a deep impact worldwide, both in terms of the loss of human
life and the economic and social disruption. The use of digital technologies
has been seen as an important effort to combat the pandemic and one of such
technologies is contact tracing applications. These applications were
successfully employed to face other infectious diseases, thus they have been
used during the current pandemic. However, the use of contact tracing poses
several privacy concerns since it is necessary to store and process data which
can lead to the user/device identification as well as location and behavior
tracking. These concerns are even more relevant when considering nationwide
implementations since they can lead to mass surveillance by authoritarian
governments. Despite the restrictions imposed by data protection laws from
several countries, there are still doubts on the preservation of the privacy
of the users. In this article, we analyze the privacy features in national
contact tracing COVID-19 applications considering their intrinsic
characteristics. As a case study, we discuss in more depth the Brazilian
COVID-19 application Coronavírus-SUS, since Brazil is one of the most impacted
countries by the current pandemic. Finally, as we believe contact tracing will
continue to be employed as part of the strategy for the current and potential
future pandemics, we present key research challenges.

###### Index Terms:

Contact Tracing; Coronavírus-SUS; COVID-19; SARS-CoV-2; Privacy.

##  I Introduction

The current global health crisis – i.e., the 2019 Coronavirus disease
(COVID-19) pandemic – was caused by the spread of the Severe Acute Respiratory
Syndrome Coronavirus 2 (SARS-CoV-2) [1]. This virus was discovered in late
2019 in China, from where it quickly disseminated worldwide. The form of
transmission is through contaminated secretions that spread through the air or
contact with a contaminated object. This transmission can occur even before
the manifestation of the first symptoms. Currently, most countries are unable
to carry out tests on their entire population in a short time because it
requires large financial investment, qualified personnel, and the manufacture
of products. In this context, governments and the private sector come together
to attempt to take advantage of digital technologies in efforts to combat the
pandemic [2].

Contact tracing111In this article, we use “contact tracing” as a synonym for
“digital contact tracing” is a technological method of monitoring the progress
of an infectious disease on a large population by detecting the contacts
between infected and healthy individuals [3]. These contacts are traced
through applications (apps) installed on mobile devices, usually smartphones.
These apps use different techniques to trace proximity: Global Positioning
System (GPS), triangulation of cellular operator antennas, electronic
transaction data (e.g., credit card data), and Bluetooth. In addition, tracing
data (i.e., contact history) can be treated in a centralized manner (processed
and stored on a central entity) or in a decentralized one (using users’
devices for processing and storage).

Contact tracing apps have been employed to face the COVID-19 pandemic since
they were proven to be effective in previous pandemics [4]. Also, the huge
adoption of smartphones along with the evolution of mobile network coverage
has increased the number of users who can use such apps. In addition, the
Google/Apple Exposure Notification (GAEN) system [5] fostered the development
of contact tracing apps by health authorities in a well-defined manner, for
both Android and iOS devices. Thus, contact tracing rose to fame during the
COVID-19 pandemic [2], being implemented by some countries at different
moments of the pandemic and with varying penetration rates.

The use of contact tracing must be accompanied by a discussion on how to
preserve the privacy of users. Otherwise, users may not adopt contact tracing
apps if they are suspicious about the private information that is being shared
or, depending on the users’ privacy concerns, even leaked. As such, contact
tracing apps should be designed to avoid personally identifiable information
to be collected on user/device basis. Data protection laws from several
countries impose restrictions on the approach to perform data processing along
with the details on the security measures required on stored personal data. On
the other hand, health agencies are usually the only ones who can operate on
collected data. In any case, there are significant concerns regarding an
unrestricted digital monitoring (i.e., mass surveillance) by authoritarian
governments, which can ultimately lead to “orwellian" systems.

In this article, we analyze privacy features in national contact tracing apps
designed to combat the COVID-19 pandemic. Because of the emergence of such
apps and the impact of similar ones on other pandemics, it is possible to
conclude that contact tracing will continue to be employed as part of the
SARS-CoV-2 containment strategies. However, logging encounters and exposure
notifications may lead to undesirable privacy attacks, especially considering
large-scale deployments. We depict a perspective on the privacy of national
contact tracing COVID-19 apps considering several variables, such as the
number of users, penetration, and localization. Finally, given that Brazil is
one of the most impacted countries by the current pandemic, we evaluate in
more detail the Brazilian COVID-19 application Coronavírus-SUS.

This article is organized as follows. In the next section, we present a
background on contact tracing, to then provide, in the following section, a
general view of privacy in the context of COVID-19. Afterwards, several
contact tracing apps employed by national authorities around the world are
evaluated. We then present a case study regarding the Coronavírus-SUS app.
Finally, we conclude the article in presenting key research challenges and
final remarks.

##  II Contact Tracing in a Nutshell

National governments as well as the private sector are working towards
developing computational tools to help on the effective management of the
COVID-19 pandemic. Contact tracing apps help break the chain of virus
transmission by notifying users whenever they are close to infected
individuals, thus before they become prone to infect others [6]. There are
precedents for such apps being used on other health crises, such as Ebola,
Tuberculosis, and HIV, as a part of the strategy for disease outbreak control
[4]. Consequently, even in the early stages of the COVID-19 pandemic, it was
hypothesized that contact tracing apps could be an important tool to decrease
the basic reproduction number ($R_{0}$) of SARS-CoV-2.

Contact tracing apps enable the monitoring of interactions of infected and
healthy users, thus detecting potential infections. After such detection, it
is necessary to notify the users, so measures to limit the spread can be taken
either by the user himself or by the health authorities. The tasks performed
by these apps can be aggregated into 3 groups: detection of contact events
(proximity tests), transmission, and exposure notification. The basic
operation of contact tracing apps is depicted in Figure 1. In Day 1, user A
(infected and with no symptoms) has an contact with user B (healthy). In Day
2, after symptoms onset on user A, he is tested and shares the positive test
result with the app which triggers an exposure notification to user B.

Figure 1: Basic operation of contact tracing applications.

The detection of contact events can be performed through several methods and
technologies. The majority of these methods is related with sensor
technologies that are integrated into current mobile devices. Proximity
tracing is a method usually performed using Bluetooth Low Energy (BLE) to
transmit messages containing identifiers to nearby devices. This can be
enhanced with the use of ultrasound so as to achieve better accuracy. Location
tracking is a method that can be performed using data from the Global
Positioning System (GPS) or cell tower triangulation. Geotagging is a method
that can also be used, where users scan the QR code with their mobile device
to record their visits, and consequently their localization data. Other
additional methods include the use of consumer credit card data and Closed-
Circuit TeleVision (CCTV) with facial recognition.

Different architectures can be employed to collect users’ data and contact
events. These architectures are depicted on Figure 2. Centralized
architectures collect the raw contact history data of mobile phones through a
communication network. After that, this data is stored and processed in a
central server, which generates reports and sends exposure notifications using
the same network. However, this centralization brings concerns over
dependability and performance. Decentralized architectures, as can be seen in
Figure 2, employ local resources for data storage and processing, which is
feasible since only contact events of the last 14 or 21 days (the number of
days depends on the app) need to be preserved. In both architectures, mobile
devices are usually responsible for generating temporary (also called
ephemeral) identifications.

Figure 2: Centralized and decentralized architecture for contact-tracing
applications.

Google and Apple formed a partnership to develop an interoperable interface
(between Android and iOS) contact event detection based on BLE technology in
April 2020. The result of this partnership is the Google/Apple Exposure
Notification (GAEN) system, which presents an Application Programming
Interface (API). This system is implemented at the operating system level to
avoid privilege problems. Contact tracing apps from many countries use this
API for their notification of exposure and the generation of temporary tokens
(to preserve user privacy). However, GAEN API is not open source and its
public documentation is limited, which brings concerns about the transparency
of the API itself [7].

A contact tracing app collects and exchanges sensitive users’ data on a
regular basis. However, such ability comes at a cost: privacy concerns. For
example, the use of proximity tracing causes the smartphone to send messages
in a repetitive manner that can be inspected. Besides, characteristics of
different architectures can impact the preservation of users’ privacy, since
the transmission, processing, and storage of users’ data are performed in
distinct ways. For instance, the anonymization and encryption can be performed
either in the mobile device or in the server.

##  III Privacy in the Context of COVID-19

The COVID-19 pandemic spread worldwide quickly and aggressively and exposed
either the lack or precariousness of public health policies to combat
epidemics. Most countries are unable to carry out tests on the entire
population at first notice, as they require a large financial investment,
qualified personnel, and the manufacture of products. Before large-scale
vaccination, social distancing and isolation still remain as the most
effective measures against the spread of COVID-19 [1]. In addition to the
health of the population, this pandemic also brought new risks to the security
of personal data, as the world adapts to the novel paradigms for working and
socializing. Threats to the privacy of personal data are one of the main
issues and are sure to be felt for years to come if not addressed
appropriately from the beginning.

The privacy subject comes under scrutiny almost immediately as the majority of
the global workforce transitioned to home office due to social distancing.
Contact tracing technologies represent a new incursion into privacy issues. In
healthcare applications, the data traveling through potentially insecure
connections are especially sensitive, since this concerns the physical welfare
of people. Therefore, it is important that technology companies, healthcare
providers, and public health officials operate with the highest ethical
standards, in particular when the existing privacy regulations do not apply.

COVID-19 contact tracing apps have another relevant issue besides the
sensitive nature of its data: the extensive coverage among the citizens of a
country to reach the intended efficacy. Thus, most of the such apps so far are
developed by governmental offices [8]. This arrangement has several benefits:
in most cases, the government already has in place the necessary structure to
launch a nation-wide application, as well as the data protection mechanisms to
ensure the privacy of its users. However, even institutional offices in a
country are not immune to data leakage. For example, in November 2020, an
employee error disclosed passwords from the Ministry of Health of the
Brazilian Government, leading to massive data leakage regarding COVID-19
patients [9]. At least 16 million people have had their data exposed,
including pre-existing illnesses and COVID-19 status. Also, the passwords
still gave access to two federal government records: one for notifications of
suspected cases and the other with hospitalizations for Severe Acute
Respiratory Syndrome (SARS).

Several data protection regulations were created to ensure the safe handling
of sensitive personal information. The Health Insurance Portability and
Accountability Act (HIPAA) [10] from the United States, a pioneer in data
regulation, was created in 1996 to regulate how personally identifiable
information from healthcare industries should be maintained. More recently,
broader regulation laws were also developed. From the European Union (EU), the
General Data Protection Regulation (GDPR) [11] aims to protect an individual’s
personal data not only inside the area but in any enterprise that processes
the personal information of its citizens as well. The GDPR was later used as a
base for the Brazilian Lei Geral de Proteção de Dados (LGPD) [12], Brazil’s
data protection regulation law.

Contact tracing apps can be a corner case for privacy in the context of
COVID-19. These apps handle personally identifiable data (e.g., names, phone
number, address, COVID-19 tests), exposure notifications, and graphs of social
density (indications of close contact between users). Since the effectiveness
of such apps depends on these data, the discussion on the users privacy tends
to be put in opposition of public health. Besides, there are concerns that the
current state of epidemiological surveillance can surpass the COVID-19
pandemic and become a permanent part of governmental strategies. In next
section, we analyse the current landscape of national contact tracing COVID-19
apps and their privacy features.

##  IV National Contact Tracing COVID-19 Apps and Privacy

The majority of national governments have employed computational-based
approaches to face the COVID-19 pandemic. Regarding the specificities of this
pandemic, national strategies are of paramount importance because only they
can provide the required level of resources and logistics to avoid the
shortage of healthcare facilities (especially Intensive Care Unit beds). In
this context, the amount of sensible data collected from the users in these
approaches is significant. In addition, as discussed before, the necessary
social distancing protocols increased concerns over privacy. Such concerns are
even more distressful considering that mobile apps have access to users’
location. This is exactly the main feature required for contact tracing apps.

In Table I, we survey a selection of contact tracing apps in different
countries (For a more comprehensive view, we suggest the “MIT Technology
Review Covid Tracing Tracker" [8]). As prerequisites, the applications should
be already launched and from a governmental developer. If these requisites are
met, we aimed for a broader set of features such as user number, adherence,
and chosen technologies. Aarogya Setu from India is by far the largest in
number of users, a relevant matter for the present article since Brazil has
the 6th largest population in the world. Ehteraz from Qatar is the most
adopted contact tracing app, reaching 91% of the population. Israel’s HaMagen
and UK’s NHS COVID-19 App have satisfactory adherence. In addition, the
National Health Service (NHS), England’s publicly funded healthcare system, is
very similar to Brazil’s Sistema Único de Saúde. Both New Zealand and Japan
have unique approaches to face the pandemic. Mexico, in turn, faces several of
the same socioeconomic problems historically found in Brazil.

Location | Name | Users | Penetration | Contact Detection | Distribution
---|---|---|---|---|---
India | Aarogya Setu | 163.000.000 | 12,05% | Bluetooth, Location | Centralized
Israel | HaMagen | 2.000.000 | 22,51% | Location | Centralized
Japan | COCOA | 7.700.000 | 6,09% | Bluetooth, GAEN | Decentralized
Mexico | CovidRadar | 50.000 | 0.04% | Bluetooth | Centralized
New Zealand | NZ COVID Tracer | 588.800 | 12,10% | Bluetooth, QR codes | Centralized
Qatar | Ehteraz | 2.531.620 | 91% | Bluetooth, Location | Centralized
UK | NHS COVID-19 App | 19.000.000 | 28,51% | Bluetooth, GAEN | Decentralized
TABLE I: Selected national contact tracing apps.

BLE is the most used technology for the detection of contact events in the
analyzed applications (Table I). Geo-location technology is used exclusively
in Israel’s HaMagen app, and in conjunction with BLE in Aarogya Setu of India
and Ehteraz of Qatar. Geo-location and a centralized architecture for the data
collection data are considered more invasive, because they expose the personal
and location data collected from users to several national control
authorities. They are also more susceptible to hacks and leaks of personal
data. Data centralization is used in applications in India, Israel, Mexico,
New Zealand and Qatar. In contrast, applications from Brazil, Japan and the UK
use data decentralization. This method is considered less invasive, as the
collected data is stored on users’ devices, maintaining anonymity.

The preservation of users’ privacy, especially considering the national
coverage of the contact tracing apps depicted on Table I, must follow best
practices (as well as data protection laws) for the use of population tracking
data. In this context, when using data from a source other than the user’s own
device (e.g., telecommunications operators), the usage of such apps must be
performed on a voluntary basis. Thus, these apps should be only allowed to
collect personal information strictly related to SARS-Cov-2 tracking. For
example, BLE-enabled apps should be prohibited from collecting location data
(which is not the case for Aarogya Setu and Ehteraz). In addition, consent
must be provided by the user from health and safety agencies to share a
positive test result.

Several of the privacy issues found in these applications can also be found in
the Brazilian application Coronavírus-SUS. We purposefully left Coronavírus-
SUS outside the selection so we can conduct a more detailed study about
particular privacy issues in the next section, as well as an overview of the
obstacles for contact tracing adhesion in Brazil.

##  V Brazilian Contact Tracing App: Coronavírus-SUS

The official application for COVID-19 contact tracing in Brazil, Coronavírus-
SUS, was launched on February 28th, 2020 as a basic version, with features
such as information about symptoms, COVID-19-related official news, and a map
with the public health facilities in one’s region. Later, on the September
31st, the app was upgraded with contact tracing functions. Coronavírus-SUS was
promoted by the Brazilian National Government and developed by DATASUS, the
Department of Informatics of Brazil’s public-funded healthcare system, the
Sistema Único de Saúde (SUS). It uses GAEN API for anonymous Bluetooth sharing
of tokens.

The users of Coronavírus-SUS are informed about the application’s privacy
policy when first downloading it or after upgrading to the contact tracing
version. That policy states that no personal data is collected, no GPS data is
used, all communications are encrypted, all information is stored in data
servers in Brazil, and there is no way of finding out one’s identity or the
identity of whoever comes in contact with. It is possible to use other
features (e.g., news and health facility map) and decline to use the contact
tracing feature, but the user must agree with the privacy policy anyway.

Coronavírus-SUS can only work properly if the patient who tested positive for
COVID agrees in sharing that result, like in any other contact tracing
application. A positive patient must validate his/her test in a separate
portal (http://validacovid.saude.gov.br) to prevent false positives and abuse.
The system will cross-validate the patient’s test with the data present in the
public Rede Nacional de Dados em Saúde (RNDS, National Network of Health
Data), where information about COVID-19 patients in Brazil is stored, and
generate a token to be used for confirmation in the app. This portal has a
separate privacy policy in compliance with Brazil’s data protection
regulation, LGPD222https://validacovid.saude.gov.br/politica-privacidade.
Image 3 shows a contact notification as well as the "Validate Your Positive
Test" option.

Figure 3: Example of contact notification. On the left, the app informs the
contact-tracing feature is enabled and about potential exposure to COVID-19.
There is also the option to share a positive test. On right, all the contact
notifications the user had are displayed. The application is available in
Portuguese, English, and Spanish.

We present the information gathered from Fala.BR, Brazil’s platform for public
information access. Any Brazilian citizen can request public information from
the Federal Executive Office, according to the federal law no. 12.527 of
November 18th, 2011. The legal proceedings of our request for information, as
well as the official answer, are publicly accessible (in portuguese) through
Fala.BR website[13]. Table II resumes the obtained information. As one can see
in this table, the adhesion of Coronavírus-SUS, despite a favorable launching,
is not yet expressive enough for a country with over 210 million people.
Furthermore, active users have the option to not enable the contact tracing
feature, so the reachability of the application is even smaller than the
stated 1.81%. Currently, the developers have no way to tell whether a user of
Coronavírus-SUS application has enabled the contact tracing feature in their
smartphone because of privacy demands, as stated by DATASUS in our request for
information available [13]. The extra proceedings for the validation of a
positive result may be keeping the users from doing it, since only on November
10th, 2020 Brazil had 23,973 new coronavirus cases, a number over 15 times
greater than the total positive tests validated in the application.

Attribute | Value
---|---
Total downloads | 10.530.000
Active users (Google Play) | 1.520.000
Active users (Apple Store) | 2.290.00
Penetration | 1.81%
Total of positive tests shared | 1.573
TABLE II: Information about Coronavírus SUS adhesion, on the 10th of November,
2020.

Based on the previous analysis regarding the privacy of contact tracing
applications, Coronavírus-SUS functionality is less invasive and respects the
privacy of its users regarding anonymity, decentralization, and storage of
data. However, this decentralization impairs the developers’ analytics to a
certain degree, as stated by DATASUS [13]. The management issue might be
keeping the Coronavírus-SUS from reaching the intended population, since data
analytics plays a considerable role in a country with considerable regional
inequality of the size of Brazil. In addition, the lack of official
advertisement of the Coronavírus-SUS application from the Brazilian Government
and from the major news outlets might be the main issue preventing its
adhesion. As the COVID-19 pandemic is still far from over, is still possible
for Coronavírus-SUS developers to achieve common ground between user privacy
and application management, and for the Brazillian Government to run the
necessary marketing campaign to ensure its reachability.

##  VI Key Research Challenges and Final Remarks

The COVID-19 pandemic has caused directly millions of deaths all over the
world as well as an important number of indirect ones, due to the exhaustion
of the healthcare systems. Besides, the impact on the economy of many
countries can decrease significantly the overall quality of life of people. In
this context, Information and Communications Technologies (ICT) can be
employed to control the impact of the pandemic. For example, contact tracing
apps are expect to provide significant benefits in contagious disease. These
apps can be viewed as a special kind of healthcare app that integrates
specific communication technologies. In any case, more studies are needed to
understand the impact of contact tracing on the evolution of the pandemic,
since this will allow to include it in an evidence-based healthcare approach.

The rush for tools to face the spread of Sars-CoV-2 virus has fostered the
rise of the several novel platforms for contact tracing. In this context,
especially considering countries without public national healthcare systems,
some regional entities (e.g., states and counties) are developing regional
contact tracing apps, employing a decentralized approach. However, such
decentralization hampers the execution of processing and analytics tasks. Even
if the regional apps use the same APIs (such as the GAEN one), it is still
unclear how the collected data could feed a national healthcare strategy.
Thus, it is necessary to discuss how to employ federations of contact-tracing
apps. A platform for these federations may present features, such as data
deduplication, capacity optimization, and integrated analysis tools.

The data collected on contact tracing apps can be valuable for different
healthcare related tasks. Thus, several different apps are expected to
communicate with contact tracing ones. For example, remote patient monitoring
and telemedicine apps could be enhanced with localization data. In any case,
it would be necessary to decide how to implement privacy policies in order to
protect the healthcare users. We envision that several organizations, besides
the governmental ones, may be interested in the features supported by contact
tracing. An example of these organizations is healthcare insurance.

The research and development efforts to face the global COVID-19 health crisis
will unveil new uses for contact tracing apps. Depending on the tasks and
their intrinsic characteristics, such uses can impose challenges regarding
privacy features. An example is the monitoring of Long COVID-19, also known as
Chronic COVID Syndrome. In this case, the symptoms caused by the SARS-CoV-2
virus persist after the usual convalescence period. Thus, it is necessary to
follow the use for a longer period. This long-term approach requires a
longitudinal privacy analysis, since as more data is collected, more privacy
concerns arrive.

As future work, we intend to enhance the evaluation of Coronavírus-SUS as well
as other relevant contact tracing apps. We plan to perform an in-depth
forensic analysis of the Android version of Coronavírus-SUS app. Furthermore,
we are also looking at additional settings that could lead to important
effects. For example, it is necessary to evaluate privacy features regarding
the intersection of contact tracing and vaccination. In addition, it is
necessary to propose guidelines for protecting users in order to prepare
humanity to face similar challenges in future, without occurring in mass
surveillance.

## References

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  * [13] Controladoria-Geral da União, “Fala.br, plataforma integrada de ouvidoria e acesso à informação,” Avaliable at https://falabr.cgu.gov.br/publico/Manifestacao/DetalharManifestacaoSolicitante?id=LjbkOH1ly3Q%3D&cac=oZQJ0M3IXLfIpY11wJiE3g%3D%3D, accessed on 2020/12/31.

Jéferson C. Nobre<EMAIL_ADDRESS>is Adjunct Professor at the Institute
of Informatics, Federal University of Rio Grande do Sul (UFRGS). He holds a
Ph.D. in Computer Science (UFRGS, 2015). Prof. Nobre has been involved in
various research areas, mainly computer networks, network management, and
computer system security. He has been publishing his works in important
journals and conferences, such as the IEEE Communications Surveys and
Tutorials, the IEEE Communications Magazine, the Journal of Network and
Systems Management (Springer), the IFIP CNSM, IEEE/IFIP NOMS, and the
IFIP/IEEE IM. He also served in the organization of scientific events and in
the TPC of relevant symposia.
---
Laura R. Soares<EMAIL_ADDRESS>is concluding the Bachelor degree in
Computer Science at the Institute of Informatics, Federal University of Rio
Grande do Sul (UFRGS). She has an Academic Research Scholarship from the
National Council of Scientific and Technologic Development (CNPq). Her
research areas of interest include computer networks, network management, and
computer system security. She also volunteered at the organization of academic
events in the University.
---
Briggette R. Huaytalla<EMAIL_ADDRESS>is a student in Master
degree in Computer Science at the Institute of Informatics of the Federal
University of Rio Grande do Sul (INF-UFRGS), Brazil, and member of the
Computer Networks Group. Bachelor in Computer Science at National University
of Engineering (UNI), Perú, and member of Scientific association specialized
in computing (ACECOM-UNI). Briggette has an Academic Research Scholarship from
the National Council of Scientific and Technologic Development (CNPq). Her
research areas of interest include computer networks, computer system
security, network security, application security, programmable networks and
artificial intelligence.
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Elvandi S. Júnior<EMAIL_ADDRESS>is Professor at the Vicente
Dutra State Education Institute and Tutor at the University of Vale do Rio dos
Sinos (UNISINOS). He holds a Ph.D. in Nanosciences (UFN, 2020), M.Sc. in
Nanosciences (UFN, 2014), M.Sc. in Professional and Technological Education by
the Federal University of Santa Maria (UFSM, 2019), Bachelor in Computer
Science (UFN, 2012). Prof. Elvandi Júnior has been involved in several
research areas, mainly computer networks, nanosciences, with an emphasis on
microcontrollers and nanosensors, education and educational digital games.
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Lisandro Zambenedetti Granville<EMAIL_ADDRESS>is Full Professor at
the Institute of Informatics of the Federal University of Rio Grande do Sul
(INF-UFRGS), Brazil. Lisandro was chair of the IEEE ComSoc’s Committee on
Network Operations and Management (CNOM), co-chair of the IRTF’s Network
Management Research Group, and president of Brazilian Computer Society (SBC).
He has served and general chair and TPC chair of conferences such as NOMS, IM,
CNSM, and ICC, among others. Lisandro is editorial member of the Springer
Journal of Network and Systems Management (JNSM) and IEEE Transactions on
Network and Service Management (TNSM). He is also associate editor-in-chief of
Wiley International Journal of Network Management (IJNM) and editor-in-chief
of Springer Journal of Internet Services and Applications (JISA).
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