# Discretization of the wave equation on a metric graph

Sergei A. Avdonin1 , Aleksander S. Mikhaylov2,3, Victor S. Mikhaylov2. Abdon
E. Choque-Rivero4, CONTACT Sergei A. Avdonin. Email<EMAIL_ADDRESS>1
Department of Mathematics & Statistics, University of Alaska Fairbanks,
Fairbanks, Alaska, USA; 2 St.Petersburg Department of Steklov Mathematical
Institute, Russian Academy of Sciences, St. Petersburg, Russia; 3 Department
of Mathematics, St. Petersburg State University, St. Petersburg, Russia; 4
Institute of Physics and Mathematics, Universidad Michoacana de San Nicolas de
Hidalgo, Michoacan, Mexico

###### Abstract

The question of what conditions should be set at the nodes of a discrete graph
for the wave equation with discrete time is investigated. The variational
method for the derivation of these conditions is used. A parallel with the
continuous case is also drawn. As an example the problem of shape
controllability from the boundary is studied.

###### keywords:

Discrete graph, node conditions, wave equation.

35L05, 35Q93, 93B05, 93C20, 37M15, 37J51

##  1 Introduction

Under differential equation networks (DENs) or, in other words, quantum graphs
we understand differential equations on metric graphs coupled by certain
vertex matching conditions. These models play a fundamental role in many
problems of science and engineering. The range for applications of DENs is
enormous and continues to grow; we will mention a few of them.

– Structural Health Monitoring. DENs, classically, arise in the study of
stability, health, and oscillations of flexible structures that are made of
strings, beams, cables, and struts [1, 2, 3, 4]

– Water, Electricity, Gas, and Traffic Networks. An important example of DENs
is the Saint-Venant system of equations, which model hydraulic networks for
water supply and irrigation [5] and first-order hyperbolic equations [6, 7, 8,
9, 10, 11] and the isothermal Euler equations for describing the gas flow
through pipelines [12, 13].Other important examples of DENs include the
telegrapher equation for modeling electric networks [14], the diffusion
equations in power networks [15], and Aw-Rascle equations for describing road
traffic dynamics [16], see also [17] for traffic flow on networks and [18, 19,
20] for modeling groundwater flow.

– Nanoelectronics and Quantum Computing. Mesoscopic quasi-one-dimensional
structures such as quantum, atomic, and molecular wires are the subject of
extensive experimental and theoretical studies [21, 22] The simplest model
describing conduction in quantum wires is the Schrödinger operator on a planar
graph. For similar models appear in nanoelectronics, high-temperature
superconductors, quantum computing and studies of quantum chaos see [23, 24,
25].

– Material Science. Quantum graphs arise in analyzing hierarchical materials
like ceramic and metallic foams, percolation networks, carbon and graphene
nano-tubes, and graphene ribbons [26, 27].

– Biology. Challenging problems involving ordinary and partial differential
equations on graphs arise in signal propagation in dendritic trees, particle
dispersal in respiratory systems, species persistence and biochemical
diffusion in delta river systems [28, 29].

– Social Networks. Examples of social applications are modeling of
international trades and space-temporal patterns of information spread [30].

There are many papers in the literature devoted to study of the spectral
properties of differential operators on graphs and well-posedness of the
initial boundary value problems for differential equations on graphs and
regularity of their solutions [31, 32, 33, 34, 35, 36, 37]. On the other hand,
numerical methods for solving ODEs and PDEs on graphs have been mostly
developed only for very specific problems [11, 38, 39, 40, 41, 42, 43, 44].
Only recently more general investigations appeared directed to developing the
finite element method for elliptic and parabolic equations on graphs [45, 46,
47].

In the present paper we study the problems of discretization of the wave
equation on metric graphs. The main attention is put to discretization of the
Kirchoff – Neumann matching conditions at the internal vertices. This
important problems has not got a proper attention in the literature. To derive
the matching conditions we apply Hamilton’s principle and variational methods
in both continuous and discrete cases. Then we discuss a proper choice of the
nodal weights for the discrete model to insure the same reflection and
transmission coefficients as the original continuous model has. The paper is
organized as follows. In Section 2 we derive the matching conditions for
metric graphs and in Section 3 — for discrete graphs. In Section 4 we discuss
transmission and reflection of the wave at the internal nodes in both,
continuous and discrete, situations. In Section 5 we solve direct and control
problems for the wave equation on a discrete star graph.

##  2 Conditions at nodes on a metric graph

Let $\Omega$ be a finite connected compact graph. The graph consists of edges
$E=\\{e_{1},\ldots,e_{N}\\}$ connected at vertices
$V=\\{v_{1}\ldots,v_{M}\\}$. Each edge $e_{j}\in E$ is identified with an
interval $(0,l_{j})$ of the real line. The boundary
$\Gamma=\\{v_{1},\ldots,v_{m}\\}$ of $\Omega$ is the set of vertices whose
degree is equal to one (outer nodes). Let $E(v)$ be a set of edges incident to
$v$. In what follows, we assume that some of the boundary vertices are
clamped, and the non-homogeneous Dirichlet boundary condition is imposed on
the other part.

The space of real square-integrable functions on the graph $\Omega$ is denoted
by $L_{2}(\Omega):=\bigoplus_{i=1}^{N}L_{2}(e_{i}).$ For the function $u\in
L_{2}(\Omega)$ we will write

$u:=\left\\{u^{i}\right\\}_{i=1}^{N}\ ,\ u^{i}\in L_{2}(e_{i}).$

By C we denote the direct problem on the metric graph:

$\displaystyle u^{i}_{tt}(x,t)-u^{i}_{xx}(x,t)=0\quad x\in e_{i},$ (1)
$\displaystyle u^{i}(v,t)=u^{j}(v,t),\quad e_{i},\ e_{j}\in E(v),\quad v\in
V\backslash\Gamma,$ (2) $\displaystyle\sum_{e_{i}\in E(v)}\partial
u^{i}(v,t)=0,v\in V\backslash\Gamma,$ (3) $\displaystyle
u(v,t)=f(t)\quad\text{for}\,\,v\in\Gamma,$ (4) $\displaystyle
u(x,0)=0,u_{t}(x,0)=0,\ \quad\text{for}\,\,x\in\Omega.$ (5)

Here $\partial u^{i}(v,t)$ is the derivative of $u^{i}$ at $v$ along $e_{i}$
in the direction away from $v$. This system describes small transverse
oscillations of the graph (Figure 1), and $u(x,t)$ denotes the displacement of
the point $x\in\Omega$ of the graph at time $t$ from the equilibrium position.
It is assumed that the boundary of the graph moves according to the law
$f(t)$, and in the initial moment of time the graph was at rest in an
equilibrium state.

Figure 1: Wave equation on a metric graph with six edges and seven vertices.
On the edge $e_{1}$ (resp. $e_{4}$) the function $f^{1}(t)$ (resp. $f^{4}(t)$)
is acting. On the edge $e_{4}$ the corresponding displacement $u(x,t)$ is
represented.

In order to write a similar system of equations for a discrete graph, let us
recall how equations (1)–(5) arose. We introduce the kinetic and potential
energies:

$\displaystyle T(t)=\frac{1}{2}\int_{\Omega}u_{t}^{2}(x,t)\,dx,$ (6)
$\displaystyle U(t)=\frac{1}{2}\int_{\Omega}u_{x}^{2}(x,t)\,dx.$ (7)

According to the principles of classical mechanics (Hamilton’s principle of
least action), the system passes from state 1 at time $t_{1}$ to state 2 at
time $t_{2}$ in such a way that the variation of the action functional
vanishes

$S[u]=\int_{t_{1}}^{t_{2}}L(t)\,dt,\quad L=T-U\text{ (Lagrangian)},$ (8)

along the true trajectory of movement. It means that

$\delta S[u,h]=0\quad\text{for all}\quad
h:h|_{t=t_{1}}=0,h|_{t=t_{2}}=0,h|_{\Gamma}=0.$

Calculating the variation of the functional and integrating by parts, we get
the following:

$\displaystyle 0=$ $\displaystyle\delta
S[u,h]=\int_{t_{1}}^{t_{2}}\int_{\Omega}(u_{t}h_{t}-u_{x}h_{x})\,dx\,dt$
$\displaystyle=$
$\displaystyle-\int_{t_{1}}^{t_{2}}\int_{\Omega}(u_{tt}-u_{xx})h\,dx\,dt+\int_{t_{1}}^{t_{2}}\int_{V\backslash\Gamma}\partial
uh\,dx\,dt,$

where the integrals over the boundary $\Gamma$ and for $t=t_{1,2}$ vanish
because of the requirements on $h$. From the last equality and the main Lemma
of the Calculus of Variations (the Dubois-Reymond Lemma), we obtain the
equation (1) and the condition at the internal node (3), the so-called
Kirchhoff condition. The condition (2) is a natural continuity condition. The
first condition in (5) means that we know the position of the system at time
$t=t_{1}=0$, and instead of writing that we know the position of the system at
time $t=t_{2}$ we write the second condition in (5). Thus, the second initial
condition does not follow from the principle of least action, but replaces the
condition of knowing the position of the system at the final moment of time.

######  Remark 2.1.

If there is a point mass $m_{j}$ at the node $v_{j}\in V$, $j=m+1,\ldots,N+1$,
then the kinetic energy of this mass
$m_{j}{\displaystyle\frac{u^{2}_{t}(v_{j},t)}{2}}$ must be added to the
kinetic energy $T$. Then the Lagrangian will change and the variation of the
action functional will look like this:

$\delta S[u,h]=\ldots+\int_{t_{1}}^{t_{2}}m_{j}u_{t}(v_{j},t)h_{t}\,dt\
=\ldots-\int_{t_{1}}^{t_{2}}m_{j}u_{tt}(v_{j},t)h\,dt.$

The last term contributes to the knot, and the Kirchhoff condition takes the
following form

$m_{j}u_{tt}(v_{j},t)=\sum_{e_{i}\in E(v_{j})}\partial u^{i}(v_{j},t).$

This equality is nothing but Newton’s second law - the product of mass and
acceleration is the sum of all forces acting on the body. On the right side,
just $\partial u^{i}(v_{j},t)$ is the tension force from the side of the edge
$e^{i}$ acting on the node $v_{j}$.

In the next section we consider discrete analogue of system (1)–(5) – a
discrete wave equation on a discrete graph.

##  3 Conditions at nodes on a discrete graph

In this section, we assume that our metric graph $\Omega$ has been replaced by
a discrete graph $\Omega_{D}$. This means that the vertices have remained in
their places, and each edge has become a finite discrete set of points, or, in
other words, each edge $e_{j}\in E$ is now identified not with the interval
$(0,l_{j})$ of the real line, but with a finite set of numbers
$a^{j}_{0}<a^{j}_{1}<\ldots<a^{j}_{N_{j}}$. For simplicity, we can assume that
$a^{j}_{0}=0$, $a^{j}_{1}=1$,…$a^{j}_{N_{j}}=N_{j}$ (Figure 2).

Figure 2: Discrete Wave equation on a discrete graph with six edges and seven
vertices. On the edge $e_{4}$ is acting the function $f^{4}(t)$. On the same
edge the discrete displacement of $u^{4}_{j,t}$ is represented.

In addition, we assume that the time $t$ is also discrete, i.e. we replace the
interval $(0,T)$ with the discrete set $t_{0}<t_{1}<\dots<t_{T}$ or, for
simplicity, $0<1<\ldots<T$. The space of real square summable functions on the
graph $\Omega_{D}$ is denoted by
$L(\Omega_{D}):=\bigoplus_{i=1}^{N}\mathbb{R}^{N_{i}}.$ For the function $u\in
L(\Omega_{D})$ we will write

$u:=\left\\{u^{i}\right\\}_{i=1}^{N},\quad
u^{i}=(u^{i}_{0},\ldots,u^{i}_{N_{i}}),\quad u^{i}_{j}\in\mathbb{R},\
j=0,\ldots,N_{i}.$

In order to unify the value at the node $v_{k}$, we introduce the function
$F(k)=\left[\begin{array}[]{l}0\\\ N_{k}\end{array}\right.$ such that

$u(v_{k})=u^{j(k)}_{F(k)},\quad{\text{where }j(k)\in E(v_{k})},\quad
k=1,\ldots,M.$

Since we want to consider a discrete wave equation, the function $u$ must
depend on two variables - coordinate and time. Time is now discrete, so we
introduce this dependence through an additional index on the vector $u^{i}$.
We will write

$u(t):=\left\\{u^{i}(t)\right\\}_{i=1}^{N},\quad
u^{i}(t)=(u^{i}_{0,t},\ldots,u^{i}_{{N_{i}},t}),\quad
u^{i}_{j,t}\in\mathbb{R},\ j=0,\ldots,N_{i}.$

Before reformulating C – a direct problem on a metric graph and writing a
similar discrete one, we consider the discrete analogs of the kinetic and
potential energy functionals:

$\displaystyle T_{D}(t)=$
$\displaystyle\sum_{i=1}^{N}T^{e}_{i}(t)+\sum_{i=1}^{M}T^{v}_{i}(t)=\sum_{i=1}^{N}\sum_{j=1}^{N_{i}-1}\frac{(u^{i}_{j,t}-u^{i}_{j,t-1})^{2}}{2}+\sum_{i=1}^{M}\frac{(u^{j(i)}_{F(i),t}-u^{j(i)}_{F(i),t-1})^{2}}{2},$
(9) $\displaystyle U_{D}(t)=$
$\displaystyle\sum_{i=1}^{N}U^{e}_{i}(t)=\sum_{i=1}^{N}\sum_{j=1}^{N_{i}}\frac{(u^{i}_{j,t}-u^{i}_{j-1,t})^{2}}{2}.$
(10)

Here $T^{e}_{i}$, $U^{e}_{i}$, $i=1,\ldots N$ are the kinetic and potential
energies from the interior points of the $e_{i}$ edges, and $T^{v}_{i}$,
$i=1,\ldots N+1$ — kinetic energy from $v_{i}$ vertices. In order to write the
action functional in the discrete case, it was necessary to integrate the
Lagrangian over the interval $(t_{1},t_{2})$. In a discrete situation, the
formula looks like this:

$S[u]=\sum_{t=1}^{T}T_{D}(t)-\sum_{t=0}^{T}U_{D}(t).$

Let us apply the principle of least action to this quadratic functional.
According to this principal the variation of the action functional $S[u]$
along the true trajectory of motion vanishes:

$\displaystyle 0=$ $\displaystyle\delta
S[u,h]=\sum_{t=1}^{T}\sum_{i=1}^{N}\sum_{j=1}^{N_{i}-1}(u^{i}_{j,t}-u^{i}_{j,t-1})(h^{i}_{j,t}-h^{i}_{j,t-1})+\sum_{t=1}^{T}\sum_{i=1}^{M}(u^{j(i)}_{F(i),t}-u^{j(i)}_{F(i),t-1})(h^{j(i)}_{F(i),t}-h^{j(i)}_{F(i),t-1})-$
$\displaystyle-\sum_{t=0}^{T}\sum_{i=1}^{N}\sum_{j=1}^{N_{i}}(u^{i}_{j,t}-u^{i}_{j-1,t})(h^{i}_{j,t}-h^{i}_{j-1,t})$
$\displaystyle=$
$\displaystyle\sum_{t=1}^{T}\sum_{i=1}^{N}\sum_{j=1}^{N_{i}-1}h^{i}_{j,t}(u^{i}_{j,t}-u^{i}_{j,t-1}-(u^{i}_{j,t+1}-u^{i}_{j,t}))+\sum_{t=1}^{T}\sum_{i=1}^{M}h^{j(i)}_{F(i),t}(u^{j(i)}_{F(i),t}-u^{j(i)}_{F(i),t-1}-(u^{j(i)}_{F(i),t+1}-u^{j(i)}_{F(i),t}))$
$\displaystyle-\sum_{t=0}^{T}\sum_{i=1}^{N}\sum_{j=1}^{N_{i}-1}h^{i}_{j,t}(u^{i}_{j,t}-u^{i}_{j-1,t}-(u^{i}_{j+1,t}-u^{i}_{j,t}))-\sum_{t=0}^{T}\sum_{i=1}^{N}h^{i}_{0,t}(u^{i}_{0,t}-u^{i}_{1,t})-\sum_{t=0}^{T}\sum_{i=1}^{N}h^{i}_{N_{i},t}(u^{i}_{N_{i},t}-u^{i}_{N_{i}-1,t}).$

Now we use the fact that the factors $h^{i}_{j,t}$ in the above sum are
arbitrary except boundary $\Gamma$ and, in view of this, we obtain the
equation at the internal points of the edges:

$\displaystyle 0=$
$\displaystyle(u^{i}_{j,t}-u^{i}_{j,t-1}-(u^{i}_{j,t+1}-u^{i}_{j,t}))-(u^{i}_{j,t}-u^{i}_{j-1,t}-(u^{i}_{j+1,t}-u^{i}_{j,t}))$
$\displaystyle=$
$\displaystyle-(u^{i}_{j,t+1}+u^{i}_{j,t-1}-u^{i}_{j+1,t}-u^{i}_{j-1,t}),$
(11)

where $i=1,\ldots,N$, $t=1,\ldots,T-1$, $j=1,\ldots,N_{i}-1$. At the boundary
points we assumed the same conditions as in (4):

$u(v_{k},t)=f^{k}(t)\quad\text{for}\,\,v_{k}\in\Gamma.$

It remains to write the condition at the internal nodes. Consider vertex
$v_{k}$. Collecting together the terms with the factor $h^{j(k)}_{F(k),t}$ and
equating it to zero, we get:

$(u^{j(k)}_{F(k),t}-u^{j(k)}_{F(k),t-1}-(u^{j(k)}_{F(k),t+1}-u^{j(k)}_{F(k),t}))-\sum_{i:e_{i}\in
E(v_{k})}(u^{i}_{F(i),t}-u^{i}_{|F(i)-1|,t})=0,$ (12)

where $|F(i)-1|=\left[\begin{array}[]{l}1\\\ N_{i}-1\end{array}\right.$ and
thus $(u^{i}_{F(i),t}-u^{i}_{|F(i)-1|,t})$ is a discrete analogue of operation
$\partial$ on a metric graph.

Assume that the number of elements in $E(v_{k})$ is $p_{k}$. Then the last
equality is transformed to

$-u^{k}_{F(k),t+1}-u^{k}_{F(k),t-1}+(2-p_{k})u^{k}_{F(k),t}+\sum_{e_{i}\in
E(k)}u^{i}_{|F(i)-1|,t}=0.$ (13)

Thus, we have obtained a discrete wave equation on the graph (11) and
conditions at the nodes (12) or (13), which are analogues of (1) and (3)

######  Remark 3.1.

If at the node $v_{k}\in V$, there is a point mass $m_{k}$, then to the
kinetic energy $T_{D}$ it is necessary to add the kinetic energy of this mass
$m_{k}{\displaystyle\frac{(u(v_{k},t)-u(v_{k},t-1))^{2}}{2}}$. Then the
Lagrangian will change and the variation of the action functional and the
condition at the node will look like this:

$-(1+m_{k})u^{k}_{F(k),t+1}-(1+m_{k})u^{k}_{F(k),t-1}+(2+2m_{k}-p_{k})u^{k}_{F(k),t}+\sum_{e_{i}\in
E(v_{k})}u^{i}_{|F(i)-1|,t}=0.$ (14)

######  Remark 3.2.

If we put $m_{k}=-1$ then (14) becomes

$\sum_{e_{i}\in E(v_{k})}(u^{i}_{|F(i)-1|,t}-u^{k}_{F(k),t})=0,$ (15)

which is a direct discrete analog of the Kirchhoff condition (3). Note that
the formal choice of $m_{k}=-1$ practically means that there is zero mass at
node $v_{k}$, because by our definition, the kinetic energy functional (9)
contained the energy $T^{v}_{k}$ from the unit mass at the node $v_{k}$.

######  Remark 3.3.

If we put

$m_{k}=\frac{p_{k}-2}{2},$ (16)

then (14) become

$-\frac{p_{k}}{2}u^{k}_{F(k),t+1}-\frac{p_{k}}{2}u^{k}_{F(k),t-1}+\sum_{e_{i}\in
E(v_{k})}u^{i}_{|F(i)-1|,t}=0.$ (17)

We are faced with the choice of which of the conditions at the nodes - (13),
(15) or (17) to prefer in the mathematical study of the direct and inverse
problems. From a mathematical point of view, we can take any of these
conditions, because in the limit, when a continuous equation is approximated
by a discrete one, they will turn into the Kirchhoff conditions. It seems that
(15) looks more preferable than the others, because it is already a direct
analogue of the Kirchhoff condition (3). In the next section, we will study
this issue and come to the conclusion that the correct condition is actually
the condition $(\ref{Kirch_d_1mm})$. For example we will demonstrate the
following fact

######  Remark 3.4.

In the case when two edges $e_{1}$ and $e_{i_{2}}$ converge at the vertex
$v_{k}$ and, consequently, $p_{k}=2$, from the formula (16) we get that
$m_{k}=0$, and ( 17) will become

$-u^{1}_{F(1),t+1}-u^{1}_{F(1),t-1}+u^{1}_{|F(1)-1|,t}+u^{i_{2}}_{|F(i_{2})-1|,t}=0.$

The last equation is the same as (11), which in turn means that these two
edges $e_{1}$ and $e_{i_{2}}$ can be replaced by one by removing the $v_{k}$
vertex from the graph.

We can conclude that the choice of the mass $m_{k}$ according to the formula
(16) is more correct than, for example, the option when we choose $m_{k}=-1$.

##  4 Transmission and reflection of waves at the internal node

In this section, as a model situation, we will consider a star graph – a
connected graph in which all edges are incident to one internal vertex. A
star-graph with $k+1$ vertices is usually denoted $S_{k}$, where $k$ is called
the order of the star. In our notation, all edges $e_{1},\ldots,e_{k}$ are
incident to the vertex $v_{k+1}$, the center of the star-graph. Each edge
$e_{i}$ starts at $v_{i}$ $i=1,\ldots,k$ and ends at $v_{k+1}$.

Metric graph.

To understand the processes of transmission and reflection we recall what
happens in the case of a metric graph for the problem C when the boundary
conditions (4) have the form:

$u^{1}(0,t)=\delta(t),\quad u^{i}(0,t)=0,\quad i=2,\ldots,k.$

This statement corresponds to the situation when all but one of the boundary
vertices are clamped and the first vertex is clicked at the initial moment of
time. Due to the fact that on each edge we have the wave equation (1), this
wave will propagate inside the graph along the edge $e_{1}$ according to the
rule $u^{1}(x,t)=\delta(t-x)$ until it reaches $v_{k+1}$. Since the length of
the edge $e_{1}$ is equal to $l_{1}$, and the speed of propagation is equal to
$1$, then at time $t=l_{1}$ the delta function reaches the vertex $v_{k+1}$,
the center of the star-graph. At this moment of time, the conditions at the
node come into force – the continuity conditions (2) and the Kirchhoff
condition (3). Using them, one can see that $\frac{2}{k}\delta$ will run along
each outgoing edge $e_{i}$ $i=2,\ldots,k$, and the reflected wave will run
along the edge $e_{1}$ in the opposite direction $-\frac{k-2}{k}\delta$.
Namely:

$u^{1}(x,t)=-\frac{k-2}{k}\delta(t+x-2l_{1}),\quad
u^{i}(x,t)=\frac{2}{k}\delta(t+x-l_{1}-l_{i}),\quad i=2,\ldots,k,$

where $l_{1}<t<l_{1}+l$, $l:=\min\\{l_{1},\ldots,l_{k}\\}$. Indeed, the
continuity condition is satisfied, because for $x=l_{1}$ and $t=l_{1}$ on the
first edge the solution looks like the sum of the incident and reflected
waves, and on the other edges as the transmitted one:

$u^{1}=\delta(t-x)-\frac{k-2}{k}\delta(t+x-2l_{1}),\quad
u^{i}=\frac{2}{k}\delta(t+x-l_{1}-l_{i}),\quad i=2,\ldots,k.$

The continuity and the Kirchhoff conditions (2), (3) are easily follow from
representations above.

It will be important to us that the delta function falling on the node passes
through it and runs further along the outgoing edges with the coefficient
$2/k$ and is reflected and runs in the opposite direction along the incoming
edge with the coefficient $-(k-2)/k$.

We want to observe exactly the same picture in the situation when the graph
and equations have become discrete.

Discrete graph.

Let $k=3$ and all three edges of the star graph have the same length, i.e. are
parameterized by the set of numbers $0,1,\ldots,N$. Recall that time is
discrete and the dependence on it is taken into account through the second
subscript. So function under consideration has a form:

$u:=\left\\{u^{i}\right\\}_{i=1}^{3},\quad
u^{i}=(u^{i}_{0,t},\ldots,u^{i}_{N,t}),\quad u^{i}_{j,t}\in\mathbb{R},\
j=0,\ldots,N.$

For $i=1,2,3$, the index value $j=0$ corresponds to the graph vertex $v_{i}$,
and the index value $j=N$ corresponds to the vertex $v_{4}$ — the center of
the star. This function satisfies the equation (11), continuity, initial and
boundary conditions:

$\displaystyle u^{i}_{j,t+1}+u^{i}_{j,t-1}-u^{i}_{j+1,t}-u^{i}_{j-1,t}=0,\
j=1,\ldots,N-1,t=0,\ldots$ (18) $\displaystyle
u^{1}_{N,t}=u^{2}_{N,t}=u^{3}_{N,t},\ t=1,\ldots$ (19) $\displaystyle
u^{1}_{0,0}=1,\ u^{2}_{0,0}=u^{3}_{0,0}=u^{i}_{0,t}=0,\ i=1,2,3,\ t=1,\ldots$
(20) $\displaystyle u^{i}_{j,0}=u^{i}_{j,-1}=0,\ i=1,2,3,\ j=1,\ldots,N.$ (21)

In the last line we formally introduced the initial condition for $t=-1$. This
is done to write an analog of the condition $u_{t}(x,0)=0$.

To this system one must add a condition at the internal node (the analogous to
the Kirchhoff condition). As mentioned in the previous section, several
variants of this condition are possible. Let us take a look at each of the
following three conditions:

$\displaystyle(u^{1}_{N-1,t}-u^{1}_{N,t})+(u^{2}_{N-1,t}-u^{2}_{N,t})+(u^{3}_{N-1,t}-u^{3}_{N,t})=0,$
(22)
$\displaystyle-u^{1}_{N,t+1}-u^{1}_{N,t-1}-u^{1}_{N,t}+u^{1}_{N-1,t}+u^{2}_{N-1,t}+u^{3}_{N-1,t}=0,$
(23)
$\displaystyle-\frac{3}{2}u^{1}_{N,t+1}-\frac{3}{2}u^{1}_{N,t-1}+u^{1}_{N-1,t}+u^{2}_{N-1,t}+u^{3}_{N-1,t}=0.$
(24)

We present the solution of each of the three problems in tables. For
simplicity we assume that $N=3$. For the problem (18)–(21) and condition (22)
we get

6 | 0 | -1/3 | 0 | 0 | 0 | -1/3 | 0
---|---|---|---|---|---|---|---
5 | 0 | -2/3 | 0 | 0 | 0 | 1/3 | 0
4 | 0 | 1/3 | -2/3 | 0 | 1/3 | 1/3 | 0
3 | 0 | 0 | 1/3 | 1/3 | 1/3 | 0 | 0
2 | 0 | 0 | 1 | 1/3 | 0 | 0 | 0
1 | 0 | 1 | 0 | 0 | 0 | 0 | 0
0 | 1 | 0 | 0 | 0 | 0 | 0 | 0
-1 | 0 | 0 | 0 | 0 | 0 | 0 | 0
t/j | 0 | 1 | 2 | 3 | 2 | 1 | 0
| i | = | 1 |  | i | = | 2,3

For the condition (23) we get

6 | 0 | -1 | 2 | -4 | 2 | -1 | 0
---|---|---|---|---|---|---|---
5 | 0 | 0 | -1 | 2 | -1 | 1 | 0
4 | 0 | 0 | 0 | -1 | 1 | 0 | 0
3 | 0 | 0 | 0 | 1 | 0 | 0 | 0
2 | 0 | 0 | 1 | 0 | 0 | 0 | 0
1 | 0 | 1 | 0 | 0 | 0 | 0 | 0
0 | 1 | 0 | 0 | 0 | 0 | 0 | 0
-1 | 0 | 0 | 0 | 0 | 0 | 0 | 0
t/j | 0 | 1 | 2 | 3 | 2 | 1 | 0
| i | = | 1 |  | i | = | 2,3

For the condition (24) we get

6 | 0 | 0 | 0 | 0 | 0 | 0 | 0
---|---|---|---|---|---|---|---
5 | 0 | -1/3 | 0 | 0 | 0 | 2/3 | 0
4 | 0 | 0 | -1/3 | 0 | 2/3 | 0 | 0
3 | 0 | 0 | 0 | 2/3 | 0 | 0 | 0
2 | 0 | 0 | 1 | 0 | 0 | 0 | 0
1 | 0 | 1 | 0 | 0 | 0 | 0 | 0
0 | 1 | 0 | 0 | 0 | 0 | 0 | 0
-1 | 0 | 0 | 0 | 0 | 0 | 0 | 0
t/j | 0 | 1 | 2 | 3 | 2 | 1 | 0
| i | = | 1 |  | i | = | 2,3
(25)

Here, boundary and initial conditions are highlighted in bold text. The tables
were filled from the bottom to the top with the values of the $u^{i}_{j,t}$
functions calculated according to the equations (18)–(21) and the specified
conditions (22)–(24) at the internal node.

In a discrete situation, the analog of the delta function is the vector
$(1,0,0,\ldots)$. This is exactly what we see at the first vertex $v_{1}$ (in
the tables, this is the first column). Note that only in the last case we have
got a complete analogue of the continuous situation, namely, that the delta
function falling on the node passes through it and runs along the outgoing
edges with a coefficient of $2/3$ and is reflected and runs in the opposite
direction along the incoming edge with coefficient $-1/3$. In the first case
this statement is true only “in the mean", i.e. if we sum the coefficients
over two time intervals $t$ and $t+1$. In other words, the support of the
initial perturbation “spreads" and the reflection from the node occurs in “two
cycles". While when choosing the correct third condition, the reflection from
the node occurs in “one cycle" and the support does not “spread".

In the second case, we see that the resonance begins at the node and such
condition does not model continuous situation well.

Summing up, we can say that

######  Conclusion 4.1.

The appropriate correct formulation of the direct discrete problem for a
discrete graph would be the equations (18)-(21) and the condition at the node
(17).

$\displaystyle u^{i}_{j,t+1}+u^{i}_{j,t-1}-u^{i}_{j+1,t}-u^{i}_{j-1,t}=0,\
1\leqslant i\leqslant N,\ 1\leqslant j\leqslant N_{i}-1,t\geqslant 0,$ (26)
$\displaystyle u^{i}_{F(i),t}=u^{j}_{F(j),t},\ e_{i},e_{j}\in E(v_{k}),\quad
t\geqslant 1,$ (27) $\displaystyle
u(v_{k},t)=f^{k}(t)\quad\text{for}\,\,v_{k}\in\Gamma,\quad t\geqslant 1,$ (28)
$\displaystyle u^{i}_{j,0}=0,\quad u^{i}_{j,-1}=0,\ 1\leqslant i\leqslant N,\
1\leqslant j\leqslant N_{i}-1,$ (29)
$\displaystyle-\frac{p_{k}}{2}u^{k}_{F(k),t+1}-\frac{p_{k}}{2}u^{k}_{F(k),t-1}+\sum_{e_{i}\in
E(v_{k})}u^{i}_{|F(i)-1|,t}=0.$ (30)

In (28) and (30), $k=m+1,\ldots,M$.

##  5 Control Problems of the Wave equations on Discrete Graph

In this section we present algorithms solving the direct and control problems
for the discrete wave equation on simple graphs. We start with a finite
interval (finite number of points). Then we construct a solution to the direct
problem on a star-shaped graph, and finally we solve a control problem.

###  5.1 Finite interval

Let $\Omega_{D}$ be a graph consisting of two vertices $v_{1}$ and $v_{2}$ and
one edge $e$. For discrete problem we identify it with a finite set of numbers
$0,\ldots,N$, the vertex $v_{1}$ is identified with $x=0$ and the vertex
$v_{2}$ is identified with $x=N$. The control is applied at $x=0$ and the
system takes the form

$\displaystyle u_{j,t+1}+u_{j,t-1}-u_{j+1,t}-u_{j-1,t}$ $\displaystyle=0,\
j=1,\ldots,N-1,t=0,\ldots$ (31) $\displaystyle u_{0,t}=f_{t},\quad u_{N,t}$
$\displaystyle=0,\quad\text{for $t=0,\ldots$}$ (32) $\displaystyle
u_{j,0}=0,\quad u_{j,-1}$ $\displaystyle=0,\ \quad\text{for $l=1,\ldots$}$
(33)

If $T<N$, the solution $u^{f-}$ to (31)–(33) has a form

$u^{f-}_{j,t}=\left\\{\begin{array}[]{l}0,\quad 0<t<j,\\\ f_{t-j},\quad
j\leqslant t.\end{array}\right.$

If $T=N$, a reflected wave occurs. If $T>N$, the solution $u^{f-}$ to
(31)–(33) has a form

$u^{f-}_{j,t}=\sum_{n=0}^{\left\lfloor{\frac{t-j}{2N}}\right\rfloor}f_{t-j-2nN}-\sum_{n=1}^{\left\lfloor{\frac{t+j}{2N}}\right\rfloor}f_{t+j-2nN},$
(34)

where $\lfloor{\cdot}\rfloor$ denotes the floor function. In case that the
upper index of the second sum of (34) is zero, then we set it equal to zero.

Now we consider the case when the control is applied at $x=N$, then the system
takes the form

$\displaystyle u_{j,t+1}+u_{j,t-1}-u_{j+1,t}-u_{j-1,t}$ $\displaystyle=0,\
j=1,\ldots,N-1,t=0,\ldots$ (35) $\displaystyle u_{0,t}=0,\quad u_{N,t}$
$\displaystyle=g_{t},\quad\text{for $t=0,\ldots$}$ (36) $\displaystyle
u_{j,0}=0,\quad u_{j,-1}$ $\displaystyle=0,\ \quad\text{for $l=1,\ldots$}$
(37)

If $T<N$, solution $u^{g+}$ to (35)–(37) has a form

$u^{g+}_{j,t}=\left\\{\begin{array}[]{l}0,\quad 0<t<j,\\\ g_{t-N+j},\quad
j\leqslant t.\end{array}\right.$

As in the previous case, if $T=N$, a reflected wave occurs. If $T>N$, the
solution $u^{g+}$ to (35)–(37) has a form

$u^{g+}_{j,t}=\sum_{n=0}^{\left\lfloor{\frac{t+j-N}{2N}}\right\rfloor}g_{t+j-(2n+1)N}-\sum_{n=0}^{\left\lfloor{\frac{t-j-N}{2N}}\right\rfloor}g_{t-j-(2n+1)N}.$
(38)

###  5.2 Star-shaped graph

Let us consider a three star graph:

$u:=\left\\{u^{i}\right\\}_{i=1}^{3},\quad
u^{i}=(u^{i}_{0,t},\ldots,u^{i}_{N_{i},t}),\quad u^{i}_{j,t}\in\mathbb{R},\
j=0,\ldots,N_{i}.$

For $i=1,2,3$, the index value $j=0$ corresponds to the graph vertex $v_{i}$,
and the index value $j=N_{i}$ corresponds to the vertex $v_{4}$ — the center
of the star. In this case the system (26)–(30) takes the form:

$\displaystyle u^{i}_{j,t+1}+u^{i}_{j,t-1}-u^{i}_{j+1,t}-u^{i}_{j-1,t}=0,\
j=1,\ldots,N_{i}-1,t=0,\ldots$ (39) $\displaystyle
u^{1}_{N_{1},t}=u^{2}_{N_{2},t}=u^{3}_{N_{3},t},\ t=1,\ldots$ (40)
$\displaystyle-\frac{3}{2}u^{1}_{N_{1},t+1}-\frac{3}{2}u^{1}_{N_{1},t-1}+u^{1}_{N_{1}-1,t}+u^{2}_{N_{2}-1,t}+u^{3}_{N_{3}-1,t}=0.$
(41) $\displaystyle u^{i}_{0,t}=f^{i}_{t},\ u^{3}_{0,t}=0,\ i=1,2,\
t=1,\ldots$ (42) $\displaystyle u^{i}_{j,0}=u^{i}_{j,-1}=0,\ i=1,2,3,\
j=1,\ldots,N.$ (43)

Below we use the notation

$g_{t}:=u^{1}_{N_{1},t}=u^{2}_{N_{2},t}=u^{3}_{N_{3},t}=\left\\{\begin{array}[]{l}0,\quad\text{if
$t<\min\\{N_{1},N_{2},N_{3}\\}$},\\\ \not=0\quad\text{if
$t\geqslant\min\\{N_{1},N_{2},N_{3}\\}$}.\end{array}\right.$

Now it is easy to write down the solution of the forward problem to the 3-star
graph using formulas obtained for the finite interval (34), (38). Namely,

$u^{1}_{j,t}=u_{j,t}^{f^{1}-}+u_{j,t}^{g+},\quad
u^{2}_{j,t}=u_{j,t}^{f^{2}-}+u_{j,t}^{g+},\quad u^{3}_{j,t}=u_{j,t}^{g+}.$
(44)

Since boundary controls are known, it remains to find $g_{t}$ to solve the
forward problem on the discrete graph. Formula for the function $g_{t}$ can be
derived using Table 25, formula (34) and linearity of the system:

$g_{t}=\frac{2}{3}(f^{1}_{t-N_{1}}+f^{2}_{t-N_{2}}).$ (45)

This equality is valid for
$t<N_{1}+N_{2}+N_{3}+\min{N_{1},N_{2},N_{3}}-\max{N_{1},N_{2},N_{3}}$.

###  5.3 Shape control problem

The _shape control_ problem on a 3-star graph is to determine the boundary
controls $f^{1}_{t},f^{2}_{t}$ at $v_{1}$ and $v_{2}$ such that $u$, the
solution of (39)–(43), satisfies the equalities $u^{i}_{j,T}=\varphi^{i}_{j}$,
$i=1,2,3$ and $j=1,...,N_{i}$, where
$\varphi=(\varphi^{1},\varphi^{2},\varphi^{3})$ is a prescribed shape at time
$t=T$. The solution of the shape control problem is based on the solution of
the forward problem (44). We will demonstrate that the optimal control time is

$T=\min\\{L_{1},L_{2}\\},\quad L_{1}:=\max\\{N_{1}+N_{3},N_{2}\\},\quad
L_{2}:=\max\\{N_{1},N_{2}+N_{3}\\}.$

Since no control acts on $v_{3}$, edge $e_{3}$ should be controlled through
$g_{t}$. Therefore,

$g_{T-j}=\left\\{\begin{array}[]{l}\varphi^{3}_{N_{3}-j},\quad
j=0,\ldots,N_{3},\\\ 0,\quad j=N_{3},\ldots,T.\end{array}\right.$

Note that $g_{t}$ can be controlled both from $v_{1}$ and $v_{2}$ (45). Let us
consider the case when $L_{1}\leqslant L_{2}$. Then we can find control
$\tilde{f}^{1}$ on $v_{1}$ such that if $f^{2}=0$ on $v_{2}$ then
$u^{i}_{N_{i},t}=g_{t}$, $i=1,2,3$ and $t=0,....$:

$\tilde{f}^{1}_{t-N_{1}}=\frac{3}{2}g_{t},\quad t\leq N_{1}.$

The control we interested in has a form $f^{1}=\tilde{f}^{1}+\hat{f}^{1}$ and
$\hat{f}^{1}$ can be found as a solution to the equation

$u^{\tilde{f}^{1}-}+u^{\hat{f}^{1}-}+u^{g+}=\varphi^{1}\quad\text{on
$e_{1}$}.$ (46)

Such control $f^{1}$ and $f^{2}=0$ makes solution to (39)–(43) satisfying
$u^{i}_{j,T}=\varphi^{i}_{j}$ $i=1,2$. It remains to satisfy the third
relation $u^{3}_{j,T}=\varphi^{3}_{j}$ by appropriate choice of $f_{2}$.
Exactly, $f^{2}$ should be a solution to the equation

$u^{f^{2}-}+u^{g+}=\varphi^{2}\quad\text{on $e_{2}$}.$ (47)

Solving the equations (46) and (47) with respect to $\hat{f}^{1}$ and $f^{2}$
we solve the shape control problem. The case $L_{1}>L_{2}$ can be considered
similarly.

For solutions of equations (46) and (47) an explicit formulas can be written,
we present ones in the case when $N_{1}=N_{2}=N_{3}$:

$f^{1}_{j}=\left\\{\begin{array}[]{l}\frac{3}{2}\varphi^{3}_{j+1},\quad
j=0,1,\ldots,N-1,\\\
\varphi^{1}_{2N-j-1}+\frac{1}{2}\varphi^{3}_{2N-j-1},\quad
j=N,N+1,\ldots,2N-1,\par\end{array}\right.$
$f^{2}_{j}=\left\\{\begin{array}[]{l}0,\quad j=0,1,\ldots,N-1,\\\
\varphi^{2}_{2N-j-1}-\varphi^{3}_{2N-j-1},\quad
j=N,N+1,\ldots,2N-1.\end{array}\right.$

In a general situation due to the different lengths of the edges of the graph,
waves can reflect not only from the inner vertex, but also from the vertex
$v_{1}$ or $v_{2}$. In such a case the formulas are more involved and contain
terms taking into account these reflections. In our subsequent publications we
plan to study control and inverse problems on more general discrete graphs and
compare the results with their continuous analogs.

## Acknowledgments

The research of Sergei Avdonin was supported in part by the National Science
Foundation, grant DMS 1909869, and by Moscow Center for Fundamental and
Applied Mathematics. The work of Victor Mikhaylov and Alexander Mikhaylov was
supported by the RFBR grant 20-01-00627. The research of Abdon Choque-Rivero
was supported by CONACYT Project A1-S-31524 and CIC-UMSNH, Mexico.

### Conflict of interest

The authors declare no potential conflict of interests.

## References

  * [1] Lagnese J, Leugering G, Schmidt EJPG. Modelling, Analysis, and Control of dynamical elastic multilink structures.  Basel: Birkhauser .  1994\.  ISBN 9781461202738.
  * [2] Lagnese JE, Leugering G, Schmidt EJPG. Modelling of dynamic networks of thin thermoelastic beams. Mathematical Methods in the Applied Sciences 1993; 16(5): 327–358.
  * [3] Kiik JC, Kurasov P, Usman M. On vertex conditions for elastic systems. Physics Letters A 2015; 379(34): 1871-1876.
  * [4] Berkolaiko G, Ettehad M. Three-dimensional elastic beam frames: Rigid joint conditions in variational and differential formulation. Studies in Applied Mathematics 2022; 148(4): 1586–1623.
  * [5] Gugat M, Leugering G. Global boundary controllability of the Saint-Venant system for sloped canals with friction. Ann. Inst. H. Poincaré Anal. Non Lineaire 2009; 26(1): 257–270.
  * [6] Steinbach MC. On pde solution in transient optimization of gas networks. Journal of Computational and Applied Mathematics 2007; 203(2): 345–361.
  * [7] Domschke P, Kolb O, Lang J. Adjoint-based error control for the simulation and optimization of gas and water supply networks. Applied Mathematics and Computation 2015; 259: 1003–1018.
  * [8] Egger H, Schobel-Krohn L. Chemotaxis on networks: analysis and numerical approximation. ESAIM: Mathematical Modelling and Numerical Analysis 2020; 54(4): 1339–1372.
  * [9] Bretti G, Natalini R, Ribot M. A hyperbolic model of chemotaxis on a network: a numerical study. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 2014; 48(1): 231–258.
  * [10] Borsche R, Klar A, Pham TNH. Nonlinear flux-limited models for chemotaxis on networks. Networks and Heterogeneous Media 2017(1556-180L2017_3_381): 381–401.
  * [11] Herty M, Mohring J, Sachers V. A new model for gas flow in pipe networks. Mathematical Methods in the Applied Sciences 2011; 33(7): 845–855.
  * [12] Bastin G, Coron JM, Novel dB. Using hyperbolic systems of balance laws for modeling, control and stability analysis of physical networks. In: Lecture notes for the Pre-Congress Workshop on Complex Embedded and Networked Control Systems 17th IFAC World Congress. IFAC. ; 2008; Seoul, Korea.
  * [13] Hante FM, Leugering G, Martin A, Schewe L, Schmidt M. Challenges in optimal control problems for gas and fluid flow in networks of pipes and canals: From modeling to industrial applications.  Singapore: 122, Ind. Appl. Math., Springer .  2017\.  ISBN 9048154464.
  * [14] Alì G, Bartel A, Günther M. Parabolic Differential-Algebraic Models in Electrical Network Design. Multiscale Model. Simul. 2005; 4(3): 813–838.
  * [15] Cheng X, Scherpen JMA. Clustering approach to model order reduction of power networks with distributed controllers. Advances in Computational Mathematics volume 2018; 44: 1917–1939.
  * [16] Colombo RM, Guerra G, Herty M, Schleper V. Optimal control in networks of pipes and canals. SIAM J. Control Optim. 2009; 48(3): 2032–2050.
  * [17] Garavello M, Piccoli B. Traffic Flow on Networks, volume 1 of AIMS Series on Applied Mathematics.  Springfield, MO: American Institute of Mathematical Sciences .  2006\.
  * [18] Oppenheimer SF. A convection-diffusion problem in a network. Applied Mathematics and Computation 2000; 112(2): 223–240.
  * [19] Herty M, Ringhofer C. Averaged kinetic models for flows on unstructured networks. Kinetic and Related Models 2011; 4(4): 1081–1096.
  * [20] García L, Barreiro-Gomez J, Escobar E, Téllez D, Quijano N, Ocampo-Martinez N. Modeling and real-time control of urban drainage systems: A review. Advances in Water Resources 2015; 85: 120–132.
  * [21] Hurt NE. Mathematical physics of quantum wires and devices. From spectral resonances to Anderson localization.  Dordrecht: Kluwer Academic .  2000\.  ISBN 9048154464.
  * [22] Joachim C, Roth S. Atomic and Molecular Wires.  Dordrecht: Springer .  1997\.  ISBN 9780792346289.
  * [23] Melnikov YuB, Pavlov BS. Two-body scattering on a graph and application to simple nanoelectronic devices. J. Math. Phys. 1995; 36(6): 2813–2825.
  * [24] Duca A. Bilinear quantum systems on compact graphs: Well-posedness and global exact controllability. Automatica 2021; 123: 123:109324.
  * [25] Kottos T, Smilansky U. Quantum chaos on graphs. Phys. Rev. Lett. 1997; 79(24): 4794–4797.
  * [26] Adam S, Hwang E, Galits VM, Das Sarma S. A self-consistent theory for graphene transport. Proceeding of the National Academy of Sciences 2007; 104(47): 18392–18397.
  * [27] Kuchment P, Post O. On the spectra of carbon nano-structures. Commun. Math. Phys 2007; 275: 805–826.
  * [28] Bell J, Cracium G. A distributed parameter identification problem in neuronal cable theory models. Math. Biosci. 2005; 194(1): 1–19.
  * [29] Avdonin S, Bell J. Determining a distributed conductance parameter for a neuronal cable model defined on a tree graph. Inverse Problems and Imaging 2016; 9(3): 645–659.
  * [30] Du B, Lian X, Cheng X. Partial differential equation modeling with Dirichlet boundary conditions on social networks. Boundary Value Problems 2018; 50: 2035–2052.
  * [31] Solomon J. PDE approaches to graph analysis. ArXiv 2015; abs/1505.00185.
  * [32] Mercier D, Régnier V. Spectrum of a network of Euler-Bernoulli beams. Journal of Mathematical Analysis and Applications 2008; 337(1): 174–196.
  * [33] Kottos T, Smilansky U. Periodic orbit theory and spectral statistics for quantum graphs. Annals of Physics 1999; 274(1): 76–124.
  * [34] von Below J. A characteristic equation associated to an eigenvalue problem on c2-networks. Linear Algebra and its Applications 1985; 71: 309–325.
  * [35] Band R, Lévy G. Quantum graphs which optimize the spectral gap. Annales Henri Poincaré 2017; 18: 3269–3323.
  * [36] Laurent M, Piovesan T. Conic approach to quantum graph parameters using linear optimization over the completely positive semidefinite cone. SIAM Journal on Optimization 2015; 25(4): 2461–2493.
  * [37] Berkolaiko G, Kuchment P. Introduction to Quantum Graphs (Mathematical Surveys and Monographs vol. 186).  Providence, RI: American Mathematical Society .  2013\.  ISBN 0821892118.
  * [38] Domschke P, Dua A, Stolwijk J, Lang J, Mehrmann V. Adaptive refinement strategies for the simulation of gas flow in networks using a model hierarchy. Electronic Transactions on Numerical Analysis 2018; 48: 97–113.
  * [39] Egger H, Philippi N. A hybrid discontinuous galerkin method for transport equations on networks. In: In Robert Klofkorn, Eirik Keilegavlen, Florin A. Radu, and Jürgen Fuhrmann, editors, Finite Volumes for Complex Applications IX - Methods, Theoretical Aspects, Examples. Dynamic Publishers. ; 2020; Springer International Publishing: 487–495.
  * [40] Grundel S, Herty M. Hyperbolic discretization of simplified Euler equation via Riemann invariants. Applied Mathematical Modelling 2022; 106: 60–72.
  * [41] Grundel S, Hornung N, Roggendorf S. Numerical aspects of model order reduction for gas transportation networks. In: In Slawomir Koziel, Leifur Leifsson, and Xin-She Yang, editors, Simulation Driven Modeling and Optimization. Dynamic Publishers. ; 2016; Springer International Publishing: 1–28.
  * [42] Gugat M, Herty M, Schleper V. Flow control in gas networks: Exact controllability to a given demand. Mathematical Methods in the Applied Sciences 2011; 34(7): 745–757.
  * [43] Pesenson I. Polynomial splines and eigenvalue approximations on quantum graphs. Journal of Approximation Theory 2005; 132(2): 203–220.
  * [44] Wybo WA, Boccalini D, Torben-Nielsen B, Gewaltig MO. Asparse reformulation of the green’s function formalism allows efficient simulations of morphological neuron models. Neural Computation 2015; 27(12): 2587–2622.
  * [45] Arioli M, Benzi M. A finite element method for quantum graphs. IMA Journal of Numerical Analysis 2017; 38(3): 1119–1163.
  * [46] García L, Barreiro-Gomez J, Escobar E, Téllez D, Quijano N, Ocampo-Martinez C. Nonoverlapping domain decomposition for optimal control problems governed by semi-linear models for gas flow in networks. Control and Cybernetics 2017; 46(3): 191–225.
  * [47] Stoll M, Winkler M. Optimal Dirichlet control of partial differential equations on networks. Electronic transactions on Numerical Analysis 2021; 54: 392–419.


#  $q$-birational morphisms and divisors

Donghyeon Kim  Department of mathematics, Yonsei University, 50 Yonsei-Ro,
Seodaemun-Gu, Seoul 03722, Korea<EMAIL_ADDRESS>
(Date: August 28, 2024)

###### Abstract.

In this paper, we define the notions $q$-birational morphism and
$q$-birational divisor and develop the theory about them. We state and prove
versions of Kodaira-type vanishing theorem and Zariski decomposition theorem
for $q$-birational divisors.

###### Key words and phrases:

$q$-birational map, $q$-birational divisor

##  1\. Introduction

Throughout the paper, $q\geq 2$ is a positive integer, $k$ is an algebraically
closed field of characteristic $0$ and all varieties are quasi-projective,
separated, finite type, reduced and irreducible schemes over $k$. Any divisor
is assumed to be Cartier unless otherwise stated.

For any normal projective varieties $X,X^{\prime}$ and any proper birational
morphism $f:X^{\prime}\to X$, the following property is well-known:

$f_{*}\mathcal{O}_{X^{\prime}}=\mathcal{O}_{X}.$

Moreover, by extending the property, one can prove a vanishing of the higher
direct image of structure sheaf as the following:

######  Proposition 1.1 (See good).

igood Let $X$ be any normal variety with $(RS_{q})$. Then the following are
equivalent:

  * (a)

$X$ is $(S_{q+1})$.

  * (b)

For any proper birational morphism $f:X^{\prime}\to X$ with $X^{\prime}$
smooth, $R^{i}f_{*}\mathcal{O}_{X^{\prime}}=0$ for $1\leq i<q$.

For the definition of $(R_{q})$, $(RS_{q})$ and $(S_{q+1})$, see
basicdefinitions. Note that the idea of the proof of igood goes back to
[Kov99], [Ale08], [Fuj17], [Kol11] and [AH12].

Under the same conditions above, assume that the exceptional locus of $f$ has
codimension $1$ and let $F$ be any effective $f$-exceptional divisor on
$X^{\prime}$. Then we have the following property:

$f_{*}\mathcal{O}_{X^{\prime}}(F)=\mathcal{O}_{X}.$

Considering igood, we can ask what condition on $f$ is sufficient for the
following vanishing to hold:

$R^{i}f_{*}\mathcal{O}_{X^{\prime}}(F)=0\,\text{ for }1\leq i<q.$

To answer the question, we define the following notion of $q$-birationality
for a birational morphism.

######  Definition 1.2 (See qbir).

iqbir Let $X,X^{\prime}$ be any normal varieties and $f:X^{\prime}\to X$ be
any proper birational morphism.

  * (a)

The _center_ of $f$ is the reduced closed subscheme $C$ of $X$ which is the
image of the exceptional locus along $f$.

  * (b)

We say $f$ is a _$q$ -birational morphism_ if the exceptional locus has
codimension $1$ and the center of $f$ has codimension $\geq q+1$.

Then we can prove the following theorems.

######  Theorem 1.3 (See zerothapplication).

izerothapplication Let $X$ be a variety with $(R_{q}),(S_{q+1})$ and
$f:X^{\prime}\to X$ be any $q$-birational morphism from a smooth variety
$X^{\prime}$. Then $R^{i}f_{*}\mathcal{O}_{X^{\prime}}(E)=0$ for any $1\leq
i<q$ and any effective $f$-exceptional divisor $E$.

######  Theorem 1.4 (See secondzerothapplication).

isecondzerothapplication Let $X$ be a Cohen-Macaulay variety with $(R_{q})$
and $f:X^{\prime}\to X$ be any proper birational morphism from a smooth
variety $X^{\prime}$. Suppose that $R^{i}f_{*}\mathcal{O}_{X^{\prime}}(E)=0$
for any $1\leq i<q$ and any effective $f$-exceptional divisor $E$. Then $f$ is
$q$-birational.

Note that a special case of izerothapplication is proved in [ST08]. Now, we
define the following notion of (weak) $q$-birationality for a divisor.

######  Definition 1.5 (See birationaldivisor).

ibirationaldivisor Let $X$ be any smooth projective variety and $D$ be any
divisor on $X$. We say $D$ is _weakly $q$-birational_ if there are smooth
projective varieties $X^{\prime}$ and $X_{0}$, a diagram

${X^{\prime}}$${X}$${X_{0},}$$\scriptstyle{\pi}$$\scriptstyle{f^{\prime}}$

an ample divisor $H$ on $X_{0}$ and a positive integer $p$ such that

  * (a)

$f^{\prime}$ is $q$-birational and,

  * (b)

$\pi^{*}(pD)-(f^{\prime})^{*}H$ is a (not necessarily effective)
$f^{\prime}$-exceptional divisor on $X^{\prime}$.

If $\pi^{*}(pD)-(f^{\prime})^{*}H$ in $(b)$ is effective, we say $D$ is _$q$
-birational_. If $D$ is weakly $q$-biratonal, we say the minimum of $p$ is the
_index_ of $D$ and denote it by $p_{D}$.

Using izerothapplication, we can prove a Kodaira-type vanishing theorem for
$q$-birational divisors as follows:

######  Theorem 1.6 (See stronglyqbirational1).

istronglyqbirational1 Let $X$ be any smooth projective variety and $D$ be any
$q$-birational divisor on $X$. Then

$H^{i}(X,\mathcal{O}_{X}(K_{X}+p_{D}D))=0\,\,\,\text{ for }1\leq i<q.$

Moreover, any $q$-birational divisor admits a form of Zariski decomposition.

######  Theorem 1.7 (See stronglyqbirationl2).

istronglyqbirational2 Let $X$ be any smooth projective variety, $D$ be any
$q$-birational divisor on $X$ and $p_{D}D=P+N$ be the weak decomposition of
$D$ for the index $p_{D}$ of $D$. Then

$H^{i}(X,\mathcal{O}_{X}(mp_{D}D))=H^{i}(X,\mathcal{O}_{X}(mP+mN))=H^{i}(X,\mathcal{O}_{X}(mP))$

for any $m\geq 0$ and $0\leq i<q$.

One can consider the Kawamata-Viehweg vanishing theorem as a generalization of
the Kodaira vanishing theorem. Moreover, divisorial Zariski decomposition
exists for any pseudo-effective divisor. Hence, along the same line, we will
discuss how to generalize istronglyqbirational1 and istronglyqbirational2. We
also pose questions concerning such generalizations, which remain open.

The rest of the paper is organized as follows. We begin Section 2 by defining
some notations and stating some basic theorems. Section 3 is devoted to
defining $q$-birational morphism and proving izerothapplication and
isecondzerothapplication. In Section 4, we define $q$-birational divisors,
prove istronglyqbirational1 and istronglyqbirational2 and discuss how to
generalize those theorems.

######  Acknowledgments.

The author thanks his advisor Sung Rak Choi and his colleague Dae-Won Lee for
comments, questions, and discussions. He is grateful for their hospitality.

##  2\. Preliminaries

In the section, we collect properties and theorems used in the next sections.

From this section and to the end of the paper, for any variety $X$ and any
(not necessarily closed) point $x\in X$, we write $\dim x$ for the dimension
of the closure of $x$. In addition to this, set
$\operatorname{codim}_{X}x:=\dim X-\dim x$ and
$X_{x}:=\operatorname{Spec}\mathcal{O}_{X,x}$. Moreover, we assume $2\leq
q<\dim X$.

For a proper morphism $f:X^{\prime}\to X$ between two varieties, set
$X^{\prime}_{x}:=X^{\prime}\times_{X}X_{x}$, $\mathcal{O}_{X^{\prime},x}$ for
the structure sheaf of $X^{\prime}_{x}$ and $\omega_{X^{\prime},x}$ for the
dualizing sheaf of $X^{\prime}_{x}$.

Now, we recall a well-known result.

######  Theorem 2.1 (See [KM98], Theorem 5.10).

rational singularities Let $X,X^{\prime}$ be any smooth projective varieties
and $f:X^{\prime}\to X$ be any proper birational morphism. Suppose that
$\mathcal{E}$ is a vector bundle on $X$. Then we have

$H^{i}(X,\mathcal{E})=H^{i}(X^{\prime},f^{*}\mathcal{E})\,\,\,\,\,\text{for
}i\geq 0.$

The following theorem is called the _relative Kawamata-Viehweg vanishing
theorem_.

######  Theorem 2.2 (See [KMM87], Theorem 1-2-3).

Let $X^{\prime}$ be any smooth projective variety, $X$ be any projective
variety, and $f:X^{\prime}\to X$ be any proper birational morphism. Suppose
that $D$ is a $f$-nef divisor on $X^{\prime}$. Then,
$R^{i}f_{*}\mathcal{O}_{X^{\prime}}(K_{X^{\prime}}+D)=0$ for $i\geq 1$.

Let us introduce the following notions.

######  Definition 2.3.

basicdefinitions Let $X$ be any normal variety.

  * (a)

We call $X$ $(R_{q})$ if the singular locus of $X$ has codimension $\geq q+1$.

  * (b)

We call $X$ $(RS_{q})$ if
$\operatorname{codim}\mathrm{Supp}\,\left(\bigcup_{i\geq
1}R^{i}f_{*}\mathcal{O}_{X^{\prime}}\right)\geq q+1$ for any resolution
$f:X^{\prime}\to X$.

  * (c)

We call $X$ $(S_{q+1})$ if $H^{i}_{x}(X_{x},\mathcal{O}_{X,x})=0$ for any
$x\in X$ and any $i<\min\\{q+1,\operatorname{codim}_{X}x\\}$.

We need the theory of derived categories to prove ssg. Hence, it is worth
stating the _Grothendieck duality_.

######  Theorem 2.4 (See 0AU3, (4) in [Stacks]).

Let $X,X^{\prime}$ be any noetherian schemes and $f:X^{\prime}\to X$ be any
proper birational morphism. For any $K\in D(X^{\prime})$,

$R\mathcal{H}om_{\mathcal{O}_{X}}(Rf_{*}K,\omega^{\bullet}_{X})\cong
Rf_{*}R\mathcal{H}om_{\mathcal{O}_{X^{\prime}}}(K,\omega^{\bullet}_{X^{\prime}})\text{
in }D(X),$

where $D(X)$ denotes the bounded derived category of coherent sheaves on $X$
and $\omega^{\bullet}_{X}$ denotes the normalized dualizing complex on $X$.

Let us prove the following corollary of the Kawamata-Viehweg vanishing theorem
and the Grothendieck duality.

######  Corollary 2.5 (See [Fuj17], Lemma 7.1.2).

ssg Let $X,X^{\prime}$ be any varieties and $X^{\prime}$ be smooth. For any
proper birational morphism $f:X^{\prime}\to X$ and any point $x\in X$ with
$\operatorname{codim}_{X}x\geq c$,
$H^{i}_{f^{-1}(x)}(X^{\prime}_{x},\mathcal{O}_{X^{\prime},x})=0$ for $i<c$.

###### Proof.

Let $E$ be the injective hull of the residue field of $\mathcal{O}_{X,x}$ as
an $\mathcal{O}_{X,x}$-module. Then we have

$\displaystyle
R\Gamma_{f^{-1}(x)}(X^{\prime}_{x},\mathcal{O}_{X^{\prime}_{x}})$
$\displaystyle=R\Gamma_{x}(X_{x},Rf_{*}\mathcal{O}_{X^{\prime}_{x}})$
$\displaystyle=\operatorname{\mathrm{Hom}}_{\mathcal{O}_{X,x}}(R\mathcal{H}om_{\mathcal{O}_{x}}(Rf_{*}\mathcal{O}_{X^{\prime}_{x}},\omega^{\bullet}_{X,x}),E),$

where we used the Leray spectral sequence for the first equality and the local
duality for the second equality (for the statement of the local duality, see
Lemma 0AAK in [Stacks]). Moreover,

$\displaystyle
R\mathcal{H}om_{\mathcal{O}_{x}}(Rf_{*}\mathcal{O}_{X^{\prime}_{x}},\omega^{\bullet}_{X,x})$
$\displaystyle=Rf_{*}R\mathcal{H}om_{\mathcal{O}_{X^{\prime},x}}(\mathcal{O}_{X^{\prime}_{x}},\omega_{X^{\prime},x}[\operatorname{codim}_{X}x])$
$\displaystyle=Rf_{*}\omega_{X^{\prime},x}[\operatorname{codim}_{X}x]$
$\displaystyle=\omega_{X^{\prime},x}[\operatorname{codim}_{X}x],$

where we used the Grothendieck duality on the first equality and the Kawamata-
Viehweg vanishing theorem for the third equality. Hence we have the assertion.
∎

For any normal projective variety $X$ and a big and finitely generated divisor
$D$ on $X$, we may consider the following birational contraction

$X\dashrightarrow X_{0}:=\operatorname{Proj}\bigoplus_{m\geq
0}H^{0}(X,\mathcal{O}_{X}(mD))$

and we say that the map is the _birational contraction induced by $D$_.

The following lemma is for an equivalent notion of bigness and finite
generation. The following lemma explains why we define ibirationaldivisor in
such a way.

######  Lemma 2.6.

fg Let $X$ be any normal projective variety and $D$ be any effective divisor
on $X$. Then the following are equivalent:

  * (a)

$D$ is big and finitely generated.

  * (b)

There are proper birational morphisms $f^{\prime},\pi$, a diagram

${X^{\prime}}$${X}$${X_{0},}$$\scriptstyle{\pi}$$\scriptstyle{f^{\prime}}$

an ample divisor $H$ on $X_{0}$ and a positive integer $p$ such that
$X^{\prime},X_{0}$ are normal projective varieties and
$\pi^{*}(pD)-(f^{\prime})^{*}H$ is an effective $f^{\prime}$-exceptional
divisor on $X^{\prime}$.

###### Proof.

For $(a)\implies(b)$, let $f:X\dashrightarrow X_{0}$ be the contraction
induced by $D$. Then we have an ample divisor $H$ on $X_{0}$ and a positive
integer $p$ such that $pD=f^{*}H+F$ holds for some effective $f$-exceptional
divisor $F$ (see Lemma 1.6 in [HK00]). By the resolution of indeterminacies,
we have a normal projective variety $X^{\prime}$ and the following diagram

${X^{\prime}}$${X}$${X_{0}.}$$\scriptstyle{\pi}$$\scriptstyle{f^{\prime}}$$\scriptstyle{f}$

From this, $\pi^{*}(pD)-(f^{\prime})^{*}H=\pi^{*}F$ is an effective
$f^{\prime}$-exceptional divisor on $X^{\prime}$.

For $(b)\implies(a)$, set $F^{\prime}=\pi^{*}(pD)-(f^{\prime})^{*}H$. Then we
have $\pi^{*}(pD)=(f^{\prime})^{*}H+F^{\prime}$ and hence $D$ is big. For the
proof of finite generation, we have

$\displaystyle H^{0}(X_{0},\mathcal{O}_{X_{0}}(mH))$
$\displaystyle=H^{0}(X^{\prime},\mathcal{O}_{X^{\prime}}(m(f^{\prime})^{*}H))$
$\displaystyle=H^{0}(X^{\prime},\mathcal{O}_{X^{\prime}}(m(f^{\prime})^{*}H+mF^{\prime}))$
$\displaystyle=H^{0}(X^{\prime},\mathcal{O}_{X^{\prime}}(m\pi^{*}(pD)))$
$\displaystyle=H^{0}(X,\mathcal{O}_{X}(mpD)).$

Hence the proof is complete. ∎

The following result is devoted to defining the notion of _weak decomposition_
, which is needed in formulating istronglyqbirational2.

######  Lemma 2.7.

the above Let $X$ be any normal projective variety. Suppose that there are two
birational contractions $f:X\dashrightarrow X_{0},f^{\prime}:X\dashrightarrow
X^{\prime}_{0}$, two ample divisors $H,H^{\prime}$ on $X_{0},X^{\prime}_{0}$,
effective $f,f^{\prime}$-exceptional divisors $F,F^{\prime}$ respectively and
$f^{*}H+F=(f^{\prime})^{*}H^{\prime}+F^{\prime}$. Then $X_{0}\cong
X^{\prime}_{0}$, $F=F^{\prime}$ and $H=H^{\prime}$.

###### Proof.

By Lemma 1.7 in [HK00], the map $f\circ g^{-1}:X^{\prime}_{0}\dashrightarrow
X_{0}$ is a proper birational morphism and $F=F^{\prime}$. If it is not an
isomorphism, then $H^{\prime}=(f\circ g^{-1})^{*}H$ is not ample, leading to a
contradiction. Thus $f\circ g^{-1}$ is an isomorphism and $H=H^{\prime}$. ∎

######  Definition 2.8.

????? Let $X$ be any normal projective variety, $D$ be any big and finitely
generated divisor on $X$ and $f:X\dashrightarrow X_{0}$ be the birational
contraction induced by $D$. Suppose $H$ is an ample divisor, $F$ is an
effective $f$-exceptional divisor and $p$ is a positive integer such that
$pD=f^{*}H+F$. The decomposition $pD=f^{*}H+F$ is called the _weak
decomposition_ of $pD$.

######  Remark 2.9.

Under the same conditions of ?????, note that the weak decomposition of $pD$
is unique. In the rest of the paper, we write the weak decomposition of $pD$
as $pD=P+N$, where $P:=f^{*}H$ and $N:=F$.

##  3\. $q$-birational morphisms

This section aims to define the notion of $q$-birational morphisms and to
prove izerothapplication and isecondzerothapplication. Let us define the
following notions.

######  Definition 3.1.

qbir Let $X,X^{\prime}$ be any normal varieties and $f:X^{\prime}\to X$ be any
proper birational morphism.

  * (a)

The _center_ of $f$ is the reduced closed subscheme $C$ of $X$ which is the
image of the exceptional locus along $f$.

  * (b)

We say $f$ is a _$q$ -birational morphism_ if the exceptional locus has
codimension $1$ and the center of $f$ has codimension $\geq q+1$.

Let us prove the following birational criterion of $(S_{q+1})$.

######  Proposition 3.2.

good Let $X$ be any normal variety with $(RS_{q})$. Then the following are
equivalent:

  * (a)

$X$ is $(S_{q+1})$.

  * (b)

For any proper birational morphism $f:X^{\prime}\to X$ with $X^{\prime}$
smooth, $R^{i}f_{*}\mathcal{O}_{X^{\prime}}=0$ for $1\leq i<q$.

###### Proof.

For $(a)\implies(b)$, choose any resolution $f:X^{\prime}\to X$ of $X$ which
is an isomorphism outside the singular locus of $X$. For any point $x\in X$,
there is a spectral sequence

(3.1)
$E^{st}_{2}=H^{s}_{x}(X_{x},(R^{t}f_{*}\mathcal{O}_{X^{\prime}})_{x})\implies
H^{s+t}_{f^{-1}(x)}(X^{\prime}_{x},\mathcal{O}_{X^{\prime},x}).$

Suppose that $x\in C$ is a point of the center $C$ of $f$.

Let us assume that $2\leq n\leq q$ and $R^{i}f_{*}\mathcal{O}_{X^{\prime}}=0$
for $1\leq i<n-1$. It suffices to show that $x$ is not an associated point of
$R^{n-1}f_{*}\mathcal{O}_{X^{\prime}}$. Note that we may assume
$\operatorname{codim}_{X}x\geq q+1$, because
$(R^{n-1}f_{*}\mathcal{O}_{X^{\prime}})_{x}=0$ for any point $x\in C$ with
$\operatorname{codim}_{X}x\leq q$ and any $i\geq 1$.

Consider the spectral sequence (3.1). For any $0\leq i\leq n$, $E^{i0}_{2}=0$
since $X$ is $(S_{q+1})$. Using the our assumption leads to $E^{(n-j)j}_{2}=0$
for $1\leq j<n-1$. Moreover, $E^{n-1}=0$ by ssg. Hence, we have

$E^{0(n-1)}_{2}=E^{0(n-1)}_{3}=\cdots=E^{0(n-1)}_{\infty}=E^{n-1}=0.$

Therefore $H^{0}_{x}(X_{x},(R^{n-1}f_{*}\mathcal{O}_{X^{\prime}})_{x})=0$ and
hence $R^{n-1}f_{*}\mathcal{O}_{X^{\prime}}$ does not have associated point as
$x$. Thus, we have the assertion.

For $(b)\implies(a)$, consider the spectral sequence (3.1). What we have to
prove is for any point $x\in X$, $H^{i}_{x}(X_{x},\mathcal{O}_{X,x})=0$ for
$i<\min\\{q+1,\operatorname{codim}_{X}x\\}$. Indeed, by
$R^{i}f_{*}\mathcal{O}_{X^{\prime}}=0$ for $1\leq i<q$,
$H^{0}_{x}(X_{x},(R^{t}f_{*}\mathcal{O}_{X^{\prime}})_{x})=0$ for $0\leq t<q$.
Thus, by the spectral sequence argument, we obtain

$E^{i0}_{2}=E^{i0}_{3}=\cdots=E^{i0}_{\infty}\subseteq
H^{i}_{f^{-1}(x)}(X^{\prime}_{x},\mathcal{O}_{X^{\prime},x})$

for any $i\leq q$. Using ssg, we have the assertion. ∎

######  Remark 3.3.

Let us note that the above proof is similar to the proof of Lemma 5.12 in
[KM98]. Furthermore, for $q=2$, Proposition 7.1.7 in [Fuj17] induces our
proposition. It is worth noting that good is similar to Lemma 4.1 in [AH12].
In [AH12], the lemma is for $(C_{q+1})$ because the proof used sheaf
cohomology, not local cohomology. Our theorem is for $(S_{q+1})$ because we
used local cohomology as in [Fuj17].

Let us prove the main results of this section, izerothapplication and
isecondzerothapplication.

######  Theorem 3.4.

zerothapplication Let $X$ be a variety with $(R_{q}),(S_{q+1})$ and
$f:X^{\prime}\to X$ be any $q$-birational morphism from a smooth variety
$X^{\prime}$. Then $R^{i}f_{*}\mathcal{O}_{X^{\prime}}(E)=0$ for any $1\leq
i<q$ and any effective $f$-exceptional divisor $E$.

###### Proof.

The main idea of the proof of zerothapplication goes back to the proof of
Lemma 3.3 in [Kov99]. Let us note that $R^{i}f_{*}\mathcal{O}_{X^{\prime}}=0$
for $1\leq i<q$ by good.

For any closed point $x\in X$, set $F:=f^{-1}(x)$ and let $\mathcal{J}$ be the
ideal sheaf of $F$ in $X^{\prime}$. Then the theorem of formal functions gives

$(R^{i}f_{*}\mathcal{O}_{X^{\prime}}(E))^{\wedge}_{x}=\lim_{\longleftarrow}H^{i}(X^{\prime},\mathcal{O}_{X^{\prime}}(E)\otimes\mathcal{O}_{X^{\prime}}/\mathcal{J}^{n})$

and by the Serre duality,
$H^{i}(X^{\prime},\mathcal{O}_{X^{\prime}}(E)\otimes\mathcal{O}_{X^{\prime}}/\mathcal{J}^{n})$
is the dual of

$\operatorname{\mathrm{Ext}}^{\dim
X-i}_{X^{\prime}}(\mathcal{O}_{X^{\prime}}/\mathcal{J}^{n},\omega_{X^{\prime}}\otimes\mathcal{O}_{X^{\prime}}(-E))=\operatorname{\mathrm{Ext}}^{\dim
X-i}_{X^{\prime}}(\mathcal{O}_{X^{\prime}}(E)\otimes\mathcal{O}_{X^{\prime}}/\mathcal{J}^{n},\omega_{X^{\prime}}).$

By the definition of local cohomology, the direct limit

$\lim_{\longrightarrow}\operatorname{\mathrm{Ext}}^{\dim
X-i}_{X^{\prime}}(\mathcal{O}_{X^{\prime}}(E)\otimes\mathcal{O}_{X^{\prime}}/\mathcal{J}^{n},\omega_{X^{\prime}})$

coincides with $H^{\dim
X-i}_{F}(X^{\prime},\omega_{X^{\prime}}\otimes\mathcal{O}_{X^{\prime}}(-E))$.
Therefore, we have

(3.2) $(R^{i}f_{*}\mathcal{O}_{X^{\prime}}(E))^{\wedge}_{x}\text{ is dual to
}H^{\dim X-i}_{F}(X^{\prime},\omega_{X}\otimes\mathcal{O}_{X^{\prime}}(-E)).$

By the Leray spectral sequence

$E^{st}_{2}=H^{s}_{x}(X,R^{t}f_{*}(\omega_{X^{\prime}}\otimes\mathcal{O}_{X^{\prime}}(-E)))\implies
H^{s+t}_{F}(X^{\prime},\omega_{X^{\prime}}\otimes\mathcal{O}_{X^{\prime}}(-E)),$

and the relative Kawamata-Viehweg vanishing theorem, we have

$H^{\dim
X-i}_{F}(X^{\prime},\omega_{X^{\prime}}\otimes\mathcal{O}_{X^{\prime}}(-E))=H^{\dim
X-i}_{x}(X,f_{*}(\omega_{X^{\prime}}\otimes\mathcal{O}_{X^{\prime}}(-E))).$

In particular, $H^{\dim X-i}_{x}(X,f_{*}\omega_{X^{\prime}})=0$ for any $1\leq
i<q$, since $R^{i}f_{*}\mathcal{O}_{X^{\prime}}=0$ holds for such $i$.

By considering the injection
$\omega_{X^{\prime}}\otimes\mathcal{O}_{X^{\prime}}(-E)\to\omega_{X^{\prime}}$,
we obtain an injection
$f_{*}(\omega_{X^{\prime}}\otimes\mathcal{O}_{X^{\prime}}(-E))\to
f_{*}\omega_{X^{\prime}}$ and an exact sequence

$0\to f_{*}(\omega_{X^{\prime}}\otimes\mathcal{O}_{X^{\prime}}(-E))\to
f_{*}\omega_{X^{\prime}}\to\mathcal{Q}\to 0$

for some coherent sheaf $\mathcal{Q}$ on $X$. Note that the dimension of the
support of $\mathcal{Q}$ is $\leq\dim X-q-1$. Hence the long exact sequence of
local cohomology implies that

$H^{\dim
X-i}_{x}(X,f_{*}(\omega_{X^{\prime}}\otimes\mathcal{O}_{X^{\prime}}(-E)))=H^{\dim
X-i}_{x}(X,f_{*}\omega_{X^{\prime}})=0$

for $1\leq i<q$. Note that $H^{\dim X-j}_{x}(X,\mathcal{Q})=0$ for $j\leq q$
which follows from dimension count. Hence
$(R^{i}f_{*}\mathcal{O}_{X^{\prime}}(E))^{\wedge}_{x}=0$ for any closed point
$x\in X$ and $R^{i}f_{*}\mathcal{O}_{X^{\prime}}(E)=0$ for $1\leq i<q$. ∎

######  Remark 3.5.

[ST08] proved our theorem for the Cohen-Macaulay case. Indeed, under the
settings of zerothapplication, let $C$ be the center of $f$ and $\mathfrak{a}$
be an ideal sheaf on $X$ which is contained in the ideal sheaf of $C$. Suppose
that $f$ is a log resolution of $(X,\mathfrak{a})$ and $E$ is the
$f$-exceptional divisor on $X^{\prime}$ with
$\mathfrak{a}\mathcal{O}_{X^{\prime}}=\mathcal{O}_{X^{\prime}}(-E)$. Then
$(X,\mathfrak{a})$ has rational singularities outside of $C$ and we have
$R^{i}f_{*}\mathcal{O}_{X}(E)=0$ by Lemma 4.8 in [ST08].

######  Theorem 3.6.

secondzerothapplication Let $X$ be a Cohen-Macaulay variety with $(R_{q})$ and
$f:X^{\prime}\to X$ be any proper birational morphism from a smooth variety
$X^{\prime}$. Suppose that $R^{i}f_{*}\mathcal{O}_{X^{\prime}}(E)=0$ for any
$1\leq i<q$ and any effective $f$-exceptional divisor $E$. Then $f$ is
$q$-birational.

###### Proof.

Let $c\geq 2$ be the codimension of the center of $f$. For any effective
$f$-exceptional divisor $E$, we may consider the injection

$\varphi_{0}:\omega_{X^{\prime}}\otimes\mathcal{O}_{X^{\prime}}(-E)\to\omega_{X^{\prime}}.$

Denote by $\mathcal{Q}$ the cokernel. By the relative Kawamata-Viehweg
vanishing theorem,
$R^{1}f_{*}(\omega_{X^{\prime}}\otimes\mathcal{O}_{X^{\prime}}(-E))=0$, and
there is an exact sequence

$0\to f_{*}(\omega_{X^{\prime}}\otimes\mathcal{O}_{X^{\prime}}(-E))\to
f_{*}\omega_{X^{\prime}}\to f_{*}\mathcal{Q}\to 0.$

Hence the support of $f_{*}\mathcal{Q}$ has codimension $c$ for some $E$. Let
us consider the map $\varphi_{1}:f_{*}\omega_{X^{\prime}}\to\omega_{X}$ in
Proposition 5.77 in [KM98], the composition

$\varphi:f_{*}(\omega_{X^{\prime}}\otimes\mathcal{O}_{X^{\prime}}(-E))\overset{f_{*}\varphi_{0}}{\to}f_{*}\omega_{X^{\prime}}\overset{\varphi_{1}}{\to}\omega_{X},$

and the cokernel $\mathcal{Q}^{\prime}$ of $\varphi$. Since
$\mathrm{Supp}\,f_{*}\mathcal{Q}\subseteq\mathrm{Supp}\,\mathcal{Q}^{\prime}$
holds and the support of the cokernel of $\varphi_{1}$ has codimension $\geq
c$, the support of $\mathcal{Q}^{\prime}$ has codimension $c$ for such $E$.

From the injection, we have an exact sequence

(3.3) $0\to
f_{*}(\omega_{X^{\prime}}\otimes\mathcal{O}_{X^{\prime}}(-E))\overset{\varphi}{\to}\omega_{X}\to\mathcal{Q}^{\prime}\to
0.$

Take local cohomology to (3.3) as in the proof of zerothapplication. Then we
have an exact sequence

$H^{\dim X-c}_{x}(X,\omega_{X})\to H^{\dim X-c}_{x}(X,\mathcal{Q})\to H^{\dim
X-c+1}_{x}(X,f_{*}(\omega_{X^{\prime}}\otimes\mathcal{O}_{X^{\prime}}(-E)))\to
H^{\dim X-c+1}_{x}(X,\omega_{X})$

for any closed point $x\in X$. Since $X$ is Cohen-Macaulay (note that if $X$
is Cohen-Macaulay, then $\omega_{X}$ is also a Cohen-Macaulay sheaf on $X$ by
Lemma 0AWS in [Stacks]), we obtain

$H^{\dim X-c}_{x}(X,\mathcal{Q})\cong H^{\dim
X-c+1}_{x}(X,f_{*}(\omega_{X^{\prime}}\otimes\mathcal{O}_{X^{\prime}}(-E)))$

for any closed point $x\in X$.

By Theorem 3.5.7 in [BH98], there is a closed point $x\in X$ such that

$H^{\dim
X-c+1}_{x}(X,f_{*}(\omega_{X^{\prime}}\otimes\mathcal{O}_{X^{\prime}}(-E)))\neq
0.$

For such $x$, using the Kawamata-Viehweg vanishing theorem and the Leray
spectral sequence as in the proof of zerothapplication, we have

$H^{\dim
X-c+1}_{F}(X,\omega_{X^{\prime}}\otimes\mathcal{O}_{X^{\prime}}(-E))\neq 0,$

where $F:=f^{-1}(x)$. Hence,
$(R^{c-1}f_{*}\mathcal{O}_{X^{\prime}}(E))^{\wedge}_{x}\neq 0$ by (3.2) and
the theorem follows. ∎

######  Theorem 3.7.

firstapplication Let $X,X^{\prime}$ be any projective varieties, $X$ be
$(R_{q})$ and Cohen-Macaulay, $X^{\prime}$ be smooth and $f:X^{\prime}\to X$
be any proper birational morphism. Then $f$ is a $q$-birational morphism if
and only if for any effective $f$-exceptional divisor $E$ on $X^{\prime}$,
$R^{i}f_{*}\mathcal{O}_{X^{\prime}}(E)=0$ for $1\leq i<q$.

We might expect that firstapplication is true for any variety $X$ with
$(R_{q})$ and $(S_{q+1})$. However, we cannot prove or disprove it.

We may write firstapplication as an absolute cohomology vanishing.

######  Corollary 3.8.

pll Let $X,X^{\prime}$ be any normal projective varieties and $f:X^{\prime}\to
X$ be any proper birational morphism. Suppose that $X^{\prime}$ is smooth and
$X$ is $(R_{q})$ and $(S_{q+1})$.

  * (a)

Suppose $f$ is $q$-birational. Then for any Cartier divisor $D$ on $X$ and
effective $f$-exceptional divisor $F$,

(3.4)
$H^{i}(X^{\prime},\mathcal{O}_{X^{\prime}}(f^{*}D+F))=H^{i}(X,\mathcal{O}_{X}(D))=H^{i}(X^{\prime},\mathcal{O}_{X^{\prime}}(f^{*}D))$

for $1\leq i<q$.

  * (b)

Suppose $X$ is Cohen-Macaulay, $f$ has divisorial exceptional locus and (3.4)
holds for $1\leq i<q$. Then $f$ is $q$-birational.

###### Proof.

For $(a)$, the second equality follows from good. Hence it suffices to prove
the first equality only.

If $f$ is $q$-birational, by the Leray spectral sequence

$E^{ij}_{2}=H^{i}(X,R^{j}f_{*}\mathcal{O}_{X^{\prime}}(f^{*}D+F))\implies
H^{i+j}(X^{\prime},\mathcal{O}_{X^{\prime}}(f^{*}D+F))$

and zerothapplication, the assertion is proved.

For $(b)$, we follow the idea of the proof of Theorem 1-2-3 in [KMM87]. Let
$m$ be any nonnegative integer and $H$ be any ample divisor on $X$. Consider
the Leray spectral sequence

$E^{ij}_{2}=H^{i}(X,R^{j}f_{*}(\mathcal{O}_{X^{\prime}}(mf^{*}H+F)))\implies
H^{i+j}(X^{\prime},\mathcal{O}_{X^{\prime}}(mf^{*}H+F)).$

For $1\leq j<q$, if $m\gg 0$, then

$H^{j}(X^{\prime},\mathcal{O}_{X^{\prime}}(mf^{*}H+F))=H^{j}(X,\mathcal{O}_{X}(mH))=0$

by our assumption and the Serre vanishing theorem. By the same way, if $i\geq
1$, then $E^{ij}_{2}=0$ for any $j\geq 0$ and $m\gg 0$. Hence,

(3.5)
$H^{0}(X,R^{j}f_{*}\mathcal{O}_{X^{\prime}}(F)\otimes\mathcal{O}_{X}(mH))=E^{0j}_{2}=E^{0j}_{\infty}=0\,\,\text{
for }1\leq j<q\text{ and }m\gg 0.$

Let $1\leq t<q$. If $R^{t}f_{*}\mathcal{O}_{X^{\prime}}(F)\neq 0$, then for
$m\gg 0$, $R^{t}f_{*}\mathcal{O}_{X^{\prime}}(F)\otimes\mathcal{O}_{X}(mH)$ is
a globally generated sheaf on $X$ and hence

$H^{0}(X,R^{t}f_{*}\mathcal{O}_{X^{\prime}}(F)\otimes\mathcal{O}_{X}(mH))\neq
0.$

It contradicts (3.5). Therefore $R^{t}f_{*}\mathcal{O}_{X^{\prime}}(F)=0$ and
by firstapplication, we obtain the assertion. ∎

##  4\. $q$-birational divisors

In this section we introduce the notion of $q$-birational divisors and prove
istronglyqbirational1 and istronglyqbirational2. Moreover, we discuss the
questions about the generalizations of those theorems.

######  Definition 4.1.

birationaldivisor Let $X$ be any smooth projective variety and $D$ be any
divisor on $X$. We say $D$ is _weakly $q$-birational_ if there are smooth
projective varieties $X^{\prime}$ and $X_{0}$, a diagram

${X^{\prime}}$${X}$${X_{0},}$$\scriptstyle{\pi}$$\scriptstyle{f^{\prime}}$

an ample Cartier divisor $H$ on $X_{0}$ and a positive integer $p$ such that

  * (a)

$f^{\prime}$ is $q$-birational and,

  * (b)

$\pi^{*}(pD)-(f^{\prime})^{*}H$ is a (not necessarily effective)
$f^{\prime}$-exceptional divisor on $X^{\prime}$.

If $\pi^{*}(pD)-(f^{\prime})^{*}H$ in $(b)$ is effective, we say $D$ is _$q$
-birational_. If $D$ is weakly $q$-biratonal, we say the minimum of $p$ is the
_index_ of $D$ and denote it by $p_{D}$.

######  Remark 4.2.

By putting $X^{\prime}=X,X_{0}=X$ and $\pi,f^{\prime}=\mathrm{id}$, any ample
divisor is $q$-birational for any $q\geq 2$. In birationaldivisor, if $D$ is
$q$-birational, such ample divisor $H$ is unique by the above.

######  Remark 4.3.

wasteremark The notion of $q$-birationality is stronger than that of bigness
and finite generation. If $X$ is a smooth projective variety and $D$ is an
effective divisor on $X$, $D$ is $q$-birational if and only if it is big and
finitely generated, $X^{\prime},X_{0}$ are smooth, and $f^{\prime}$ is
$q$-birational if we use the notions in fg.

The main purposes of introducing $q$-birationality are to state and prove the
following Kodaira-type vanishing theorem and Zariski decomposition theorem.

######  Theorem 4.4.

stronglyqbirational1 Let $X$ be any smooth projective variety and $D$ be any
$q$-birational divisor on $X$. Then

$H^{i}(X,\mathcal{O}_{X}(K_{X}+p_{D}D))=0\,\,\,\text{ for }1\leq i<q.$

###### Proof.

Let

${X^{\prime}}$${X}$${X_{0}}$$\scriptstyle{\pi}$$\scriptstyle{f^{\prime}}$

be the diagram in the definition of $q$-birationality and $H$ be the ample
divisor on $X_{0}$ such that $p_{D}\pi^{*}D=(f^{\prime})^{*}H+F_{0}$ for some
effective $f^{\prime}$-exceptional divisor $F_{0}$ on $X^{\prime}$.

By Serre duality and rational singularities,

$\displaystyle H^{i}(X,\mathcal{O}_{X}(K_{X}+p_{D}D))$ $\displaystyle=H^{\dim
X-i}(X,\mathcal{O}_{X}(-p_{D}D))$ $\displaystyle=H^{\dim
X-i}(X^{\prime},\mathcal{O}_{X^{\prime}}(-p_{D}\pi^{*}D))$
$\displaystyle=H^{i}(X^{\prime},\mathcal{O}_{X^{\prime}}(K_{X^{\prime}}+p_{D}\pi^{*}D)).$

Since $f^{\prime}$ is $q$-birational and pll holds, we have

$H^{i}(X^{\prime},\mathcal{O}_{X^{\prime}}(K_{X^{\prime}}+p_{D}\pi^{*}D))=H^{i}(X^{\prime},\mathcal{O}_{X^{\prime}}(K_{X^{\prime}}+p_{D}(f^{\prime})^{*}H)).$

Note that we used the fact that $K_{X^{\prime}}=(f^{\prime})^{*}K_{X}+F$ for
some effective $f^{\prime}$-exceptional divisor $F$ on $X^{\prime}$. Now, we
can apply the Kawamata-Viehweg vanishing theorem to prove the assertion. ∎

######  Theorem 4.5.

stronglyqbirationl2 Let $X$ be any smooth projective variety, $D$ be any
$q$-birational divisor on $X$ and $p_{D}D=P+N$ be the weak decomposition of
$p_{D}D$ for the index $p_{D}$ of $D$. Then

$H^{i}(X,\mathcal{O}_{X}(mp_{D}D))=H^{i}(X,\mathcal{O}_{X}(mP+mN))=H^{i}(X,\mathcal{O}_{X}(mP))$

for any $m\geq 0$ and $0\leq i<q$.

###### Proof.

Let

${X^{\prime}}$${X}$${X_{0}}$$\scriptstyle{\pi}$$\scriptstyle{f^{\prime}}$

be the diagram in the definition of $q$-birationality. By rational
singularities,

$H^{i}(X,\mathcal{O}_{X}(mP+mN))=H^{i}(X^{\prime},\mathcal{O}_{X^{\prime}}(m\pi^{*}P+m\pi^{*}N)).$

We see that there is an ample divisor $H$ on $X_{0}$ such that
$\pi^{*}P=(f^{\prime})^{*}H$ and $\pi^{*}(mN)$ is an effective
$f^{\prime}$-exceptional divisor on $X^{\prime}$. Hence pll gives

$H^{i}(X^{\prime},\mathcal{O}_{X^{\prime}}(m\pi^{*}P+m\pi^{*}N))=H^{i}(X^{\prime},\mathcal{O}_{X^{\prime}}(m(f^{\prime})^{*}H)),$

if we use $q$-birationality of $f^{\prime}$. Therefore, by rational
singularities, we have the assertion. ∎

The rest of the paper discusses generalizations of stronglyqbirational1 and
stronglyqbirationl2. To do this, let us prove the numerical invariance of weak
$q$-birationality.

######  Proposition 4.6.

Let $X$ be any smooth projective variety and $D$ be any weakly $q$-birational
divisor on $X$. Then for some $m\geq 1$, $mD$ is also weakly $q$-birational.
Conversely, for any divisor $D^{\prime}$ on $X$, if $mD^{\prime}$ is weakly
$q$-birational for some $m\geq 1$, then $D^{\prime}$ is weakly $q$-birational.

###### Proof.

For the first assertion, under the same setting of birationaldivisor,
$\pi^{*}(pmD)-{f^{\prime}}^{*}(mH)$ is an $f^{\prime}$-exceptional divisor on
$X^{\prime}$. The second assertion follows from the definition immediately. ∎

The following shows that weak $q$-birationality is preserved by numerical
equivalence.

######  Theorem 4.7.

num Let $X$ be any smooth projective variety, $D$ be any weakly $q$-birational
divisor on $X$ and $D^{\prime}$ be any divisor on $X$. If $D\equiv
D^{\prime}$, then $D^{\prime}$ is weakly $q$-birational.

###### Proof.

Let

${X^{\prime}}$${X}$${X_{0}}$$\scriptstyle{\pi}$$\scriptstyle{f^{\prime}}$

be the diagram in the definition and $H$ be an ample $\mathbb{Q}$-divisor on
$X_{0}$ such that $\pi^{*}D-(f^{\prime})^{*}H$ is $f$-exceptional. By Lemma
2.40 in [Laz04], there is a very ample divisor $N$ such that $N+P$ is globally
generated for any numerically trivial divisor $P$ on $X$.

It suffices to show that $\pi^{*}D^{\prime}-(f^{\prime})^{*}H^{\prime}$ is
$f$-exceptional for some ample $\mathbb{Q}$-divisor $H^{\prime}$ on $X_{0}$.
If we set $P:=D^{\prime}-D$, then $D^{\prime}=D+P$.

By

$\displaystyle\pi^{*}(mD^{\prime})$
$\displaystyle-(f^{\prime})^{*}(mH+(f^{\prime})_{*}\pi^{*}(N+mP)-(f^{\prime})_{*}\pi^{*}N)$
$\displaystyle\sim_{\mathbb{Q}}(\pi^{*}(mD)-(f^{\prime})^{*}(mH))+(\pi^{*}(N+mP)-(f^{\prime})^{*}(f^{\prime})_{*}\pi^{*}(N+mP))$
$\displaystyle\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-(\pi^{*}N-(f^{\prime})^{*}f^{\prime}_{*}\pi^{*}N)$

and the fact that $mH+(f^{\prime})_{*}\pi^{*}(N+mP)-(f^{\prime})_{*}\pi^{*}N$
is ample for $m\gg 0$, we have $mD^{\prime}$ is $q$-birational for $m\gg 0$.
Therefore $D^{\prime}$ is $q$-birational. ∎

######  Remark 4.8.

This proof does not work for $q$-birational divisors because we do not know
whether

$(\pi^{*}(N+mP)-(f^{\prime})^{*}(f^{\prime})_{*}\pi^{*}(N+mP))-(\pi^{*}N-(f^{\prime})^{*}f^{\prime}_{*}\pi^{*}N)$

is effective or not. We can not expect numerical invariance for $q$-birational
divisors because finite generation is not preserved by numerical equivalence
in general.

######  Remark 4.9.

We may define (weak) $q$-birationality with weaker condition that $X_{0}$ in
birationaldivisor is required to be $(R_{q})$ and $(S_{q+1})$ in which $X_{0}$
is not necessarily smooth. If we use such weaker definition of (weak)
$q$-birationality instead of the definition in birationaldivisor, we can prove
the analogous statements of stronglyqbirational1 and stronglyqbirationl2.
However, we do not know whether the analog of num is true. The main issue is
unless $X_{0}$ is $\mathbb{Q}$-factorial, the notion
$(f^{\prime})^{*}(f^{\prime})_{*}\pi^{*}N$ does not make sense because
$(f^{\prime})_{*}\pi^{*}N$ is not necessarily $\mathbb{Q}$-Cartier.

num gives a new class of divisors related to nefness.

######  Definition 4.10.

qnef Let $X$ be any smooth projective variety and $N^{1}(X)$ be the Néron-
Severi group of $X$ as a finitely generated $\mathbb{Z}$-module.

  * (a)

We call $D\in N^{1}(X)$ _weakly $q$-birational_ if an inverse of $D$ along
$\operatorname{Pic}(X)\to N^{1}(X)$ is weakly $q$-birational.

  * (b)

For $D\in N^{1}(X)\otimes_{\mathbb{Z}}\mathbb{Q}$, we call $D$ _weakly
$q$-birational_ if there is a positive integer $m$ such that $mD$ is a weakly
$q$-birational integral divisor on $X$.

  * (c)

For $D\in N^{1}(X)\otimes_{\mathbb{Z}}\mathbb{R}$, we call $D$ _$q$ -nef_ if
it is pseudo-effective and an element of the closure of the set of weakly
$q$-birational $\mathbb{Q}$-divisors on $X$.

  * (d)

For $D\in\operatorname{Pic}(X)$, we call $D$ _$q$ -nef_ if the image of $D$
along $\operatorname{Pic}(X)\to N^{1}(X)$ is $q$-nef.

Now, we can formulate the following two questions, which are generalizations
of stronglyqbirational1 and stronglyqbirationl2. For the definition of
divisorial Zariski decomposition, see Definition 3.1.12 in [Nak04].

######  Question 4.11.

conj1 Let $X$ be any smooth projective variety and $D$ be any big and $q$-nef
$\mathbb{Q}$-divisor on $X$. Does the following vanishing

$H^{i}(X,\mathcal{O}_{X}(K_{X}+\left\lceil D\right\rceil))=0$

hold for $1\leq i<q$?

######  Question 4.12.

conj2 Let $X$ be any smooth projective variety, $D$ be any big integral
divisor on $X$ and $D=P+N$ be the divisorial Zariski decomposition of $D$. If
$D$ is $q$-nef, does the map $\mathcal{O}_{X}(\left\lfloor
mP\right\rfloor)\to\mathcal{O}_{X}(mD)$ induce isomorphisms

$H^{i}(X,\mathcal{O}_{X}(\left\lfloor mP\right\rfloor))\cong
H^{i}(X,\mathcal{O}_{X}(mD))$

for all $m\geq 0$ and $0\leq i<q$?

######  Remark 4.13.

For $q=\dim X$, conj1 and conj2 have affirmative answers, because any $q$-nef
divisors are nef. Note that for any big and nef divisor, the Kawamata-Viehweg
vanishing theorem holds and a nef divisor does not have the negative part in
its Zariski decomposition.

## References

  * [Ale08] V. Alexeev: Limits of stable pairs, Pure Appl. Math. Q. 4 (2008), no. 3, 767–783.
  * [AH12] V. Alexeev and C. Hacon: Non-rational centers of log canonical singularities, J. Algebra. 369 (2012), 1–15.
  * [BH98] W. Bruns and H. Herzog: Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1998.
  * [Fuj17] O. Fujino: Foundations of the minimal model program, MSJ Memoirs, vol. 35, Mathematical Society of Japan, Tokyo, 2017\.
  * [HK00] Y. Hu and S. Keel: Mori dream spaces and GIT, Michigan Math. J. 48 (2000), 331–348.
  * [KMM87] Y. Kawamata, K. Matsuda and K. Matsuki: Introduction to the minimal model problem, Adv. Stud. Pure Math. 10 (1987), 283–360.
  * [Kol11] J. Kollár: A local version of the Kawamata-Viehweg vanishing theorem, Pure Appl. Math. Q. 7 (2011), no. 4, 1477–1494.
  * [KM98] J. Kollár and S. Mori: Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134. Cambridge University Press, Cambridge, 1998.
  * [Kov99] S. Kovács: Rational, log canonical, Du Bois singularities: on the conjectures of Kollár and Steenbrink, Compositio Math. 118 (1999), no. 2, 123–133.
  * [Laz04] R. Lazarsfeld: Positivity in algebraic geometry I: Classical setting: line bundles and linear series, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 48, Springer, Beriln, 2004.
  * [Nak04] N. Nakayama: Zariski-decomposition and abundance, Mathematical Society of Japan, vol. 14, Tokyo, 2004.
  * [ST08] K. Schwede and S. Takagi: Rational singularities associated to pairs, Special volume in honor of Melvin Hochster, Michigan Math. J. 57 (2008), 625–658.
  * [Stacks] A. J. de Jong et al.: The Stacks Project, Available at http://stacks.math.columbia.edu.