# Castelnuovo-Mumford regularity of unprojections and the Eisenbud-Goto
regularity conjecture

Junho Choe  School of Mathematics, Korea Institute for Advanced Study (KIAS),
85 Hoegiro Dongdaemun-gu, Seoul 02455, Republic of Korea<EMAIL_ADDRESS>
###### Abstract.

McCullough and Peeva found counterexamples to the Eisenbud-Goto conjecture on
the Castelnuovo-Mumford regularity by using Rees-like algebras. In this paper,
we suggest another method to construct counterexamples to the conjecture and
show that for nondegenerate projective varieties of a fixed dimension $n\geq
3$ and a fixed codimension $e\geq 2$, the regularity can not be bounded above
by any polynomial function of the degree whose order is about half of the
dimension $n$. Our strategy is an unprojection process and utilizes the
possible complexity of homogeneous ideals with three generators and
exceptional-fundamental schemes. Also, our counterexamples exhibit how the
singularity affects the regularity.

###### Key words and phrases:

Castelnuovo–Mumford regularity, unprojections

###### 2020 Mathematics Subject Classification:

Primary: 14N05; Secondary: 13D02

###### Contents

  1. 1 Introduction
  2. 2 Unprojections
  3. 3 The regularity and other invariants
  4. 4 Families of homogeneous ideals with high regularity
  5. 5 Proof of the main theorem

##  1\. Introduction

We work over an algebraically closed field $\Bbbk$ of any characteristic. Let
$X\subseteq\mathbb{P}^{r}$ be a projective variety which is nondegenerate,
that is, it lies in no smaller linear spaces; every variety is always
irreducible and reduced, and if its dimension is $n$, then it is called an
$n$-fold.

The Castelnuovo-Mumford regularity, or simply the regularity, is one of the
most important invariants in projective algebraic geometry. It is used to
express homological complexity as follows: Let $S$ be the homogeneous
coordinate ring of $\mathbb{P}^{r}$, and consider a finitely generated graded
$S$-module $M$ with minimal free resolution

${0}$${M}$${F_{0}}$${F_{1}}$${\cdots}$${F_{i}}$${\cdots,}$

where the graded $S$-free modules $F_{i}$ can be written as

$F_{i}=\bigoplus_{j\in\mathbb{Z}}S^{\beta_{i,j}(M)}(-i-j).$

Then the regularity of $M$ is defined to be

$\textup{reg\,}M=\max\\{j\in\mathbb{Z}:\beta_{i,j}(M)\neq 0\text{ for some
}i\\},$

and the regularity of $X\subseteq\mathbb{P}^{r}$, denoted by
$\textup{reg\,}X$, is that of its homogeneous ideal $I_{X}$.

Note that the regularity also measures cohomological complexity in relation of
the Serre vanishing theorem; refer to the well-known fact that if the section
module

$M=\bigoplus_{j\in\mathbb{Z}}H^{0}(\mathbb{P}^{r},F(j))$

of a coherent sheaf $F$ on $\mathbb{P}^{r}$ is finitely generated over $S$,
then an integer $q$ satisfies

$H^{i}(\mathbb{P}^{r},F(q-i))=0\quad\text{for every $i>0$}$

precisely when $q\geq\textup{reg\,}M$.

The Eisenbud-Goto regularity conjecture, or the regularity conjecture for
short, says that the regularity of $X\subseteq\mathbb{P}^{r}$ has an upper
bound in terms of the degree and the codimension of $X$ as follows:

######  Conjecture 1.1 (Eisenbud-Goto [10]).

Let $X\subseteq\mathbb{P}^{r}$ be any nondegenerate projective variety. Then
we have

$\textup{reg\,}X\leq\deg X-\textup{codim\,}X+1.$

This conjecture holds true for

  1. (1)

any projective variety $X\subseteq\mathbb{P}^{r}$ such that $S/I_{X}$ is
Cohen-Macaulay ([10]);

  2. (2)

any curves ([4] and [14]);

  3. (3)

smooth or mildly singular surfaces over $\mathbb{C}$ ([23], [17] and [20]);
and

  4. (4)

smooth or mildly singular threefolds in $\mathbb{P}^{5}$ over $\mathbb{C}$
([15] and [21]),

where $\mathbb{C}$ is the field of complex numbers. For other related results,
see for example [8], [16] and [18].

But counterexamples to the regularity conjecture have been found by McCullough
and Peeva [18]. Their method uses both the Rees-like algebra and the existence
of homogeneous (nonprime) ideals having extremely large regularity. In fact,
McCullough and Peeva showed that for any univariate (real) polynomial $p$, one
can find a nondegenerate projective variety $X\subset\mathbb{P}^{r}$ for some
$r$ such that

$\textup{reg\,}X>p(\deg X).$

Moreover, they presented a computational example which is a nondegenerate
singular projective threefold in $\mathbb{P}^{5}$ of degree $31$ and
regularity $38$.

Our main result is an unprojection process that is able to construct a
collection of counterexamples to the regularity conjecture with a fixed
dimension $n\geq 3$ and a fixed codimension $r-n\geq 2$. More specifically, we
prove that there are no polynomial bounds for the regularity in terms of the
degree whose order is about half of the dimension $n$. Our method works due to
the possible complexity of homogeneous ideals with three generators and
exceptional-fundamental schemes.

We use the following asymptotic notation: If $(a_{d})$ and $(b_{d})$ are
sequences of positive numbers, then we write

$a_{d}=\Omega(b_{d})\quad\text{when}\quad
0<\liminf_{d\to\infty}\frac{a_{d}}{b_{d}}\leq\infty.$

Now let us state our main theorem. It is interesting to compare this result
with [6, Theorem 7.2].

######  Theorem 1.2.

Fix integers $n\geq 3$ and $e\geq 2$. Then there exists a sequence

$(X_{d}\subset\mathbb{P}^{n+e}:d\gg 0\textup{ in }\mathbb{Z})$

of nondegenerate projective $n$-folds such that
${\displaystyle\lim_{d\to\infty}\deg X_{d}=\infty}$ and

$\textup{reg\,}X_{d}=\Omega((\deg X_{d})^{k})\quad\text{with}\quad
k=\left\lfloor\frac{n+1}{2}\right\rfloor.$

This paper is organized as follows: In Section 2, we introduce the
unprojection and the partial elimination ideal theory. In Section 3, we give
preliminary results to compute significant invariants such as the degree, the
regularity and the projective dimension. In Section 4, we list certain
families of homogeneous ideals whose values of the regularity seem to rapidly
increase; some of them become ingredients for the proof of the main theorem.
In Section 5, we finally prove the main theorem, and a specific counterexample
to the regularity conjecture is given in detail.

The following assumptions and notations are used:

  * •

$X\subseteq\mathbb{P}^{r}$ and $Y\subseteq\mathbb{P}^{r-1}$ are nondegenerate
projective varieties.

  * •

$S_{X}=S/I_{X}$ and $S_{Y}=R/I_{Y}$ are homogeneous coordinate rings and
homogeneous ideals of $X\subseteq\mathbb{P}^{r}$ and
$Y\subseteq\mathbb{P}^{r-1}$, respectively.

  * •

$K(D)$ means the field of fractions of an integral domain $D$.

  * •

The missing entries in a matrix or a table are usually zero.

##  2\. Unprojections

An unprojection is the reverse process of a projection. In this section, we
explain about unprojections associated to birational linear projections whose
centers are points. For more complicated and theoretical approach to the
unprojection, consult with [24].

Projections from a point. Fix a point $z\in\mathbb{P}^{r}$, and consider the
projection $\pi_{z}:\mathbb{P}^{r}\setminus\\{z\\}\to\mathbb{P}^{r-1}$ from
$z$. In order to analyze geometry of the restricted map
$\pi_{z}|_{X}:X\dashrightarrow\mathbb{P}^{r-1}$, one uses partial elimination
ideals introduced by Green [13]. We collect some information on them below and
refer to [7, Appendix] for their applications.

Taking suitable homogeneous coordinates $x_{0},x_{1},\ldots,x_{r}\in S_{1}$ in
$\mathbb{P}^{r}$, we may assume that the point $z\in\mathbb{P}^{r}$ is given
as

$(x_{0}:x_{1}:\cdots:x_{r})=(1:0:\cdots:0).$

One thus has
$\pi_{z}(x_{0}:x_{1}:\cdots:x_{r})=(x_{1}:\cdots:x_{r})\in\mathbb{P}^{r-1}$.
Set $R=\Bbbk[x_{1},\ldots,x_{r}]\subset S$ so that it forms the homogeneous
coordinate ring of $\mathbb{P}^{r-1}$ with $R[x_{0}]=S$. For an integer $i$
and the graded $R$-module

$\widetilde{K}_{i}(X,z)=\\{f\in I_{X}:\deg_{x_{0}}f\leq i\\},$

the $i$-th partial elimination ideal of the pair $(X,z)$ is defined to be

$K_{i}(X,z)=\\{\text{coefficients of $x_{0}^{i}$ in any
}f\in\widetilde{K}_{i}(X,z)\\}\subseteq R.$

Partial elimination ideals fit into the following short exact sequence and
chain:

(1)
${0}$${\widetilde{K}_{i-1}(X,z)}$${\widetilde{K}_{i}(X,z)}$${K_{i}(X,z)(-i)}$${0}$
$K_{0}(X,z)\subseteq K_{1}(X,z)\subseteq\cdots\subseteq
K_{i}(X,z)\subseteq\cdots$

We denote by $X_{z}\subseteq\mathbb{P}^{r-1}$ the projective scheme given by
$K_{0}(X,z)\subset R$. Then since $K_{0}(X,z)$ is prime,
$X_{z}\subseteq\mathbb{P}^{r-1}$ is a nondegenerate projective variety with
$I_{X_{z}}=K_{0}(X,z)$, and one may see that $X_{z}$ has support
$\overline{\pi_{z}(X\setminus\\{z\\})}\subseteq\mathbb{P}^{r-1}$. By abuse of
language, both the induced map $\pi_{z}|_{X}:X\dasharrow X_{z}$ and the
projective variety $X_{z}\subseteq\mathbb{P}^{r-1}$ are called the projection
of $X$ from $z$. And if $z$ lies in $X$, then the projective scheme defined by

$K_{\infty}(X,z)=\bigcup_{i=0}^{\infty}K_{i}(X,z)\subseteq R$

is called the projectivized tangent cone $\mathbb{P}C_{z}X$ to $X$ at $z$, and
it is known that $\mathbb{P}C_{z}X$ is equidimensional with
$\dim\mathbb{P}C_{z}X=\dim X-1$. Unless $z\in X$, we have $K_{\infty}(X,z)=R$.

Partial elimination ideals have the following set-theoretic description:

######  Proposition 2.1 (cf. [13, Proposition 6.2]).

Let $Z_{i},Z_{\infty}\subseteq\mathbb{P}^{r-1}$ be the projective schemes
corresponding to $K_{i}(X,z),K_{\infty}(X,z)\subseteq R$, respectively. Then
the support of $Z_{i}$ is given as

$Z_{i}=\\{p\in\mathbb{P}^{r-1}:\textup{length}(X\cap\pi_{z}^{-1}(p))>i\\}\cup
Z_{\infty}\subseteq\mathbb{P}^{r-1},$

where $\pi_{z}^{-1}(p)$ is the affine line being the fiber of $p$ under the
projection $\pi_{z}:\mathbb{P}^{r}\setminus\\{z\\}\to\mathbb{P}^{r-1}$. In
particular, the projection $\pi_{z}|_{X}:X\dasharrow X_{z}$ has degree

$\deg(\pi_{z}|_{X})=\max\\{i\in\mathbb{Z}:Z_{i}=X_{z}\\}+1.$

This is related to the well-known degree formula

(2) $\deg X=\deg(\pi_{z}|_{X})\deg
X_{z}+\begin{cases}\deg\mathbb{P}C_{z}X,&z\in X\\\ 0,&z\not\in X\end{cases}$

for the case where the projection $\pi_{z}|_{X}:X\dasharrow X_{z}$ is
generically finite.

For simplicity, we use the following terminology:

######  Definition 2.2.

Suppose that $\pi_{z}|_{X}:X\dasharrow X_{z}$ is birational, and let $Z\subset
X_{z}\subseteq\mathbb{P}^{r-1}$ be the projective scheme defined by
$K_{1}(X,z)/K_{0}(X,z)\subset S_{X_{z}}$. We call $Z$ the exceptional-
fundamental scheme of the pair $(X,z)$.

Now let us introduce the unprojection we need. It depends on a given fake
linear form that plays the role of the so-called unprojection variable.

######  Definition 2.3.

If a homogeneous element $\lambda\in K(S_{Y})$ of degree one is not in
$S_{Y}$, then one says that $\lambda$ is a fake linear form on
$Y\subseteq\mathbb{P}^{r-1}$. For a fake linear form $\lambda$ on $Y$, the
ideal of denominators of $\lambda$ is defined to be

$D(\lambda)=\\{f\in S_{Y}:f\lambda\in S_{Y}\\}\subset S_{Y}.$

######  Theorem 2.4 (Unprojections).

Fix both a projection $\pi_{z}:\mathbb{P}^{r}\dasharrow\mathbb{P}^{r-1}$ from
a point $z\in\mathbb{P}^{r}$ and a linear form $x_{0}$ on $\mathbb{P}^{r}$
such that $x_{0}(z)\neq 0$. Then for any nondegenerate projective variety
$Y\subseteq\mathbb{P}^{r-1}$, there is a one-to-one correspondence

$\\{X\subset\mathbb{P}^{r}:\pi_{z}|_{X}\textup{ is birational onto
}Y\\}\longleftrightarrow\\{\textup{fake linear forms $\lambda$ on $Y$}\\},$

where $X$ stands for nondegenerate projective varieties. In addition, under
this correspondence, we have

$K_{0}(X,z)=I_{Y}\quad\textup{and}\quad\frac{K_{1}(X,z)}{K_{0}(X,z)}=D(\lambda).$

###### Proof.

We describe the correspondence as follows:

Let $X\subset\mathbb{P}^{r}$ be a nondegenerate projective variety satisfying
that the projection $\pi_{z}|_{X}:X\dashrightarrow X_{z}$ is birational with
$X_{z}=Y$ in $\mathbb{P}^{r-1}$. By 2.1, there is a homogeneous form $f\in
K_{1}(X,z)$ not in $K_{0}(X,z)=I_{Y}$. By definition, we obtain $fx_{0}-a\in
I_{X}$ for some homogeneous form $a\in R$ with $\deg a=\deg f+1$. Consider the
degree one homogeneous element

$\lambda=a/f\in K(S_{Y}).$

If $\lambda$ were contained in $S_{Y}$ with a representative $\lambda_{0}\in
R_{1}$, then we would have $fx_{0}-a=f(x_{0}-\lambda_{0})$ modulo
$I_{Y}[x_{0}]\subset I_{X}$, hence $x_{0}-\lambda_{0}\in I_{X}$, a
contradiction. So $\lambda$ is a fake linear form on $Y$.

On the other hand, let $\lambda=a/f$ be any fake linear form on $Y$. Then we
define an $R$-algebra homomorphism

$\phi:S\to S_{Y}[\lambda]\subset
K(S_{Y})\quad\text{by}\quad\phi(x_{0})=\lambda.$

Its kernel is a homogeneous prime ideal and has no linear forms, for $\lambda$
is a fake linear form. Take $X\subset\mathbb{P}^{r}$ to be the nondegenerate
projective variety defined by $\ker\phi$. We have $K_{0}(X,z)=I_{Y}$, and
$K_{1}(X,z)/K_{0}(X,z)$ contains the nonzero element $f$ in $S_{Y}$. By 2.1,
the projection $\pi_{z}|_{X}$ is birational onto $Y$. ∎

For convenience, we use the following notation. It is well defined up to
projective equivalence.

######  Definition 2.5.

After choosing a point $z\in\mathbb{P}^{r}$, when 2.4 associates
$X\subset\mathbb{P}^{r}$ to a fake linear form $\lambda$ on
$Y\subseteq\mathbb{P}^{r-1}$, we write

$(X,z)=\textup{Unpr}(Y,\lambda)$

and call it the unprojection of the pair $(Y,\lambda)$.

On the other hand, to compute invariants of the unprojections, we usually need
to determine when the partial elimination ideals stabilize, that is, for which
values of $i$, the equality $K_{i}(X,z)=K_{\infty}(X,z)$ holds. The following
helps us to recognize the situation for $i=1$:

######  Proposition 2.6.

Let $\lambda=a/f$ be a fake linear form on $Y\subseteq\mathbb{P}^{r-1}$ with
unprojection $(X,z)=\textup{Unpr}(Y,\lambda)$. Take
$Z_{i}\subseteq\mathbb{P}^{r-1}$ and $W_{i}\subset Y\subseteq\mathbb{P}^{r-1}$
to be the projective schemes given by $K_{i}(X,z)\subseteq R$ and
$(a,f)^{i}\subset S_{Y}$, respectively. With $n=\dim Y$, we set

$\chi_{n-1}(W)=\begin{cases}\deg W,&\dim W=n-1\\\ 0,&\dim W<n-1\end{cases}$

for any projective scheme $W\subsetneq Y\subseteq\mathbb{P}^{r-1}$, and
consider the sequence

$(d_{0},d_{1},d_{2},\ldots)=(0,\chi_{n-1}(W_{1}),\chi_{n-1}(W_{2}),\ldots).$

Then this sequence is convex and satisfies that for each $i\geq 1$, we have

$d_{i}\leq d_{i-1}+\deg f\deg Y$

with equality if and only if $\dim Z_{i}<\dim Y-1$. Furthermore, in the case
$S_{Y}$ Cohen-Macaulay, the sequence $(d_{i})$ is linear if and only if $z\in
X$ and

$K_{1}(X,z)=K_{\infty}(X,z).$

###### Proof.

It is easy to see that

$K_{i}(X,z)/K_{0}(X,z)=(f(a,f)^{i-1}:a^{i})\quad\text{in}\quad S_{Y}.$

So we have the following diagram:

${0}$${\dfrac{R}{K_{i}(X,z)}(-i(\deg
f+1))}$${\dfrac{S_{Y}}{f(a,f)^{i-1}}}$${\dfrac{S_{Y}}{(a,f)^{i}}}$${0}$${0}$${\dfrac{S_{Y}}{(a,f)^{i-1}}(-\deg
f)}$${\dfrac{S_{Y}}{f(a,f)^{i-1}}}$${\dfrac{S_{Y}}{(f)}}$${0}$$\scriptstyle{a^{i}}$$\scriptstyle{f}$

From Hilbert polynomials of the graded $R$-modules in this diagram, it follows
that

$\chi_{n-1}(Z_{i})=\deg f\deg Y-(d_{i}-d_{i-1})$

for every integer $i\geq 1$. Refer to the fact that

$\chi_{n-1}(Z_{1})\geq\chi_{n-1}(Z_{2})\geq\cdots\geq 0.$

Therefore, what remains to check is the last assertion.

Assume that $S_{Y}$ is Cohen-Macaulay. Keep in mind that for any unmixed
homogeneous ideal $I\subseteq R$ and another homogeneous ideal $J\supseteq I$
with the same codimension as $I$, we attain $\deg R/I=\deg R/J$ if and only if
$I=J$; it is because of the associativity formula (see [9, Exercise 12.11.e]).
Consider the containments

$(f)\subsetneq(a,f)\quad\text{and}\quad K_{1}(X,z)\subseteq K_{\infty}(X,z).$

Here since $S_{Y}$ is Cohen-Macaulay and $K_{1}(X,z)/K_{0}(X,z)=((f):a)$ in
$S_{Y}$, both the preimage of $(f)\subset S_{Y}$ in $R$ and the homogeneous
ideal $K_{1}(X,z)\subset R$ are unmixed. Having $\chi_{n-1}(Z_{1})>0$, we
conclude that $z\in X$ and $K_{1}(X,z)=K_{\infty}(X,z)$ if the sequence
$(d_{i})$ is linear, and vise verse. ∎

##  3\. The regularity and other invariants

This section is devoted to the computation of invariants by using partial
elimination ideals. To this end, we deal with Koszul cohomology groups
$K_{i,j}(M,S_{1})$ (see [12]). Notice that with the residue field $\Bbbk$ of
$S$,

$K_{i,j}(M,S_{1})\cong\textup{Tor}^{S}_{i}(M,\Bbbk)_{i+j}$

as functors for any finitely generated graded $S$-module $M$, hence
$\dim_{\Bbbk}K_{i,j}(M,S_{1})=\beta_{i,j}(M)$. Consider the $i$-th partial
elimination module of $(X,z)$ defined to be

$M_{i}(X,z)=\frac{K_{i}(X,z)}{K_{i-1}(X,z)}(-i).$

######  Lemma 3.1 (cf. [13, Proposition 6.6]).

Let $z\in\mathbb{P}^{r}$ be a point. For a fixed integer $k$, there is a
spectral sequence $E=E(k)$ such that

$E^{1}_{i,j}=K_{i+j,k-i-j}(M_{i}(X,z),R_{1})\Rightarrow_{i}K_{i+j,k-i-j}(I_{X},S_{1}).$

###### Proof.

By the well-known Lefschetz type theorem, we have

$K_{i,j}(I_{X},S_{1})\cong K_{i,j}(I_{X}/x_{0}I_{X},R_{1})$

for all integers $i$ and $j$. Thus it suffices to take account of the graded
$R$-module $I_{X}/x_{0}I_{X}$.

Now let us see the following Koszul type complex and its ascending filtration:

$C_{j}=\bigwedge^{j}R_{1}\otimes(I_{X}/x_{0}I_{X})_{k-j}\quad\text{and}\quad
F_{i}C_{j}=\bigwedge^{j}R_{1}\otimes(\widetilde{K}_{i}(X,z)/x_{0}\widetilde{K}_{i-1}(X,z))_{k-j}.$

This filtration is given by the short exact sequence

${0}$${\dfrac{\widetilde{K}_{i-1}(X,z)}{x_{0}\widetilde{K}_{i-2}(X,z)}}$${\dfrac{\widetilde{K}_{i}(X,z)}{x_{0}\widetilde{K}_{i-1}(X,z)}}$${M_{i}(X,z)}$${0}$

that comes from the short exact sequence (1). Consider the spectral sequence
$E=E(k)$ induced by the filtered complex $C_{\ast}$. One has

$\displaystyle E^{0}_{i,j}$
$\displaystyle=G_{i}C_{i+j}=\bigwedge^{i+j}R_{1}\otimes M_{i}(X,z)_{k-i-j};$
$\displaystyle E^{1}_{i,j}$
$\displaystyle=H_{j}(G_{i}C_{i+\ast})=K_{i+j,k-i-j}(M_{i}(X,z),R_{1});\text{
and}$ $\displaystyle E^{\infty}_{i,j}$
$\displaystyle=G_{i}H_{i+j}(C_{\ast})=G_{i}K_{i+j,k-i-j}(I_{X}/x_{0}I_{X},V_{z}),$

where $G$ means the associated graded object. We have constructed the desired
spectral sequence. ∎

We suggest a simple trick for partial computation of the spectral sequence
above. We first adopt some terminology.

######  Definition 3.2.

A table is a $\mathbb{Z}\times\mathbb{Z}$ matrix of nonnegative integers; for
instance, the Betti table $\mathbb{B}(M)$ of a finitely generated graded
$S$-module $M$ is a table defined by

$\mathbb{B}(M)_{i,j}=\beta_{i,j}(M).$

A table sequence is a collection of tables indexed by $\mathbb{Z}$. For a
table sequence $\mathcal{B}=(B_{k}:k\in\mathbb{Z})$, its cancellation is
another table sequence that can be obtained from $\mathcal{B}$ by repeating
the following a finite number of times: for a tuple $(i,j,k,l)$ of integers
with $l>0$, subtract a common nonnegative integer from both

$(B_{k})_{i,j}\quad\text{and}\quad(B_{k-l})_{i-1,j+1}.$

For a table $B$, its decomposition is a table sequence
$(B_{k}:k\in\mathbb{Z})$ such that the entries $B_{k}$ sum up to $B$, namely
$\sum_{k\in\mathbb{Z}}B_{k}=B$.

######  Theorem 3.3 (Cancellation principle).

For any fixed point $z\in\mathbb{P}^{r}$, the Betti table $\mathbb{B}(I_{X})$
has a decomposition that is a cancellation of the table sequence
$(\mathbb{B}(M_{k}(X,z)):k\in\mathbb{Z})$.

###### Proof.

For convenience, write

$K^{l}_{i,j,k}=E^{l}_{k,i-k}\quad\text{and}\quad
K^{\infty}_{i,j,k}=E^{\infty}_{k,i-k}$

for the spectral sequence $E=E(i+j)$ in 3.1. Now the differentials of $E$
induce the complexes

${\cdots}$${K^{l}_{i+1,j-1,k+l}}$${K^{l}_{i,j,k}}$${K^{l}_{i-1,j+1,k-l}}$${\cdots,}$

and the homology group at $K^{l}_{i,j,k}$ is $K^{l+1}_{i,j,k}$. For the tables

$B^{l}_{k}=(\dim_{\Bbbk}K^{l}_{i,j,k}:(i,j)\in\mathbb{Z}\times\mathbb{Z})\quad\text{and}\quad
B^{\infty}_{k}=(\dim_{\Bbbk}K^{\infty}_{i,j,k}:(i,j)\in\mathbb{Z}\times\mathbb{Z}),$

consider the table sequences

$\mathcal{B}^{l}=(B^{l}_{k}:k\in\mathbb{Z})\quad\text{and}\quad\mathcal{B}^{\infty}=(B^{\infty}_{k}:k\in\mathbb{Z}).$

By the property of $E$, one finds that $\mathcal{B}^{l+1}$ is a cancellation
of $\mathcal{B}^{l}$, and the summation $\sum_{k\in\mathbb{Z}}B^{\infty}_{k}$
equals $\mathbb{B}(I_{X})$. Consequently, the table sequence

$\mathcal{B}^{1}=(\mathbb{B}(M_{k}(X,z)):k\in\mathbb{Z})$

is a cancellation of $\mathcal{B}^{\infty}$, and $\mathcal{B}^{\infty}$ is a
decomposition of $\mathbb{B}(I_{X})$. ∎

For the theorem above, the following can be regarded as the first nontrivial
example:

######  Example 3.4.

Let $X\subset\mathbb{P}^{3}$ be an elliptic normal curve $E$ (of degree $4$),
and take a general point $z\in E$. Then

  1. (1)

$E_{z}\subset\mathbb{P}^{2}$ is a cubic plane curve; and

  2. (2)

$K_{1}(E,z)$ is the homogeneous ideal of a point in $\mathbb{P}^{2}$ with
$K_{1}(E,z)\supsetneq K_{0}(E,z)$.

We have

$\mathbb{B}(M_{0}(E,z))=\begin{array}[]{c|cc}j\backslash i&0&1\\\ \hline\cr
2&&\\\
3&1&\end{array},\quad\mathbb{B}(M_{1}(E,z))=\begin{array}[]{c|cc}j\backslash
i&0&1\\\ \hline\cr 2&2&1\\\
3&&1\end{array}\quad\text{and}\quad\mathbb{B}(M_{k}(E,z))=0$

for any $k>1$. However, any two linearly independent quadratic forms in
$I_{E}$ have no nontrivial relations; the cancellation in 3.3 must occur
between $\beta_{1,2}(M_{1}(E,z))$ and $\beta_{0,3}(M_{0}(E,z))$. Therefore, we
have shown that the decomposition of $\mathbb{B}(I_{E})$ induced by the
cancellation principle consists of $\mathbb{B}(I_{E})$ itself and zero tables.
Note that $E\subset\mathbb{P}^{3}$ is a complete intersection of two quadrics:

$\mathbb{B}(I_{E})=\begin{array}[]{c|cc}j\backslash i&0&1\\\ \hline\cr 2&2&\\\
3&&1\end{array}$

######  Remark 3.5.

Take $i,j\in\mathbb{Z}$ to be arbitrary integers, and let us consider the
spectral sequence $E=E(i+j)$ in 3.1. The differential

$E^{1}_{k,i-k}=K_{i,j}(M_{k}(X,z),R_{1})\to
E^{1}_{k-1,i-k}=K_{i-1,j+1}(M_{k-1}(X,z),R_{1})$

is related to the Griffiths-Harris second fundamental form for $X$ at $z$;
this is derived from a map of the form

$K_{i}(X,z)(-1)\to R/K_{i-1}(X,z).$

Refer to [13].

Let $M$ be a finitely generated graded $S$-module. Recall that the regularity
of $M$ is related to the height of the Betti table
$\mathbb{B}(M)=(\beta_{i,j}(M):(i,j)\in\mathbb{Z}\times\mathbb{Z})$ when the
rows (resp. columns) are indexed by $j$ (resp. $i$), and that the projective
dimension of $M$, denoted by $\textup{pd\,}M$, is known to measure the width
of $\mathbb{B}(M)$, namely

$\textup{pd\,}M=\max\\{i\in\mathbb{Z}:\beta_{i,j}(M)\neq 0\text{ for some
$j\in\mathbb{Z}$}\\},$

and to satisfy the Auslander-Buchsbaum formula

$0ptM+\textup{pd\,}M=0ptS,$

where $0ptM$ is the depth of the irrelevant ideal on $M$.

######  Definition 3.6.

One says that $M_{i}(X,z)$ dominates the others

  1. (1)

by the regularity if

    1. (a)

$\textup{reg\,}M_{i}(X,z)\geq\textup{reg\,}M_{j}(X,z)$ for every $j<i$; and

    2. (b)

$\textup{reg\,}M_{i}(X,z)\geq\textup{reg\,}M_{j}(X,z)+2$ for every $j>i$; and

  2. (2)

by the projective dimension if

    1. (a)

$\textup{pd\,}M_{i}(X,z)\geq\textup{pd\,}M_{j}(X,z)+2$ for every $j<i$; and

    2. (b)

$\textup{pd\,}M_{i}(X,z)\geq\textup{pd\,}M_{j}(X,z)$ for every $j>i$.

With these conditions, the cancellation principle (3.3) directly implies the
result below:

######  Proposition 3.7.

Let $i\geq 0$ be an integer.

  1. (1)

If $M_{i}(X,z)$ dominates the others by the regularity, then we have

$\textup{reg\,}X=\textup{reg\,}M_{i}(X,z).$

  2. (2)

If $M_{i}(X,z)$ dominates the others by the projective dimension, then we have

$\textup{pd\,}S_{X}=\textup{pd\,}M_{i}(X,z)+1.$

######  Corollary 3.8.

Let $\lambda=a/f$ be a fake linear form on $Y$ with the homogeneous ideal
$I_{Y}\subset T\subset R$ such that $T/I_{Y}=(a,f)$ in $S_{Y}$, and put
$(X,z)=\textup{Unpr}(Y,\lambda)$. Suppose that $M_{i}(X,z)=0$ for all integers
$i\geq 2$.

  1. (1)

For the projective scheme $W\subset\mathbb{P}^{r-1}$ defined by $T\subset R$,
if $\dim W=\dim Y-1$, then

$\deg X=(\deg f+1)\deg Y-\deg W,$

and otherwise $\deg X=(\deg f+1)\deg Y$.

  2. (2)

If $\textup{reg\,}T>\textup{reg\,}Y+\deg f$, then

$\textup{reg\,}X=\textup{reg\,}T-\deg f+1.$

  3. (3)

If $\textup{pd\,}R/T>\textup{pd\,}S_{Y}+2$, then

$\textup{pd\,}S_{X}=\textup{pd\,}R/T-1.$

Moreover, when $\\{f,a\\}$ is a regular sequence on $S_{Y}$, we have

$\deg X=(\deg f+1)\deg Y,\quad\textup{reg\,}X=\textup{reg\,}Y+\deg
f\quad\text{and}\quad\textup{pd\,}S_{X}=\textup{pd\,}S_{Y}+1$

together with $M_{i}(X,z)=0$ for every integer $i\geq 2$.

###### Proof.

We begin with the following short exact sequences:

${0}$${M_{1}(X,z)}$${S_{Y}(-1)}$${\dfrac{R}{K_{1}(X,z)}(-1)}$${0}$
${0}$${\dfrac{R}{K_{1}(X,z)}(-1)}$${\dfrac{S_{Y}}{(f)}(\deg
f)}$${\dfrac{R}{T}(\deg f)}$${0}$$\scriptstyle{a}$ ${0}$${S_{Y}}$${S_{Y}(\deg
f)}$${\dfrac{S_{Y}}{(f)}(\deg f)}$${0}$$\scriptstyle{f}$

(1) It is by the degree formula (2). Note that $K_{1}(X,z)$ equals
$K_{\infty}(X,z)$ with $z\in X$, and so the second short exact sequence above
computes the degree of $\mathbb{P}C_{z}X$. (2) Tensoring the short exact
sequences above with the residue field $\Bbbk$ of $R$, we see that under the
assumption, $M_{1}(X,z)$ dominates the others by the regularity. Apply 3.7.
(3) Similarly, in this case, $M_{1}(X,z)$ dominates the others by the
projective dimension.

Now to accomplish the proof, suppose given a regular sequence $\\{f,a\\}$ on
$S_{Y}$. Then we have $K_{1}(X,z)/K_{0}(X,z)=((f):a)=(f)$ and

$\frac{K_{i}(X,z)}{K_{0}(X,z)}=(f(a,f)^{i-1}:a^{i})\subseteq((f):a^{i})=((f):a)=\frac{K_{1}(X,z)}{K_{0}(X,z)}$

for every $i\geq 1$. These imply that

$M_{1}(X,z)\cong S_{Y}(-\deg f-1)\quad\text{and}\quad M_{i}(X,z)=0\text{ for
all }i\geq 2.$

By part (1), we have $\deg X=(\deg f+1)\deg Y$. Since
$\textup{reg\,}M_{1}(X,z)=\textup{reg\,}S_{Y}(-\deg
f-1)=\textup{reg\,}S_{Y}+\deg f+1=\textup{reg\,}Y+\deg
f\geq\textup{reg\,}M_{0}(X,z)$, 3.7 tells us that
$\textup{reg\,}X=\textup{reg\,}M_{1}(X,z)=\textup{reg\,}Y+\deg f$. For the
projective dimension part, put $p=\textup{pd\,}M_{1}(X,z)$ and
$q=\max\\{j\in\mathbb{Z}:\beta_{p,j}(M_{1}(X,z))\neq 0\\}$ so that
$\beta_{p,q}(M_{1}(X,z))$ is a positive integer. However, we obtain
$\beta_{p-1,q+1}(M_{0}(X,z))=\beta_{p,q}(S_{Y})=\beta_{p,q+\deg
f+1}(M_{1}(X,z))=0$, and also
$\textup{pd\,}M_{1}(X,z)=\textup{pd\,}S_{Y}=\textup{pd\,}I_{Y}+1=\textup{pd\,}M_{0}(X,z)+1$.
As a consequence, the cancellation in 3.3 can not happen between

$\beta_{p,q}(M_{1}(X,z))\quad\text{and}\quad\beta_{p-1,q+1}(M_{0}(X,z)),$

and thus
$\textup{pd\,}S_{X}=\textup{pd\,}I_{X}+1=\textup{pd\,}M_{1}(X,z)+1=\textup{pd\,}S_{Y}+1$.
∎

######  Remark 3.9.

Notice that if $Y\subset\mathbb{P}^{r-1}$ is a hypersurface, then the
homogeneous ideal $T\subset R$ above becomes an ideal with three generators,
namely a three-generated ideal. Such ideals have a wide range of complexity as
shown in [3], hence so do exceptional-fundamental schemes. This is a crucial
fact for the validity of our counterexamples to the regularity conjecture.

##  4\. Families of homogeneous ideals with high regularity

In this section, we focus on sequences of homogeneous ideals whose values of
the regularity grow fast.

######  Example 4.1.

In [5, Section 4.2], Caviglia considered homogeneous ideals

$I_{d}=(z_{2}^{d-1}y_{1}-y_{2}^{d-1}z_{1},y_{1}^{d},z_{1}^{d})\subset\Bbbk[y_{1},z_{1},y_{2},z_{2}].$

Finding its Gröbner basis with respect to the reverse lexicographic order, he
showed that

$\textup{reg\,}I_{d}=d^{2}-1$

for every $d\geq 2$. Furthermore, for the homogeneous ideals

$I_{k,d}=(z_{i+1}^{d-1}y_{i}-y_{i+1}^{d-1}z_{i}:i=1,\ldots,k-1)+(y_{1}^{d},z_{1}^{d})\subset\Bbbk[y_{1},z_{1},\ldots,y_{k},z_{k}],$

he expected with computational evidence that there exist some univariate
(rational) polynomials $p_{k}$ such that each $p_{k}$ has order $k$ and yields

$\textup{reg\,}I_{k,d}=p_{k}(d).$

Notice that in [2, Remark 2.7], Borna and Mohajer made the same claim for the
homogeneous ideals

$(z_{i+1}^{d-1}y_{i}-y_{i+1}^{d-1}z_{i},y_{i}^{d},z_{i}^{d}:i=1,\ldots,k-1)\subset\Bbbk[y_{1},z_{1},\ldots,y_{k},z_{k}].$

######  Example 4.2.

Beder, McCullough, Núñez-Betancourt, Seceleanu, Snapp and Stone constructed
interesting three-generated homogeneous ideals denoted by
$I_{2,(2,\ldots,2,1,d)}$, and [1, Example 4.2] raised the speculation that for
the $k$-tuple $(2,\ldots,2,1,d)$, the estimate

$\textup{reg\,}I_{2,(2,\ldots,2,1,d)}=\Omega(d^{k})$

holds. With minor modifications, they are formed as follows:

Let $k\geq 2$ and $d\geq 0$ be integers. We first set

$F_{k-1}=y_{k-1}^{d+2},\quad
G_{k-1}=z_{k}^{d+1}y_{k-1}-y_{k}^{d+1}z_{k-1}\quad\text{and}\quad
H_{k-1}=z_{k-1}^{d+2},$

and then inductively put

$F_{i}=y_{i}^{d+2k-2i},\quad
G_{i}=F_{i+1}y_{i}^{2}+G_{i+1}y_{i}z_{i}+H_{i+1}z_{i}^{2}\quad\text{and}\quad
H_{i}=z_{i}^{d+2k-2i}$

for each $i=k-2,k-3,\ldots,1$. Now the homogeneous ideal
$I_{2,(2,\ldots,2,1,d)}$ is defined by

$I_{2,(2,\ldots,2,1,d)}=(F_{1},G_{1},H_{1})\subset\Bbbk[y_{1},z_{1},\ldots,y_{k},z_{k}].$

The following partially verifies Borna-Mohajer’s assertion in 4.1 and is the
starting point for our counterexample construction. Its proof is inspired by
that of [6, Proposition 7.1].

######  Theorem 4.3.

Fix an integer $k\geq 2$, and let
$a_{1},b_{1},c_{1},\ldots,a_{k-1},b_{k-1},c_{k-1}$ and $a_{k}$ be positive
integers satisfying

$a_{1}<b_{1},c_{1}\quad\text{and}\quad 2\leq a_{i}\leq
b_{i},c_{i}\quad\text{for all }i=2,\ldots,k-1.$

Then the homogeneous ideal

$I=(z_{i+1}^{a_{i+1}}y_{i}-y_{i+1}^{a_{i+1}}z_{i},y_{i}^{b_{i}},z_{i}^{c_{i}}:i=1,\ldots,k-1)\subset
Q=\Bbbk[y_{1},z_{1},\ldots,y_{k},z_{k}]$

has invariants

$\textup{reg\,}I\geq\prod_{i=1}^{k}a_{i}+b_{1}+c_{1}-2a_{1}+\sum_{i=1}^{k}(a_{i}-1)\quad\text{and}\quad\textup{pd\,}Q/I=2k.$

###### Proof.

Put $A=\Bbbk[y_{k},z_{k}]\subset Q$, and consider the $Q$-module $Q/I$. Note
that $Q/I$ is finitely generated as an $A$-module. This means that two module
structures of $Q/I$ give the same local cohomology groups. Let $M$ be the
$A$-direct summand of $Q/I$ generated by monomials $m\in Q$ with

$\deg^{\prime}m=(b_{1}+c_{1}-a_{1}-1,a_{2}-1,\ldots,a_{k-1}-1,0),$

where $\deg^{\prime}$ is a new grading such that $\deg^{\prime}y_{i}$ and
$\deg^{\prime}z_{i}$ are the $i$-th standard basis vector. Due to the
vanishing of local cohomology groups (see [8, Corollary 4.5 and Proposition
A1.16]), we have

$\textup{reg\,}I=\textup{reg\,}Q/I+1\geq\textup{reg\,}M+1\quad\text{and}\quad\textup{pd\,}Q/I\geq\textup{pd\,}M+2k-2.$

Thanks to these inequalities, it would be enough to compute $\textup{reg\,}M$
and $\textup{pd\,}M$.

Let us find the minimal free resolution of $M$. Any integer $0\leq
d<a_{1}\cdots a_{k-1}$ is uniquely written as

$d=d_{1}a_{2}\cdots a_{k-1}+d_{2}a_{3}\cdots a_{k-1}+\cdots+d_{k-1}$

with integers $0\leq d_{i}<a_{i}$ for $1\leq i\leq k-1$, and then we set

$m_{d}=y_{1}^{b_{1}-a_{1}}z_{1}^{c_{1}-a_{1}}\prod_{j=1}^{k-1}y_{j}^{d_{j}}z_{j}^{a_{j}-d_{j}-1}.$

Apparently, the $m_{d}$ generate the $A$-module $M$. To obtain the $A$-linear
relations on them, suppose that

$f=\sum_{d}f_{d}m_{d}\in I$

for some $f_{d}\in A$. But comparing monomials, one sees that

$f\in(z_{i+1}^{a_{i+1}}y_{i}-y_{i+1}^{a_{i+1}}z_{i}:i=1,\ldots,k-1)+(y_{1}^{b_{1}},z_{1}^{c_{1}}).$

Plug

$y_{i}=y_{k-1}^{a_{i+1}\cdots a_{k-1}}\quad\text{and}\quad
z_{i}=z_{k-1}^{a_{i+1}\cdots a_{k-1}}$

for $1\leq i\leq k-1$ into $f$ so that we reach to

$\sum_{d=0}^{\alpha-1}f_{d}y_{k-1}^{d+\beta-\alpha}z_{k-1}^{\gamma-d-1}\in(z_{k}^{a_{k}}y_{k-1}-y_{k}^{a_{k}}z_{k-1},y_{k-1}^{\beta},z_{k-1}^{\gamma})\subset
A[y_{k-1},z_{k-1}],$

where $\alpha=a_{1}a_{2}\cdots a_{k-1}$, $\beta=b_{1}a_{2}\cdots a_{k-1}$ and
$\gamma=c_{1}a_{2}\cdots a_{k-1}$. Or equivalently, the vector
$(f_{0},f_{1},f_{2},f_{3},\ldots)$ is spanned by

$(-z_{k}^{a_{k}},0,0,0,\ldots),(y_{k}^{a_{k}},-z_{k}^{a_{k}},0,0,\ldots),(0,y_{k}^{a_{k}},-z_{k}^{a_{k}},0,\ldots),\ldots$

over $A$. In fact, from the nature of generators
$z_{i+1}^{a_{i+1}}y_{i}-y_{i+1}^{a_{i+1}}z_{i}$, it follows that

$y_{k}^{a_{k}}m_{d}=z_{k}^{a_{k}}m_{d+1}\quad\text{in}\quad Q/I$

for every $0<d<\alpha-1$ together with $0=z_{k}^{a_{k}}m_{0}$ and
$y_{k}^{a_{k}}m_{\alpha-1}=0$ modulo $I$. By the computation we have just
done, the minimal free resolution of $M$ has differentials

$\begin{pmatrix}-z_{k}^{a_{k}}&y_{k}^{a_{k}}&&&\\\
&-z_{k}^{a_{k}}&y_{k}^{a_{k}}&&\\\ &&\ddots&\ddots&\\\
&&&-z_{k}^{a_{k}}&y_{k}^{a_{k}}\end{pmatrix}\quad\text{and}\quad\begin{pmatrix}(y_{k}^{a_{k}})^{\alpha}\\\
(y_{k}^{a_{k}})^{\alpha-1}z_{k}^{a_{k}}\\\
(y_{k}^{a_{k}})^{\alpha-2}(z_{k}^{a_{k}})^{2}\\\ \vdots\\\
(z_{k}^{a_{k}})^{\alpha}\end{pmatrix},$

and the Betti table of $M$ looks like

$\begin{array}[]{c|ccc}j\backslash i&0&1&2\\\ \hline\cr j_{0}&\alpha&&\\\
j_{1}&&\alpha+1&\\\ j_{2}&&&1\end{array}$

in the setting of

  1. (1)

$j_{0}=b_{1}+c_{1}-2a_{1}+\sum_{i=1}^{k-1}(a_{i}-1)$;

  2. (2)

$j_{1}=b_{1}+c_{1}-2a_{1}+\sum_{i=1}^{k}(a_{i}-1)$; and

  3. (3)

$j_{2}=b_{1}+c_{1}-2a_{1}+\sum_{i=1}^{k}(a_{i}-1)+\prod_{i=1}^{k}a_{i}-1$.

We are done. ∎

##  5\. Proof of the main theorem

Generalizing a technique of Caviglia, Chardin, McCullough, Peeva and Varbaro,
we are ready to show the main theorem.

######  Proposition 5.1 (cf. [6, Proposition 7.1]).

Let $Q\subset R$ be a polynomial ring obtained by eliminating two variables in
$R$, say $y_{0}$ and $z_{0}$, and take a homogeneous ideal
$U=(g_{0},\ldots,g_{s})\subset Q$ and positive integers $\Delta$ and $\delta$.
Suppose that

  1. (1)

$g_{s}$ is nonzero;

  2. (2)

$\deg g_{i}=mi+\deg g_{0}$ for a common integer $m\geq 0$; and

  3. (3)

$\Delta>s$ and $\delta>(m+1)s$.

We put

$G=g_{s}y_{0}^{s}+g_{s-1}y_{0}^{s-1}z_{0}^{m+1}+\cdots+g_{0}z_{0}^{(m+1)s}\quad\text{and}\quad
T=(G,y_{0}^{\Delta},z_{0}^{\delta}).$

Then we have

$\deg R/T\leq\delta
s,\quad\textup{reg\,}T\geq\textup{reg\,}U+\Delta+\delta-2\quad\text{and}\quad\textup{pd\,}R/T\geq\textup{pd\,}Q/U+2.$

Moreover, $(m+1)(\Delta-s+1)\geq\delta$ implies $\deg R/T=\delta s$, and if
$g_{s}$ is irreducible, then so is $G$ provided that the $g_{i}$ are linearly
independent in the case $m=0$.

###### Proof.

Mimicking the proof of [6, Proposition 7.1] with the monomial
$y^{\Delta-1}z^{\delta-1}$, we have the desired lower bounds of
$\textup{reg\,}T$ and $\textup{pd\,}R/T$. Also, since the radical of $T$ is
$(y_{0},z_{0})$, we obtain

$\deg R/T=\dim_{K(Q)}R/T\otimes_{Q}K(Q).$

Let $V$ be the $K(Q)$-vector space $R/T\otimes_{Q}K(Q)$. By division by $G$,
the vector space $V$ is obviously generated by the monomials

(3)
$\begin{array}[]{ccc}y_{0}^{0}z_{0}^{\delta-1},&\ldots,&y_{0}^{s-1}z_{0}^{\delta-1},\\\
\vdots&&\vdots\\\
y_{0}^{0}z_{0}^{0},&\ldots,&y_{0}^{s-1}z_{0}^{0}.\end{array}$

Hence, the upper bound $\deg R/T\leq\delta s$ holds.

Next, assume that $(m+1)(\Delta-s+1)\geq\delta$. Introduce a new grading
$\deg^{\prime}$ having values $\deg^{\prime}y_{0}=m+1$, $\deg^{\prime}z_{0}=1$
and $\deg^{\prime}=0$ on $Q$. According to the values of this grading, we get
the $K(Q)$-vector space decomposition

$V=\bigoplus_{j\in\mathbb{Z}}V_{j}.$

We divide into two cases $j<\Delta(m+1)$ and $j\geq\Delta(m+1)$. If
$j<\Delta(m+1)$, and if we express $j$ as $j=a(m+1)+b$ with integers $0\leq
b<m+1$ and $0\leq a<\Delta$, then the summand $V_{j}$ is generated by the set
$\\{y_{0}^{a}z_{0}^{b},y_{0}^{a-1}z_{0}^{b+m+1},\ldots\\}$ and becomes the
cokernel of a matrix of the form

$\begin{pmatrix}g_{s}&&\\\ g_{s-1}&\ddots&\\\ \vdots&\ddots&g_{s}\\\
&&g_{s-1}\\\ &&\vdots\end{pmatrix}.$

However, in the range $j\geq\Delta(m+1)$, some matrix of the form

$\begin{pmatrix}\cdots&g_{s-1}&g_{s}&&&\\\ &\cdots&g_{s-1}&g_{s}&&\\\
&&&\ddots&\ddots&\\\ &&&&g_{s-1}&g_{s}\end{pmatrix}$

works for a presentation of $V_{j}$; it is due to the assumption
$(m+1)(\Delta-s+1)\geq\delta$. So one may observe that the monomials (3) form
a basis of $V$.

Finally, let us see the irreducibility of $G$. Refer to the fact that $G$ is
homogeneous with respect to both the standard grading and another grading
$\deg^{\prime}$. Suppose that

$G=(a_{t}y_{0}^{t}+\cdots+a_{0}z_{0}^{(m+1)t})(b_{u}y_{0}^{u}+\cdots+b_{0}z_{0}^{(m+1)u})$

for integers $t,u\geq 0$ with $t+u=s$ and homogeneous forms $a_{i},b_{j}\in Q$
with $\deg a_{i}=mi+\deg a_{0}$ and $\deg b_{j}=mj+\deg b_{0}$. We find that
$a_{t}b_{u}=g_{s}$, and so we may assume that $a_{t}$ is a constant. Since
$\deg a_{i}=mi+\deg a_{0}$, and since the $g_{i}$ are linearly independent
when $m=0$, we have $t=0$, and thus the first factor above is a constant,
which means that $G$ is irreducible. ∎

######  Proof of 1.2..

Fix an integer $k\geq 2$, set $d$ to be any sufficiently large integer, and
let $R=\Bbbk[y_{0},z_{0},\ldots,y_{k},z_{k}]$ stand for the homogeneous
coordinate ring of $\mathbb{P}^{2k+1}$. Note that 4.3 produces a homogeneous
ideal

$U_{d}=(g_{0},\ldots,g_{3k-4})\subset Q=\Bbbk[y_{1},z_{1},\ldots,y_{k},z_{k}]$

such that

  1. (1)

$g_{3k-4}$ is irreducible;

  2. (2)

$\deg g_{i}=i+d$; and

  3. (3)

we have $\textup{reg\,}U_{d}=\Omega(d^{k})$ and $\textup{pd\,}Q/U_{d}=2k$.

By 5.1, we get an irreducible $(d+6k-8)$-form $G_{d}$ with

$a=y_{0}^{6k-6}\quad\text{and}\quad f=z_{0}^{6k-7}$

so that the three-generated homogeneous ideal $T_{d}=(G_{d},a,f)\subset R$
gives

$\deg
R/T_{d}=(6k-7)(3k-4),\quad\textup{reg\,}T_{d}=\Omega(d^{k})\quad\text{and}\quad\textup{pd\,}R/T=2k+2.$

Let $Y_{d}\subset\mathbb{P}^{2k+1}$ be the nondegenerate projective variety
given by $G_{d}$, write $\lambda=a/f\in K(S_{Y_{d}})$, and take the
unprojection $(X_{d},z)=\textup{Unpr}(Y_{d},\lambda)$ in $\mathbb{P}^{2k+2}$.
Notice that part of 5.1 induces

$\deg W_{i}\leq\deg R/(G_{d},a^{i},f^{i})\leq i\deg R/T_{d}$

for the subscheme $W_{i}\subset Y_{d}\subset\mathbb{P}^{2k+1}$ defined by
$(a,f)^{i}\subset S_{Y}$. By 2.6, we have $M_{i}(X_{d},z)=0$ for any $i\geq 2$
with $z\in X_{d}$. Thus 3.8 tells us that $\deg X_{d}=\Theta(d)$,
$\textup{reg\,}X_{d}=\Omega(d^{k})$ and $\textup{pd\,}S_{X_{d}}=2k+1$, where
for positive number sequences $(a_{d})$ and $(b_{d})$,

$a_{d}=\Theta(b_{d})\quad\text{means}\quad
0<\liminf_{d\to\infty}\frac{a_{d}}{b_{d}}\leq\limsup_{d\to\infty}\frac{a_{d}}{b_{d}}<\infty.$

Here the $X_{d}\subset\mathbb{P}^{2k+2}$ have codimension $2$. For higher
codimension cases $e>2$, keep in mind that the homogeneous coordinate ring has
depth $2$, pick a fake linear form $\lambda=a/f$ such that $f$ and $a$ form a
regular sequence on the homogeneous coordinate ring, and then apply 3.8 again.
We have therefore constructed a sequence

$(X_{d}\subset\mathbb{P}^{2k+e}:d\gg 0)$

of nondegenerate projective $2k$-folds satisfying

$\deg
X_{d}=\Theta(d),\quad\textup{reg\,}X_{d}=\Omega(d^{k})\quad\text{and}\quad\textup{pd\,}S_{X_{d}}=2k+e-1.$

For $(2k-1)$-fold counterexamples, consider a general hyperplane section of
each $2k$-fold $X_{d}$; it is valid due to $0ptS_{X_{d}}=2$. We are done. ∎

######  Remark 5.2.

It can be said that our counterexamples work as they have very bad
singularities. Indeed, for instance, let us look at the codimension two case
$(X_{d},z)$ in the proof above. Then the point $z$ is contained in $X_{d}$,
and the projectivized tangent cone
$\mathbb{P}C_{z}X_{d}\subset\mathbb{P}^{2k+1}$ is nothing but the exceptional-
fundamental scheme of $(X_{d},z)$. Hence the point $z\in X_{d}$ is a
singularity whose projectivized tangent cone has extremely large regularity
compared to its degree, that is, the multiplicity of $X_{d}$ at $z$.

Finally, we present in detail a counterexample to the regularity conjecture
obtained by applying our method in consideration of 4.2. Many of the
computations below are done by the computer algebra system Macaulay2 [11] over
the field $\mathbb{Q}$ of rational numbers.

######  Example 5.3.

For simplicity, suppose that $\Bbbk$ has characteristic zero. Let
$R=\Bbbk[y_{0},z_{0},y_{1},z_{1},y_{2},z_{2}]$ be the homogeneous coordinate
ring of $\mathbb{P}^{5}$, and take $Y\subset\mathbb{P}^{5}$ to be the
hypersurface given by the irreducible homogeneous polynomial

$y_{1}^{4}y_{0}^{2}+(z_{2}^{3}y_{1}-y_{2}^{3}z_{1})y_{0}z_{0}+z_{1}^{4}z_{0}^{2}$

that appears in 4.2. Notice that the three-generated homogeneous ideal

$T=I_{Y}+(y_{0}^{4},z_{0}^{3})\subset R$

satisfies

$\deg R/T=6,\quad\textup{reg\,}T=20\quad\text{and}\quad\textup{pd\,}R/T=6.$

Now with the fake linear form

$\lambda=y_{0}^{4}/z_{0}^{3}\in K(S_{Y})$

on $Y$, consider the nondegenerate projective fourfold

$(X,z)=\textup{Unpr}(Y,\lambda)\quad\text{in}\quad\mathbb{P}^{6}.$

Then having

$\deg X=18,\quad\textup{reg\,}X=18\quad\text{and}\quad\textup{pd\,}S_{X}=5,$

we have established a fourfold counterexample to the regularity conjecture.
The Betti table $\mathbb{B}(I_{X})$ is computed to be the following:

$\begin{array}[]{c|ccccccccccccccc}i\backslash
j&4&5&6&7&8&9&10&11&12&13&14&15&16&17&18\\\ \hline\cr
0&1&&1&2&&&3&&1&1&1&&1&&1\\\ 1&&&&3&&&6&&4&2&4&&4&&4\\\
2&&&&&&&3&&5&1&6&&6&&6\\\ 3&&&&&&&&&2&&4&&4&&4\\\
4&&&&&&&&&&&1&&1&&1\end{array}$

And thanks to $0ptS_{X}=2$, a general hyperplane section

$X\cap\mathbb{P}^{5}\subset\mathbb{P}^{5}$

is a threefold counterexample with the same invariants as above. Compare it
with the projective threefold in [18, Example 4.7].

Let us describe singularities of the fourfold $X$. The singular locus of $X$
is

$\textup{Sing\,}X=C\cup\Lambda_{1}\cup\cdots\cup\Lambda_{4}\cup\mathbb{P}^{3}\subset\mathbb{P}^{6},$

where $C$ is a plane quartic curve, the $\Lambda_{i}$ are planes in a certain
order, and $\mathbb{P}^{3}$ is a $3$-plane in $\mathbb{P}^{6}$. Here the
location of $z\in\mathbb{P}^{6}$ is obtained as

$z\in(C\cap\Lambda_{1}\cap\Lambda_{2})\setminus(\Lambda_{3}\cup\Lambda_{4}\cup\mathbb{P}^{3})\subset\textup{Sing\,}X.$

Note that $\mathbb{P}C_{z}X\subset\mathbb{P}^{5}$ is of degree $12$, its
homogeneous ideal has regularity $17$, and its support is a $3$-plane.

Acknowledgements. We would like to thank Prof. Sijong Kwak for his advice. For
this study, experiments with Macaulay2 [11] were very helpful. This work was
supported by the National Research Foundation of Korea (NRF) grant funded by
the Korea government (No. 2019R1A2C3010487).

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# Framework of Fracture Network Modeling using Conditioned Data with
Sequential Gaussian Simulation

Yerlan Amanbek1, Corresponding author<EMAIL_ADDRESS>Timur
Merembayev1,2, Corresponding author<EMAIL_ADDRESS>Sanjay
Srinivasan  Department of Energy and Mineral Engineering, Pennsylvania State
University, 110 Hosler Building, University Park, State College, PA
16802-5000, USA

###### Abstract

The fracture characterization using a geostatistical tool with conditioning
data is a computationally efficient tool for subsurface flow and transport
applications.The main objective of the paper is to propose a framework of
geostatistical method to model the fracture network. In the method, we have
chosen neighborhood area to apply the Gaussian Sequential Simulation in order
to generate the fracture network in the unknown region. The angle was
propagated from the seed where direction is guided by the neighborhood data in
simulation regime. Initial seeds can be distributed by Poisson procedure. The
method is applied for geological faults from the Central Kazakhstan and for
field data from Scotland, UK. The simulation results are compatible with the
original fracture network in the flow and transport modeling setting. From the
research that has been carried out, it is possible to conclude that the
numerical simulation of fracture network is a valuable tool in the subsurface
flow and transport applications.

_K_ eywords fracture network model $\cdot$ Sequential Gaussian Simulation
$\cdot$ geostatistical method

##  1 Introduction

Fractures characterization plays vital role in flow and transport problems in
the subsurface applications such as oil and gas production, groundwater
remediation, CO2 sequestrations and etc. Due to limited information about the
subsurface property, the prediction of the fracture network in the porous
media is still a challenge.

A variety of approaches has been studied to predict behavior of fracture at
particular regions. For several years great effort has been devoted to
numerical and experimental studies of fracture propagation described using
complicated geomechanical equations using various methods [1, 2, 3, 4, 5].
Such geomechanical models require more computational resources to model a
fracture network with provided subsurface parameters. However, the key reasons
for fracture propagation can be detected from such analysis. These reasons are
necessary to adopt in the design of the fracture behavior. Input parameters
have a large uncertainty due to spatial variations of subsurface properties
such as stress field, existing fracture dimensions, fracture toughness,
physical properties and etc.

Permeability and porosity profiles are typically interpreted from the fracture
geometric network, which can be generated stochastically. This structure might
not be exactly resemble a real fracture structure, however, it provides a
reasonable configuration for the flow and transport models. On the basis of
the available data, for example data collected near wellbore and other
techniques, a geostatistical technique can be trained based on known regions
to model the fracture network in unknown regions. In the literature,
stochastic approaches have attracted much attention from researchers in
reduction of subsurface uncertainty.

In [6] the dfnWorks workflow was introduced to simulate single or multi-phase
flow and transport problem coupled with geostatistically generated discrete
fracture network (DFN). A combination of different types of tools in single
platform is necessary to validate modeling uncertainty.

In [7] the geostatistical modeling of fracture system was developed using
pattern statistics from pattern bar chart. The multiple point statistics(MPS)
was used with pixel based data in the growth-based fracture network method. As
reported in [8], the model was trained from satellite images which were
obtained by detecting faults in the surface. The training and testing were
performed for chosen surface samples.

Authors [9] have proposed the fracture network system simulation based on MPS
[8, 10] using the sophisticated relations among cracks and faults. In [10] the
method was developed based on MPS honoring surrounding data and multiple point
histogram. The propagation orientation is utilized in the MPS simulation. This
method was applied for real-world datasets such as Teapot Dome, Wyoming.
Another application of discrete fracture network simulation work using a
commercial software MoFrac were presented in [11] for the Canadian Shield
dataset.

An interesting approach to this issue has been proposed by [12]. This method
is adaptable for additional user defined constraints which honors other types
of subsurface data. Algorithms including Sequential Gaussian Simulation (SGS)
is used to model with simple kriging approach of the fracture orientation. In
the model, the process is similar to a propagation of fracture where in each
iteration the orientation and step length is computed honoring provided
subsurface information. However, there is a necessity on systematic workflow
in the fracture pattern prediction.

In this paper, we present the fracture characterization model using
geostatistical analysis with conditioned data based on the approach in [12].
The workflow of algorithm is provided for the natural fracture network
simulation in the undiscovered zone. Numerical simulations were conducted for
different examples such as realistic faults from the Central Kazakhstan and
the natural fracture dataset from South Glasgow, Scotland, UK. To verify the
realization, we have compared major variables such as the orientation
histogram and the concentration profile at production well, where the adaptive
numerical homogenization method was applied to simulate the single-phase and
incompressible flow and transport model in the fractured reservoir due to
multiscale feature in this setting, see [13, 14] for more details.

The remaining part of the paper proceeds as follows. Section 2 describes the
numerical method. The process of modeling is shown in the flow-chart.
Numerical results are presented in Section 3. In Section 4, the results were
discussed and the conclusion is reported in Section 5.

##  2 Numerical Methods

We first illustrate the fracture network simulation in 2D with neighbor
selection procedure, and then the Sequential Gaussian Simulation (SGS) process
for simulating fracture direction. In our case, there is data from certain
regions is provided, however, the rest of the region is unknown due to
sparsity of data. By using the information of fracture network we model
fractures in unknown regions. In the simulation of fracture networks, key
parameters are fracture direction (angles) and length of propagation. The
orientation or the angle of fracture was computed by SGS method using the data
from known regions. In each iteration of simulation, fracture grows in all
domain honoring the information of neighbourhood region.

It is common to observe the family of fractures, not only single fracture.
Such family of fractures has a common features including orientations. The
changes of directions in the fracture family are usually smooth and transition
of certain patterns are soft [15]. Therefore, the average value of fracture
orientation, the intensity distribution, etc are necessary in the fracture
simulation.

###  2.1 Fracture Network Simulation with Conditioned by Neighboring Data

From a geomechanical standpoint, a fracture direction is influenced the
systematic organization of neighboring subsurface property such as
geomechanical features, fracture intersections and others [15]. The stress
field nearby the fracture tip is a key factor in the simulation of fracture
propagation. This is measured by a stress intensity factor ($K_{I}$), which
depends on loading (stress) and the fracture dimensions. The fracture keeps
its propagation as long as $K_{I}>K_{IC}$, where $K_{IC}$ is fracture
toughness. In addition, previous studies indicate that the energy minimization
is important in the systematic fracture path simulation using the phase-field
model [1, 3]. Therefore, the effect of neighbor fracture should be taken into
account fracture direction simulations.

We describe the key ingredients of algorithm in the fracture network
simulations including initiation, propagation and termination [15]. We will
address all steps below.

To illustrate the proposed method of fracture characterization, we show the
major steps in 2D for surface traces.

In our model, a fracture is a collection of fracture segments with
corresponding length and orientation. The initial fracture segment will be
growing honoring the surrounding information. We first define the initial
fracture segments (coordinates and azimuth angles) in the entire domain. In
the unknown region, we use the Poisson procedure to seed initial fracture
segments, however, these locations can be indicated by user. In known area,
fractures will be propagated from the middle of the original fracture length
and repeat their original path in both directions from the middle. Below you
can see the key steps in the method for defining the neighborhood region, as
shown in Figure 1.

  * Step 1.

The initial points will be seeded in the domain. For each point, which is the
middle of fracture segment, there are corresponding parameters such as azimuth
angles, coordinates of points and the neighboring segments. The filled point
is target point from where a new fracture segment will grow.

  * Step 2.

We define the neighborhood zone as sector of influencing points that was
created from the following parameters: the initial azimuth angle, a standard
deviation of angle and radius. From the initial point along the line, which
follows provided azimuth, we draw sector with radius r and angle of sector
equals to two standard deviations. The line generated from azimuth divides the
sector angle equally. In here, we compute azimuth of angle from fracture angle
distribution of known region. A radius of the sector is defined from the
variogram model of fracture segments in the known region. It is within this
search sector that conditioning data for fracture angles will be searched.

  * Step 3.

There will be included a virtual point additionally to the selected points in
the sector. To consider the influence of the nearest fracture segment, we draw
virtual fracture segment in the center of the segment, which connects points
between the current point and the nearest fracture segment. The virtual
fracture segment keeps the angle. The selected points in the sector including
the additional data will be used in the SGS. This approach was initially
proposed in [12].

  * Step 4.

We compute important features of a new fracture segment such as the
orientation and the fracture segment length. We use the points in the
neighbourhood sector as input data in the SGS analysis to compute a newly
simulated angle. The angle is chosen from the computed distribution in random
fashion. The length of simulated segment equals the mean of fracture segment
distribution of known region. We note that the length can be specified by
user. Combining the information of angle and length of segments we draw a new
fracture segment which is colored in blue.

By propagating the target or initial points and repeating step 2 - step 4 we
can achieve a configuration shown in Figure 1 (Final result). Standard
deviation and radius are constant for further iterations. After several
propagation iterations a newly simulated fracture will be terminated until
crossing any fractures or length of fracture reaches the constraints.

Figure 1: Schema of a fracture simulation.

###  2.2 Sequential Gaussian Simulation

We briefly go through SGS algorithm in perspective of fracture segment
orientation. In step 4 from the previous section, we need to identify a new
angle of the next fracture segment by considering angles of neighboring
fractures.

The method of SGS assumes the presence of a normal distribution of the
simulated random variable. For any point in the region, the local distribution
function will be distributed according to the normal law and will be
determined by two parameters such as mean and standard deviation. Before to
make simulation, it is assumed that the stationary is exist for random
function $Z(x)$ and a random function $Y(x)$ is exist such that

$Y(x)=\phi[Z(x)]$ (1)

where $Y(x)$ is $N(0,1)$ and $\phi$ \- the normal score of the transformation.

The ordinary kriging is the main method that helps to propagate the fracture
segment with the provided spatial distribution. The kriging is the linear
unbiased predictor and can be explicit as the linearly - weighted average
function of observations in the unknown area of fractures $x$, location of
fractures . $Y(x)$ is a set of random variables (azimuthal fault angles). The
ordinary kriging estimate $Y_{sk}^{*}(x)$ at $x$ is calculated as

$Y_{sk}^{*}(x)=\sum_{i=1}^{n(x)}\lambda_{i}^{sk}(x)Y(x_{i})$ (2)

The ordinary kriging weights $\lambda_{i}^{sk}$ that defined by by solving the
system $n(x)$ of ordinary kriging equations:

$\sum_{j=1}^{n(x)}\lambda_{j}^{sk}C_{ij}=C_{i0},\forall i=1,...,n(x)$ (3)

where $n(x)$ is the total number of neighboring points $x_{i}$ used in
estimating the point $x$. The ordinary kriging score variation is given by

$\sigma_{sk}^{2}=C(0)-\sum_{i=1}^{n(x)}\lambda_{i}^{sk}C(x_{i}-x)$ (4)

As a result, we obtain the parameters of the local normal distribution
function $N(Y_{sk}^{*}(x),\sigma_{sk}^{2}(x))$ at the estimation point $x$.
This distribution is required in the identification of the fracture
propagation angle.

###  2.3 Estimation of fracture orientation

In step 4, we have used the SGS for the modeling of fracture orientation in
each iteration of the fracture network simulation. The SGS is the most
commonly used sequential simulation algorithm for modeling continuous
variables. A detailed introduction to this method can be found in [16, 17].
For visual representation of the flow-chart of the proposed method the reader
is referred to Figure 2. In addition, the pseudocode of the proposed method is
depicted in Figure 3. In the algorithm,simulation of azimuth angles in SGS
proceeds as follows:

  1. 1.

Validate whether the distribution of the provided areas dataset is normal
distribution or not. This dataset is a collection of the fracture segment
azimuth;

  2. 2.

If necessary, transform the distribution of the data into a normal
distribution;

  3. 3.

Seed the initial fracture segments randomly using the Poisson process in
unknown region;

  4. 4.

Identify the points in the neighbor zone as described in Step 2-3, Section
2.1;

  5. 5 & 6.

If number of neighbors are less or equal 1 then the direction of the new
fracture segment will be chosen by the Monte-Carlo method for the normal
distribution of the provided dataset, as described in Step 1, Section 2.1.
This produces a new azimuth angle that is combined with the length of fracture
segment in order to propagate a new fracture segment;

  6. 5 & 7.

If number of the neighbor points are more than 1 then to add point of halway
to the nearest neighbor for the influence of the nearest fracture segment;

  7. 8 & 9.

As subroutine of SGS procedure, the ordinary kriging approach is utilized to
obtain kriged estimate and the corresponding kriging variance;

  8. 10.

Termination of the fracture segment occurs when it crosses another one. We do
not consider another option for termination a propagation of fractures,
criteria of achievement limit length of full length of fracture, the criteria
can be integrated;

  9. 11.

Transform from the normal distribution to the original univariate
distribution.

Sequential Gaussian SimulationStart 1.Dataset normal distr? 2.Transform to
norm dataset 3.Seed unknown fracture 4.Find neighbors 5.Count neighbors >=1
7.Add point halfway to the nearest neighbor 6.Calculate new strike of fracture
$randomn(\mu,\sigma)$ 8.Calculate weights by closest fracture segment
9.Simulate strike trace by invers covariation matrix $randomn(\mu,\sigma)$
10.Intersect fracture traces 11.Back transform from norm to original
Finishnoyesnoyesnoyes Figure 2: Flow-chart of the fracture algorithm.

1

Input : mean, standard deviation $std$, fracture length $len$, iteration
$iter$, matrix azimuth $A$, count of seed fractures $m$

Output : $A(m,n)$

2 for  _$i\leftarrow 1$ to $m$_ do

3 for  _$j\leftarrow 1$ to $iter$_ do

4 $[x_{id_{ij}},y_{id_{ij}}]\leftarrow$ Calculate of a mid of fracture

5 $[x_{id_{k}},y_{id_{k}}]\leftarrow$ Find neighbours $A(X,Y)$

6 if  _$count[x_{id_{k}},y_{id_{k}}]\leq 1$_ then

7 $x_{id_{i,j+1}}=len*cos(randomn(mean,std))+x_{id_{ij}}$

8 $y_{id_{i,j+1}}=len*cos(randomn(mean,std))+y_{id_{ij}}$

9

10 if  _$A(x_{id_{ij}},y_{id_{ij}})\cap A(X,Y)$_ then

11 close loop j

12 end if

13

14 else

15 $\lambda,\sigma\leftarrow$ Calculate weights($id_{k},covar^{-1}$)

16
$A(x_{id_{i,j+1}},y_{id_{i,j+1}})=$Simulation($\lambda,\sigma,randomn(mean,std)$)

17 if  _$A(x_{id_{ij}},y_{id_{ij}})\cap A(X,Y)$_ then

18 close loop j

19 end if

20

21 end if

22

23 end for

24

25 end for

Algorithm 1 Sequential Gaussian Simulation Figure 3: Pseudo code of fracture
characterization .

##  3 Numerical results

In this section, we show several numerical examples such as comparison with
previous result from [7], application for faults from Zhezkazgan, Kazakhstan
and fractures of McDonald limestone dataset from Scotland, UK. To illustrate
applications of the proposed method, we show numerical simulations for
realistic datasets in two dimensions with verifications.

###  3.1 Example 1

In this example, we have applied our method to the scenario with initial
conditions presented in [12]. We have compared the results in here to depict
the similarity of proposed method. To verify the method, our result of
simulation is compared with the result of fractures from [12], as shown in
Figure 4.

The example was digitized by using GRABIT [18]. Fractures from the gray region
is given in the right side. In the original work, fracture of the dark region
was predicted. We have used the following parameters as input data in our
algorithm. The mean of orientations is 70 degrees and the standard deviation
is 12. The variogram model is a spherical. Nugget is 1, spherical is 2, range
is 50, r = 50, fracture length is 10.

In here, the azimuth angles were generated by Monte Carlo method in the SGS
procedure. As can be seen from Figure 4(b), simulated fractures are presented
in different colors such as green, blue, red and black. It is important to
note that the fracture path in the known region was reproduced and the part of
it was honored during the simulation regime.

(a) The result of simulation in [12].

(b) Our result of simulation.

Figure 4: The results of simulation by proposed algorithm.

Various numerical examples are performed with different initial seeds in the
hidden region. Poisson procedure is applied to determine coordinates of
initial points for simulated fractures. Figure 5 presents different
realizations where red squares are middles of initial seeds.

(a) Example 1.

(b) Example 2.

Figure 5: Simulation fractures from initial points generated by Poisson
process.

In Figure 5(a) and 5(b), blue fractures (initial azimuth: 45 and 25) are
similar to the original fracture. During the simulation, we have observed that
there is a tendency of having traces in similar directions for nearby
fractures. This phenomena was discussed in Section 2.1 with perspective of
geomechanics. We observe similar pattern for a black fracture (initial azimuth
120), which connected to the known fracture, see Figure 5(a), Figure 5(b)
gives an examples of similar direction pattern for a black and blue
fractures(initial azimuth 125 and 25). After several more iterations the
resulting path of black and blue fractures become close to each other in the
center of domain.

###  3.2 Example 2

The analysis of conditioned data is necessary for the fracture
characterization simulation. We first provide the results obtained from the
FracPaq toolbox in Matlab [19]. Second, we show the one of realizations of
simulated fractures for the region in Zhezkazgan, Kazakhstan.

####  3.2.1 Fracture Data analysis

Before applying SGS algorithm we need to make data analysis of fractures given
in known area using FracPaq [19]. The software allows to obtain the
quantitative fracture patterns, digitalization of fractures and their
distribution in two dimensions for length, segment scales, and rock types.
Fracture is defined as a collection of one or more segments which creates a
continuous line in the software. It is assumed that segments at any scale are
formed because of the interaction and fusion of small cracks, so analysis of
these segments separately can be useful.

(a) Traces colored by length.

(b) Segments colored by length.

Figure 6: Result of data analysis.

The FracPaQ provides an analysis of the traces and the segment of fractures by
length so it helps to indicate major and minor fractures, analysis of segment
traces provides visual confirmation that fractures have one mean of fracture
segments. The traces colored and the segment colored of fractures by length
are shown on Figure 6.

(a) Angles of fracture segments.

(b) Angles of fracture segments.

Figure 7: Fracture orientation.

(a) Frequency-size histogram for trace lengths.

(b) Frequency-size histogram for segment lengths.

Figure 8: Histograms of analyzed data. Red lines show the minimum and maximum
lengths.

The distribution of fracture angles for the known region is shown in Figure 7
and is presented in a rose plot and a histogram. The rose plots and the
histograms show the main fracture population (about 120 degrees) in the known
region, the second peak of the fracture population is an opposite direction
(120 degrees) a fracture segment. Based on the angles shown in the rose plot,
the mean of the azimuth angle of fracture segments is 120.

Base on the data there are two frequency distributions – size plots for trace
and segment lengths can help to estimate the mean length of segment. The
length will be used for simulation of fracture propagation. Figure 8 shows
histograms of trace and segments lengths of geology faults.

FracPaQ generated the variogram (or semi-variogram) of segment lengths.
Semivariogram had been calculated as a function of the separation of two
segments from the stored coordinate positions of all the segment mid-points
(centroids) for every pair of segments.

In Figure 9 the calculated variogram model showed the correlation range is
achieved on 15000 meters. This range value is used as a radius of the sector
for searching neighbors.

Figure 9: Variogram of segment lengths.

####  3.2.2 Result of simulation for structural measurements

We examined the SGS algorithm in 7011,9 $km^{2}$ area near of Zhezkazgan city,
Kazakhstan. The study area is located in the central part of Kazakhstan. Red
beds and volcanic suites are well captured. The rocks wereformed in age from
the Paleozoic, Carboniferous, Devonain and Ordovician. They have a good
visibility in defined fold structures. This geological data is obtained from
[20]. The main purpose of such maps is the investigatigation of the potential
mines in the area, see Figure 10. We take the digital data: geology and faults
for the area of interest to make the test of fracture characterization. More
detail about the description and color of the legend is explained in source
[20].

(a) The image has faults and geology.

(b) The image has only faults.

Figure 10: Simplified geological.

Based on data analysis from the tool we get next information that is necessary
for SGS algorithm:

  * •

Mean of segment length – 2289.27 meters, The value is used as constant length
of simulated fractures segment. As previously noted the length can be
specified by an user;

  * •

Exist bimodal normal distribution of angle faults, 120 and about 300 degrees.
300 degrees direction is the opposite direction (120 degrees) so we choose 120
degrees as the mean of the azimuth angle of distribution for the known domain;

  * •

Base on the variogram model of segment length, we define lag of between
segments - 15000 m. Variogram model is represented in spherical mode.

Propagations for each fracture started from the middle of fracture length and
propagate to both sides from a middle point. As shown in Figure 11(a), a trace
indicated in red was selected to be predicted by the proposed method. After
removal of that fracture from the conditing data set, the method used
fractures in blue as conditioned data and then a fracture marked in green is
generated at the location of the removed fracture. As can be seen from Figure
11(b), there is a good match between the original fracture (red) and the
simulated fracture(green).

(a) Geology faults with middle points.

(b) The simulated chosen fault.

Figure 11: Result of SGS simulation for one fracture.

To test the algorithm for realistic condition, we hide the fracture
information in the center area (cropped from the original domain). We have
used the midpoints of the original fractures as initial seeds for simulation
of green fractures. In Figure 12(a), red color is the boundary of crop domain
from the original fractures, the blue color is original fractures or fractures
from the known domain. In Figure 12(b) the original fractures are in the blue
color. In Figure 12(c) there are the simulated fractures by the proposed
algorithm in the green color. The result of the algorithm is consistent with
the major trends of the original fracture traces.

(a) Hidden zone of fractures

(b) Original fractures

(c) Simulated fractures

Figure 12: Result of Proposed algorithm, simulation for KZ geology fractures.

Geological fractures are divided into primary and secondary features, The
primary fractures are long extensional features, the secondary fractures are
of insignificant length, and their prevails over the primary. These fractures
have different structures and origins. If the midpoint of long fractures are
not inside the hidden square we keep such faults as conditioned data. This is
to ensure that we reproduce the length distribution accurately.

###  3.3 Example 3

The realistic fracture obtained from [19] is used in the fracture
characterization to illustrate the proposed model. Collected data is a
fractured bedding plane in the McDonald (or Hosie) Limestone in the Spireslack
opencast coal pit. Overall, in Midland Valley there are mining carboniferous
coal-bearing fluvial-deltaic rocks and have been started since the 19th
century, Figure 13. The count of initial segment fractures is user-defined for
an unknown area. In this case, we choose the 50 initial segment fractures. 50
initial fracture segments randomly seed by the Poisson process in the hidden
center area.

As the previous figure, we got 3 images: hidden domain (cropped region),
original fractures and simulated in Figure 13. If we compare the result of
fractures simulation and original fractures on the figures, we can see that
the trend of fractures direction is reproduced. To prove this statement we
will resort to a histogram of fracture angles distribution.

(a) Hidden zone of fractures

(b) Original fractures

(c) Simulated fractures

Figure 13: Result of proposed algorithm, simulation for McDonald Limestone
dataset from the Spireslack open cast coal pit, south of Glasgow in Scotland,
UK.

The verification was given by comparing the distribution of fracture angles
for original fracture networks (left figure) and simulated fracture networks
(right figure) within the hidden area, see Figure 14. We have observed
similarity between histograms for original and simulated fractures. In
particular, shape and means of distribution are similar that was illustrated
in Figure 14 (vertical red lines in histogram).

---

(a) For original fractures.

(b) For simulated fractures.

Figure 14: Histograms of verification.

Analysis of the plotted histograms yielded a multimodal distribution (4
peaks), it is due to the fact that we have two main directions of fractures
and two conjugate directions for propagation of fractures. Based on histogram
we get means of trace segment angles for original fracture networks: 20; 110;
185; 290. The corresponding angles of the simulated fracture networks: 18; 85;
190; 275. These histograms give us the opportunity to assert that the proposed
method shows a good result and confirms its ability to simulate fracture
networks.

Figure 15: Location of injection and production wells.

The additional verification of the fracture network model is given by
comparing the concentration profile at the production well in a tracer test
setting. We have a single-phase , incompressible flow and transport model
which is based on work presented in [13]. Here, the flow problem solved
initially and then the transport problem. For more details we refer to earlier
work [13, 14, 21, 22, 23, 24]. We locate the production well in the top right
corner of the domain and the injection well in the bottom left of the domain,
see Figure 15. Dimension of the reservoir is 6900 ft x 5700 ft. In the
adaptive regime, the local cell dimension is 10 x 10 grids for homogenization
of permeability, i.e. domain dimension is 69 x 57 grids in coarse scale.
Porosity was taken as 0.2 and diffusion is 0.0001 $ft^{2}/day$. No flow
boundary conditions in the domain. Permeability at the fracture was chosen 200
times more than non-fracture media. Time step is 5 days, ratio between coarse
and fine scale is 10 and dx=10 ft. The injection rate at the bottom left well
is chosen 100 stb/day.

As shown in Figure 16, the flow and transport model with proposed fracture
network algorithm was able to match concentration at production well for the
model with original fracture network. There is a good agreement in
concentration profile for both scenarios as well as on breakthrough time. We
limit in this work with provided realization of fracture network in Figure 14
to illustrate the verification.

Figure 16: Location of injection and production wells.

Our verification was performed for the provided realization. Future work will
involve more flow and transport simulations for various realizations in an
efficient way using the multicore machines in parallel mode.

##  4 Discussion

One can look at the history of fracture generation in porous media. In other
words, if model allows to go back in time and afterwards to take into account
main geological events from the past which was reason for generation of
younger fractures along with older ones. Thus, our method can be seen as
similar as the fracture propagation scenario where it starts from seed nodes
obtained from the Poisson Distribution or user-specified locations. In
addition, the algorithm can be conditioned by data from available sources such
as seismic, wellbore and others.

We assumed that the fracture orientation has a gaussian distribution in the
simulation regime. The understanding of behaviour of 2D fracture network is an
important base for studying the fracture network in 3D. The extension from two
dimensional to three dimensional can be done in a similar fashion by
considering azimuth, length and polar angles. A combination of layers can be
interpreted as characterization of the subsurface volume [12]. During the
generation of a sector zone, we have used the unique and precomputed standard
deviation and radius for each sector in the simulation process. However, these
parameters can be specified by user.

##  5 Conclusion

The purpose of this paper is to provide the framework of the characterization
of fracture network in different scales using the geostatistical method. We
have applied the proposed algorithm for natural fracture path from the Central
Kazakhstan, the area includes Zhezkazgan city, and for McDonald limestone from
Scotland, UK. To our best knowledge, this is the first study to deal with the
fracture network modeling of the Central Kazakhstan faults. In the growth-
based fracture network model, the Gaussian Sequential Simulation is used for
orientation of fracture honoring the neighborhood information. We currently
believe that the computation time of the fracture network characterization
using the proposed algorithm is less than the CPU time of the geomechanical
models that considers the coupled partial differential equations in the
subsurface. To verify the proposed fracture network method, the distribution
of the segment angles for the original fracture path is compared with the same
distribution for the simulated fracture path. The main features of both
distributions are in good agreement. In addition, we have compared the
concentration profiles at the production well for different fracture networks
in the single-phase and incompressible flow and transport models. There is a
good match between the concentration history used simulated fracture network
and the concentration history used original fracture network. In our future
research we intend to concentrate on machine learning algorithms for fracture
characterization using image training approaches for real-field dataset.

## Acknowledgement

Financial support from the Nazarbayev University Faculty Development
Competitive Research Grant, No 110119FD4502, is gratefully acknowledged.

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