# Multigraded regularity of complete intersections

Marc Chardin  Institut de mathématiques de Jussieu, CNRS & Sorbonne
Université, 4 place Jussieu, 75005 Paris , France<EMAIL_ADDRESS>and
Navid Nemati  Université Côte d’Azur, Inria, 2004 route des Lucioles, 06902
Sophia Antipolis, France<EMAIL_ADDRESS>
###### Abstract.

$V$ is a complete intersection scheme in a multiprojective space if it can be
defined by an ideal $I$ with as many generators as $\operatorname{codim}(V)$.
We investigate the multigraded regularity of complete intersections scheme in
$\mathbb{P}^{n}\times\mathbb{P}^{m}$. We explicitly compute many values of the
Hilbert functions of $0$-dimensional complete intersections. We show that
these values only depend upon $n,m$, and the bidegrees of the generators of
$I$. As a result, we provide a sharp upper bound for the multigraded
regularity of $0$-dimensional complete intersections.

###### Key words and phrases:

Multigraded regularity, Hilbert function, multiprojective space, complete
intersection

###### 2010 Mathematics Subject Classification:

13D02, 13D40, 14B15, 13D45

##  1\. Introduction

The theory of syzygies offers a microscope for looking at systems of
equations. Castelnuovo-Mumford regularity is an important invariant in
commutative algebra and algebraic geometry which is strongly related to
syzygies. D. Eisenbud and S. Goto in [2] showed that Castelnuovo-Mumford
regularity can be obtained from the minimal free resolution. The minimal free
resolution was first introduced by Hilbert to study Hilbert function. It is
one of the finest invariant that we can associate with a finitely generated
graded module M over a polynomial ring. Castelnuovo-Mumford regularity
measures the maximum degree of the syzygies and provides a quantitative
version of Serre vanishing theorem for the associated sheaf. In particular, it
bounds the largest degree of the minimal generators and the smallest twist for
which the sheaf is generated by its global sections. It has been used as a
measure for the complexity of computational problems in algebraic geometry and
commutative algebra. The two most frequent definitions of $\mathbb{Z}$-graded
Castelnuovo-Mumford regularity are the one in terms of graded Betti numbers
and the one using local cohomology.

An extension of Castelnuovo-Mumford regularity for the multigraded setting in
a special case was first introduced by Hoffman and Wang in [6]. Later by
Maclagan and Smith in [8], and Botbol and Chardin in [1] in a more general
setting. The main motivation for studying regularity over multigraded
polynomial rings was from toric geometry. Maclagan and Smith [8] developed a
multigraded theory of regularity of sheaves on a simplicial toric variety
$\mathcal{X}$ with an algebraic variant defined in terms of the vanishing of
graded pieces of $H^{i}_{B}(M)$, the $i$-th local cohomology module of $M$.
Here $B$ is the irrelevant ideal of the homogeneous coordinate ring of
$\mathcal{X}$.

The conceptual difficulty of computing multigraded regularity lies in the
simple fact that bounded subsets of $\mathbb{Z}^{r}$ need not have single
maximal or minimal elements. This makes it very hard to capture the vanishing
or non-vanishing of the multigraded pieces of local cohomology. One can define
a complete intersection in projective space as a subscheme defined by as many
forms as its codimension. In this case, it corresponds to complete
intersection homogeneous ideals; therefore many homological invariants are
determined by the degrees of the forms. One natural question that may arise is
to ask the same question for the complete intersection schemes in a product of
projective spaces.

Proposition 6.7 in [8] implies that the multigraded graded regularity a
$0$-dimensional schemes is the same as the stabilization region of the Hilbert
functions. Understanding the Hilbert functions and the minimal free
resolutions of the coordinate rings of points in multiprojective space is
included among the list of open problems in commutative algebra found in the
survey article of Peeva-Stillman[9].

Based on the above motivations, our main concern in this article is to study
multigraded regularity and bigraded Hilbert function complete intersection
points in $\mathbb{P}^{n}\times\mathbb{P}^{m}$. In Section 2, we start by
setting our notation and define multigraded regularity. We show that if the
ring is bigraded, then one can consider an alternative way to define the
multigraded regularity (see Theorem 2.5).

In section 3, we study the multigraded regularity of general complete
intersection schemes in $\mathbb{P}^{n}\times\mathbb{P}^{m}$. A scheme
$V\subset\mathbb{P}^{n}\times\mathbb{P}^{m}$ is complete intersection if it
can be defined by as many forms as its codimension. Our motivation is to show
which homological invariants are determined by the degrees of these forms.

In Section 4, we focus on the case of complete intersection points in
$\mathbb{P}^{n}\times\mathbb{P}^{m}$. The main result of this section is
Theorem 4.3. For a complete intersection scheme of point
$V\subseteq\mathbb{P}^{n}\times\mathbb{P}^{m}$ defined by
$f_{1},\dots,f_{n+m}$, many values of its bigraded Hilbert function are
independent from the choices of $f_{i}$’s and they only depend upon their
bidegrees. Moreover, in some regions, there is a duality among these values.

######  Example A.

(Example 4.4) Let $S=k[x_{0},x_{1},x_{2},y_{0},y_{1},y_{2}]$, and
$I=(f_{1},\dots,f_{4})$ is generated by $4$ bigraded forms of bidegree $(2,2)$
such that $\operatorname{proj}(S/I)=V$ is complete intersection scheme of
points . For any
$\mu\notin(2,6)+(-\operatorname{\mathbb{N}},\operatorname{\mathbb{N}})\cup(6,2)+(\operatorname{\mathbb{N}},-N)$,
the Hilbert function of $V$ at $\mu$ is independent from the choices of
$f_{i}$’s and can be computed explicitly via Theorem 4.3.

In Section 5, we study generic complete intersection points in
$\mathbb{P}^{n}\times\mathbb{P}^{m}$. The main Theorem of this section is the
following

######  Theorem A.

(Theorem 5.5) Let $S=k[x_{0},\dots,x_{n},y_{0},\dots,y_{m}]$ be a bigraded
polynomial ring over a field $k$ of characteristics zero. If $I$ is generated
by $n+m$ generic forms of bidegree $(d,e)$, then the scheme $V$ defined by $I$
is a set of reduced points and

$|\\{\mu\in\mathbb{N}^{2}|\operatorname{HF}_{V}(\mu)\neq\deg(V)\\}|<\infty,$

which means the natural projections are one-to-one.

##  2\. Multigraded regularity in bigraded setting

######  Notation 2.1.

From now on we use the following notation in the rest of the paper. Let
$S=k[x_{0},\dots,x_{n},y_{0},\dots,y_{m}]$ be a bigraded polynomial ring where
$\deg(x_{i})=(1,0)$ and $\deg(y_{i})=(0,1)$. Let $B_{1}=(x_{0},\dots,x_{n})$,
and $B_{2}=(y_{0},\dots,y_{m})$ be the irrelevant ideal of $\mathbb{P}^{n}$
and $\mathbb{P}^{m}$. Define $B:=B_{1}\cdot B_{2}$ the irrelevant ideals of
$\mathbb{P}^{n}\times\mathbb{P}^{m}$, and
$\operatorname{\mathfrak{m}}=B_{1}+B_{2}$ the maximal ideal of $S$ as a
standard graded ring.

We state the definition of multigraded regularity in the case of bigraded
rings; for further reading, we refer readers to [1, 8].

######  Definition 2.2.

Let $M$ be a graded $S$-module. The support of the module $M$ is

$\operatorname{Supp}(M):=\\{\gamma|M_{\gamma}\neq 0\\}.$

######  Definition 2.3.

(Multigraded regularity) Let $C\subseteq\operatorname{\mathfrak{m}}$ be a
finitely generated bigraded ideal of $S$. Define for a
$\gamma\in\operatorname{\mathbb{Z}}^{2}$, $M$ is  $(C,\gamma)$-regular if

$\gamma+\operatorname{\mathbb{N}}^{2}\cap\cup_{i}\operatorname{Supp}(H^{i}_{C}(M))+\mathcal{F}_{i-1}=\emptyset,$

where $\mathcal{F}_{i}:=\\{(i-1,0),(i-2,1),\dots,(0,i-1)\\}$ for $i>0$ and
$\mathcal{F}_{0}:=\\{0\\}$, $\mathcal{F}_{-1}=-\mathcal{F}_{1}$, and
$\mathcal{F}_{i}=0$ for $i<-1$. $C$-regularity of $M$ is

$\operatorname{reg}_{C}(M):=\\{\gamma\in\operatorname{\mathbb{Z}}^{r}\mid\text{M
is }(C,\gamma)\text{-regular}\\}.$

In particular, if $C=B$, $B$-regularity of $M$, $\operatorname{reg}_{B}(M)$,
is called the multigraded regularity of $M$.

Our approach for computing multigraded regularity relies on Mayer-Vietoris
sequence to relate the various cohomology modules.

######  Remark 2.4.

Let $S=k[x_{1},\dots,x_{m},y_{1},\dots,y_{n}]$ be a bigraded polynomial ring.
Define $B_{1},B_{2},B$ and $\operatorname{\mathfrak{m}}$ as in Notation 2.1.
Then we have the following complex between local cohomology modules:

$\cdots\rightarrow H^{i-1}_{B}(M)\rightarrow
H^{i}_{\operatorname{\mathfrak{m}}}(M)\rightarrow H^{i}_{B_{1}}(M)\oplus
H^{i}_{B_{2}}(M)\rightarrow H^{i}_{B}(M)\rightarrow\cdots$

The following Theorem provide an alternative way to define multigraded
regularity in the bigraded setting by knowing
$\operatorname{\mathfrak{m}},B_{1}$, and $B_{2}$-regularity.

######  Theorem 2.5.

Let $S=k[x_{1},\dots,x_{m},y_{1},\dots,y_{n}]$ be a bigraded polynomial ring
and $M$ is a graded $S$-module. Adopt Notations 2.1, then

$\operatorname{reg}_{B}(M)=\operatorname{reg}_{\operatorname{\mathfrak{m}}}(M)\cap\operatorname{reg}_{B_{1}}(M)\cap\operatorname{reg}_{B_{2}}(M).$

###### Proof.

By analyzing the spectral sequences correspond to Čech-Koszul double complex
$C^{\bullet}_{B}K_{\bullet}(\underline{\textbf{x}},\underline{\textbf{y}};M)$
we get

$\operatorname{Supp}(\operatorname{Tor}_{i}^{S}(M,k))\subseteq\cup_{j=0}^{i}\operatorname{Supp}(H^{j}_{B}(M))+\mathcal{E}_{i+j}$

where $\mathcal{E}_{i}$ denote the set of twists of the summands in the $i$-th
step of the minimal free resolution of $B$; see [1, Section 4.2] for more
details regarding the relation between $B$-regularity and the support of Tor
modules. By [1, Corollary 3.12], for every $i$ we have

$\displaystyle\operatorname{Supp}(H^{i}_{\operatorname{\mathfrak{m}}}(M))$
$\displaystyle\subseteq\cup\operatorname{Supp}(\operatorname{Tor}_{m+n-i}^{S}(M,k))+\operatorname{Supp}(H^{m+n}_{\operatorname{\mathfrak{m}}}(S))$
$\displaystyle\subseteq\cup\operatorname{Supp}(\operatorname{Tor}_{m+n-i}^{S}(M,k))-(m,n)-\operatorname{\mathbb{N}}^{2}$
$\displaystyle\subseteq\cup\cup_{j=0}^{i}\operatorname{Supp}(H^{j}_{B}(M))+\mathcal{E}_{m+n-(i-j)}-(m,n)-\operatorname{\mathbb{N}}^{2}$
$\displaystyle\subseteq\cup_{j=0}^{i}\operatorname{Supp}(H^{j}_{B}(M))+\mathcal{E}_{i-i}-\operatorname{\mathbb{N}}^{2}.$

Therefore

$\operatorname{Supp}(H^{i}_{\operatorname{\mathfrak{m}}}(M))+\mathcal{F}_{i-1}\subseteq\cup_{j=0}^{i}\operatorname{Supp}(H^{j}_{B}(M))+\mathcal{F}_{j-1}-\operatorname{\mathbb{N}}^{2},$

which means

(2.1)
$\operatorname{reg}_{B}(M)\subseteq\operatorname{reg}_{\operatorname{\mathfrak{m}}}(M).$

Suppose $\mu\in\operatorname{reg}_{B}(M)$. By the definition,
$\left(\mu-\mathcal{F}_{i-1}+\operatorname{\mathbb{N}}^{2}\right)\cap\operatorname{Supp}H^{i}_{B}(M)=\emptyset$
for every $i$. By 2.1,
$\left(\mu-\mathcal{F}_{i-1}+\operatorname{\mathbb{N}}^{2}\right)\cap\operatorname{Supp}H^{i}_{\operatorname{\mathfrak{m}}}(M)=\emptyset$.
By using Mayer-Vietoris exact sequence one gets
$\left(\mu-\mathcal{F}_{i-1}+\operatorname{\mathbb{N}}^{2}\right)\cap\left(\operatorname{Supp}H^{i}_{B_{1}}(M)\cap\operatorname{Supp}H^{i}_{B_{2}}(M)\right)=\emptyset$,
that implies

$\operatorname{reg}_{B}(M)\subseteq\operatorname{reg}_{\operatorname{\mathfrak{m}}}(M)\cap\operatorname{reg}_{B_{1}}(M)\cap\operatorname{reg}_{B_{2}}(M).$

Suppose
$\mu\in\operatorname{reg}_{\operatorname{\mathfrak{m}}}(M)\cap\operatorname{reg}_{B_{1}}(M)\cap\operatorname{reg}_{B_{2}}(M)$.
By the definition, for every $i$,
$\left(\mu-\mathcal{F}_{i-1}+\operatorname{\mathbb{N}}^{2}\right)\cap\left(\operatorname{Supp}H^{i}_{B_{1}}(M)\cap\operatorname{Supp}H^{i}_{B_{2}}(M)\right)=\emptyset$,
and
$\left(\mu-\mathcal{F}_{i}+\operatorname{\mathbb{N}}^{2}\right)\cap\linebreak\operatorname{Supp}H^{i+1}_{\operatorname{\mathfrak{m}}}(M)=\emptyset$.
Since
$\left(\mu-\mathcal{F}_{i-1}+\operatorname{\mathbb{N}}^{2}\right)\subset\left(\mu-\mathcal{F}_{i}+\operatorname{\mathbb{N}}^{2}\right)$,
$\left(\mu-\mathcal{F}_{i-1}+\operatorname{\mathbb{N}}^{2}\right)\cap\operatorname{Supp}H^{i+1}_{\operatorname{\mathfrak{m}}}(M)=\emptyset$.
The assertion follows from Mayer-Vietoris exact sequence and the definition of
$\operatorname{reg}_{B}(M)$.

∎

##  3\. Complete intersections in $\mathbb{P}^{n}\times\mathbb{P}^{m}$

We start this section by stating the definition of complete intersection
scheme in a product of projective that we are using in this article.

######  Definition 3.1.

A subscheme $V\subseteq\mathbb{P}^{n}\times\mathbb{P}^{m}$, is a complete
intersection if $V=\operatorname{proj}(S/I)$ where $I$ is generated by
$\operatorname{codim}(V)$ bihomogeneous elements.

Let $V$ be a complete intersection scheme in a projective space, then there
exist ideal $I$ defining $V$ with as many generators as the codimension of
$V$. In this case, ideal $I$ is complete intersection, therefore many
homological invariants are entirely determined by the degrees of the
generators. Let
$V=\operatorname{proj}(S/I)\subset\mathbb{P}^{n}\times\mathbb{P}^{m}$ be a
complete intersection scheme, where $I$ is generated by
$\operatorname{codim}(V)$ forms. In this case, $I$ is not complete
intersection unless $n,m=1$ in which case $I$ is not the defining ideal of $V$
(it is not $B$-saturated). Although the similar argument would not work in
this case, but our goal is to show that many homological invariants are only
depend upon the degrees of the generators of $I$.

######  Proposition 3.2.

[4, Example 8.4.2] Let $S=k[x_{0},\dots,x_{n},y_{0},\dots,y_{m}]$ be a
bigraded polynomial ring where $\deg(x_{i})=(1,0)$ and $\deg(y_{i})=(0,1)$.
Let $I=(f_{1},\dots,f_{n+m})$ generated by $n+m$ forms of degree
$(d_{i},e_{i})$ and $V$ be the scheme defined by $I$. If $I$ is a complete
intersection, then

$\deg(V)=\sum d_{i_{1}}\cdots d_{i_{n}}\cdot e_{j_{1}}\cdots e_{j_{m}},$

where the sum is over all permutations
$(d_{i_{1}},\dots,d_{i_{n}},e_{j_{1}},\dots,e_{j_{m}})$ of $(1,\dots,n+m)$
with $i_{1}<i_{2}<\cdots<i_{n}$ and $j_{1}<j_{2}<\cdots<j_{m}$. In particular
if $d_{i}=d$ and $e_{i}=e$ for all $i$, then

$\deg(V)=\binom{n+m}{n}d^{n}e^{m}.$

######  Proposition 3.3.

Let $I=(f_{1},\dots,f_{r})$ be a bigraded ideal defining a complete
intersection of codimension $r$ and let $\deg(f_{i})=\textbf{d}_{i}$. For
$\mu\in\cap_{i}\operatorname{reg}_{B}(H_{i}(\operatorname{\textbf{K}}(\operatorname{\mathbf{f}},S))$,

$\operatorname{HF}_{S/I}(\mu)=P(\mu),$

where $P$ is a polynomial that only depends upon
$\textbf{d}_{1},\dots,\textbf{d}_{r}$.

###### Proof.

By the Serre Grothendieck formula [1, Proposition 4.27]

(3.1) $HP_{S/I}(\mu)=\operatorname{HF}_{S/I}(\mu)+\sum_{i}(-1)^{i}\dim
H^{i}_{B}(S/I)_{\mu}.$

Hence, if $\mu\in\operatorname{reg}_{B}(S/I)$ then
$HP_{S/I}(\mu)=\operatorname{HF}_{S/I}(\mu)$. Set

(3.2)
$\chi(\mu):=\sum_{i}(-1)^{i}\dim(H_{i}(\operatorname{\textbf{K}}(\operatorname{\mathbf{f}},S))_{\mu}=\sum_{i}(-1)^{i}\dim(K_{i})_{\mu}.$

Note that the second equality shows that $\chi$ is a function that only
depends upon the degrees. Since $I$ is complete intersection,
$H_{i}(\operatorname{\textbf{K}}(\operatorname{\mathbf{f}},S)=H^{0}_{B}(H_{i}(\operatorname{\textbf{K}}(\operatorname{\mathbf{f}},S))$.
Hence,
$\operatorname{Supp}(H_{i}(\operatorname{\textbf{K}}(\operatorname{\mathbf{f}},S))\cap\operatorname{reg}_{B}(H_{i}(\operatorname{\textbf{K}}(\operatorname{\mathbf{f}},S))=\emptyset$.
Therefore, for
$\mu\in\cap_{i>0}\operatorname{reg}_{B}(H_{i}(\operatorname{\textbf{K}}(\operatorname{\mathbf{f}},S))$,
$\chi(\mu)=\operatorname{HF}_{S/I}(\mu).$ Hence, for
$\mu\in\cap_{i}\operatorname{reg}_{B}(H_{i}(\operatorname{\textbf{K}}(\operatorname{\mathbf{f}},S))$,

$HP_{S/I}(\mu)=\operatorname{HF}_{S/I}(\mu)=\chi(\mu).$

Setting $P:=HP_{S/I}$ the first equality shows that
$\operatorname{HF}_{S/I}(\mu)=P(\mu)$ and the second equality shows this
function only depends upon the degrees. ∎

######  Proposition 3.4.

With the Notation 2.1, for any bigraded free $S$-module $M$,

$H^{n+m+1}_{B}(M)\cong H^{n+m+2}_{\operatorname{\mathfrak{m}}}(M).$

Furthermore, If $n=m$ then

$H^{n+1}_{B}(M)\cong H^{n+1}_{B_{1}}(M)\oplus H^{n+1}_{B_{2}}(M),$

else if $n<m$ then,

$H^{n+1}_{B}(M)\cong H^{n+1}_{B_{1}}(M)\quad\text{and}\quad
H^{m+1}_{B}(M)\cong H^{m+1}_{B_{2}}(M).$

###### Proof.

Note that
$H^{a}_{B_{1}}(M)=H^{b}_{B_{2}}(M)=H^{c}_{\operatorname{\mathfrak{m}}}(M)=0$
if $a\neq n+1$, $b\neq m+1$ and $c\neq n+m+2$. The Mayer-Vietoris exact
sequence

$\cdots\rightarrow H^{i}_{\operatorname{\mathfrak{m}}}(M)\rightarrow
H^{i}_{B_{1}}(M)\oplus H^{i}_{B_{2}}(M)\rightarrow H^{i}_{B}(M)\rightarrow
H^{i+1}_{\operatorname{\mathfrak{m}}}(M)\rightarrow\cdots$

gives the results. ∎

######  Theorem 3.5.

Let $S=k[x_{0},\dots,x_{n},y_{0},\dots,y_{m}]$ be a bigraded polynomial ring
where $\deg(x_{i})=(1,0)$ and $\deg(y_{i})=(0,1)$. Define
$B=(x_{0},\dots,x_{n})\cdot(y_{0},\dots,y_{m})$ the irrelevant ideal of
$\mathbb{P}^{n}\times\mathbb{P}^{m}$. Assume $I=(f_{1},\dots,f_{r})$ with
$\deg(f_{i})=(d_{i},e_{i})$ and $V$ be the scheme defined by $I$. If
$\operatorname{codim}(V)=r$ then

$\begin{array}[]{lll}\bullet&\operatorname{Supp}(H_{i}(\operatorname{\textbf{K}}(\operatorname{\mathbf{f}},S))&=\operatorname{Supp}(V_{n+1+i})\cup\operatorname{Supp}(W_{m+1+i})\quad\text{for}\,\,i>0\\\
\bullet&\operatorname{Supp}(H^{i}_{B}(S/I))&=\operatorname{Supp}(V_{n+1-i})\cup\operatorname{Supp}(W_{m+1-i})\,\text{for}\,\,0\leq
i\leq\dim V,\\\ \bullet&\operatorname{Supp}(H^{\dim
V+1}_{B}(S/I))&=\\{(\mu,\nu)|H^{\dim V}(V,\mathcal{O}_{V}(\mu,\nu))\neq
0\\}\\\
&&\subseteq\operatorname{Supp}(V_{r-m})\cup\operatorname{Supp}(W_{r-n})\cup\\{\sigma-\operatorname{Supp}\left(S/I\right)\\},\end{array}$

where $V_{i}$ (resp. $W_{i}$) is a subquotient of
$H^{n+1}_{B_{1}}(\operatorname{K}_{i}(\operatorname{\mathbf{f}},S))$ (resp.
$H^{m+1}_{B_{2}}(\operatorname{K}_{i}(\operatorname{\mathbf{f}},S))$), and
$\sigma:=\sum_{i}(d_{i},e_{i})-(n+1,m+1)$.

###### Proof.

Consider the double complex
$\operatorname{\textbf{C}}^{\bullet}_{B}(\operatorname{\textbf{K}}_{\bullet}(\operatorname{\mathbf{f}},S))$
and suppose $n\leq m$. If we start taking homologies vertically, by 3.4 in the
third page we have:

$\begin{array}[]{ccccc}0&0&\cdots&0&0\\\ \vdots&\vdots&\vdots&\vdots&\vdots\\\
V_{r}&V_{r-1}&\cdots&V_{1}&V_{0}\\\ \vdots&\vdots&\vdots&\vdots&\vdots\\\
W_{r}&W_{r-1}&\cdots&W_{1}&W_{0}\\\ \vdots&\vdots&\vdots&\vdots&\vdots\\\
M&\cdots&0&0&0,\end{array}$

where

$\displaystyle M$ $\displaystyle\cong
H_{r}(H^{n+m+1}_{B}(\operatorname{\textbf{K}}(\operatorname{\mathbf{f}},S))$
$\displaystyle\cong
H_{r}\left(H^{n+m+2}_{\operatorname{\mathfrak{m}}}\left(\operatorname{\textbf{K}}(\operatorname{\mathbf{f}},S)\right)\right)$
$\displaystyle\cong
H_{r}\left((\operatorname{\textbf{K}}\left(\operatorname{\mathbf{f}},S\right)\left(\sigma)\right)^{\star}\right)$
$\displaystyle\cong(S/I)^{\star}\left(\sigma\right).$

If we start taking homology horizontally, in the second page we have

$\begin{array}[]{cccccc}\cdots&0&H_{\gamma}&\cdots&H_{1}&H^{0}_{B}(S/I)\\\
\cdots&0&0&0&0&H^{1}_{B}(S/I)\\\ \cdots&0&0&0&0&\vdots\\\
\vdots&\vdots&\vdots&\vdots&\vdots&H^{\dim(V)+1}_{B}(S/I)\\\
\vdots&\vdots&\vdots&\vdots&\vdots&0,\end{array}$

where $\gamma:=\max\\{0,r-n-1,r-m-1\\}$. The result is obtained by comparing
the two abutments. ∎

######  Definition 3.6.

Let $\textbf{d}=(d,e)\in\mathbb{N}^{2}$ and $r\in\mathbb{N}$, define

$\displaystyle v_{i}$ $\displaystyle:=i\cdot\mathbf{d}-(n+1,0),$
$\displaystyle w_{i}$ $\displaystyle:=i\cdot\mathbf{d}-(0,m+1),$
$\displaystyle\sigma$ $\displaystyle:=r\cdot\mathbf{d}-(n+1,m+1).$

######  Lemma 3.7.

Let $I=(f_{1},\dots,f_{r})\subseteq S$ and $\deg(f_{i})=\textbf{d}=(d,e)$.
Suppose $V$ is the scheme defined by $I$, if $\operatorname{codim}(V)=r$ then

$\displaystyle\operatorname{Supp}(H_{i}(\operatorname{\textbf{K}}(\operatorname{\mathbf{f}},S))\subseteq$
$\displaystyle v_{n+1+i}+(-\mathbb{N},\mathbb{N})\cup
w_{m+1+i}+(\mathbb{N},-\mathbb{N})\quad\text{for}\,\,i>0,$
$\displaystyle\operatorname{Supp}(H^{i}_{B}(S/I))\subseteq$ $\displaystyle
v_{n+1-i}+(-\mathbb{N},\mathbb{N})\cup
w_{m+1-i}+(\mathbb{N},-\mathbb{N})\quad\text{for}\,\,0\leq i\leq\dim V.$

In addition, the inclusions are sharp.

###### Proof.

With the proof of Theorem 3.5, in the second page of the spectral sequence we
have for $p<n+m+1$

${}^{v}E^{\infty}_{p,q}=^{v}E^{2}_{p,q}=\begin{cases}H_{q}(H^{n+1}_{B_{1}}\operatorname{\textbf{K}}(\operatorname{\mathbf{f}},S))\quad\mbox{if}\,\,p=n+1,\\\
H_{q}(H^{n+1}_{B_{2}}\operatorname{\textbf{K}}(\operatorname{\mathbf{f}},S))\quad\mbox{if}\,\,p=m+1,\\\
0\quad\mbox{else}.\end{cases}$

Combining with the Theorem 3.5 shows the inclusions. The sharpness follows
from the fact that for any $q$, $v_{q}$ (resp. $w_{q}$) is in the support of
$H^{n+1}_{B_{1}}K_{q}(\operatorname{\mathbf{f}},S)$ (resp.
$H^{m+1}_{B_{2}}K_{q}(\operatorname{\mathbf{f}},S)$) and it is not in the
support of $H^{n+1}_{B_{1}}K_{q-1}(\operatorname{\mathbf{f}},S)$ and
$H^{n+1}_{B_{1}}K_{q+1}(\operatorname{\mathbf{f}},S)$ (resp.
$H^{m+1}_{B_{2}}K_{q-1}(\operatorname{\mathbf{f}},S)$ and
$H^{m+1}_{B_{2}}K_{q-1}(\operatorname{\mathbf{f}},S)$). ∎

######  Corollary 3.8.

Let $I=(f_{1},\dots,f_{r})\subseteq S$ and $\deg(f_{i})=\textbf{d}=(d,e)$ and
$V$ be the scheme defined by $I$. If $\operatorname{codim}(V)=r$, then

  * $(1)$

if $\mu\notin v_{i}+(-\mathbb{N},\mathbb{N})\cup
w_{j}+(\mathbb{N},-\mathbb{N})$ for $i=r-m,r-m-1$ and $j=r-n,r-n-1$ then

$H^{\dim(V)+1}_{B}(S/I))_{\mu}=(S/I)^{\star}_{\sigma-\mu}.$

###### Proof.

Note
that$\operatorname{Supp}\left(H^{n+1}_{B_{1}}(K_{i}(\operatorname{\mathbf{f}},S))\right)=v_{i}+(-\mathbb{N},\mathbb{N})$
and
$\operatorname{Supp}H^{m+1}_{B_{2}}(K_{i}(\operatorname{\mathbf{f}},S))=w_{i}+(\mathbb{N},+\mathbb{N})$
for all $i$. Consider the double complex
$\operatorname{\textbf{C}}^{\bullet}_{B}(\operatorname{\textbf{K}}_{\bullet}(\operatorname{\mathbf{f}},S))$
and suppose $n\leq m$ as in the proof of Theorem 3.5. Since
$\operatorname{Supp}(V_{i})\cap\operatorname{Supp}(W_{j})=\emptyset$, there
will be no nonzero map in the spectral sequence among them. In this case, in
the $n+2$-th page we have an induced map $\psi_{1}:M\rightarrow W_{r-n-1}$,
and in the $m+2$-th page we have an induced map
$\psi_{2}:\ker\psi_{1}\rightarrow V_{r-m-1}$. If $\mu\notin
v_{r-m}+(-\mathbb{N},\mathbb{N})\cup w_{r-n}+(\mathbb{N},-\mathbb{N})$,
$H^{r}_{B}(S/I)_{\mu}\cong(\ker\psi_{2})_{\mu}$. In addition, if $\mu\notin
v_{r-m-1}+(-\mathbb{N},\mathbb{N})\cup w_{r-n-1}+(\mathbb{N},-\mathbb{N})$

$H^{\dim(V)+1}_{B}(S/I)_{\mu}\cong(\ker\psi_{2})_{\mu}=M_{\mu}=(S/I)^{\star}_{\sigma-\mu}.$

∎

##  4\. Complete intersection points in $\mathbb{P}^{n}\times\mathbb{P}^{m}$

In this section we apply the results in the previous section to the case of
complete intersection points in $\mathbb{P}^{n}\times\mathbb{P}^{m}$. In the
rest of this section, $I=(f_{1},\dots,f_{n+m})$ and $V$ be the complete
intersection scheme of points defined by $I$ with
$\textbf{d}:=\deg(f_{i})=(d,e)$.

######  Definition 4.1.

Set
$\Gamma_{i}:=\operatorname{Supp}(H^{n+1}_{B_{1}}(\operatorname{K}_{n+i}(\operatorname{\mathbf{f}},S)))\cup\operatorname{Supp}(H^{m+1}_{B_{2}}(\operatorname{K}_{m+i}(\operatorname{\mathbf{f}},S)))$.
Notice that $\Gamma_{i}=\emptyset$ if and only if $i\geq\max\\{n,m\\}$. If
$i\leq m$ then

$\operatorname{Supp}(H^{n+1}_{B_{1}}(\operatorname{K}_{n+i}(\operatorname{\mathbf{f}},S)))=v_{n+i}+(-\mathbb{N},\mathbb{N}),$

and similarly for
$\operatorname{Supp}(H^{m+1}_{B_{2}}(\operatorname{K}_{m+i}(\operatorname{\mathbf{f}},S)))$
if $i\leq n$.

######  Definition 4.2.

For a function $F:\mathbb{Z}^{2}\rightarrow\mathbb{Z}$ define

$\displaystyle F^{\star}(a,b)$ $\displaystyle:=F(-a,-b),$ $\displaystyle
F^{\prime}(a,b)$ $\displaystyle:=F(-a,b),\,\,\text{and}$ $\displaystyle
F^{\prime\prime}(a,b)$ $\displaystyle:=F(a,-b).$

######  Theorem 4.3.

Let $S=k[x_{0},\dots,x_{n},y_{0},\dots,y_{m}]$ be a bigraded polynomial ring
where $\deg(x_{i})=(1,0)$ and $\deg(y_{i})=(0,1)$. Assume $V$ be a complete
intersection scheme of points defined by $I=(f_{1},\dots,f_{n+m})$ with
$\deg(f_{i})=(d,e)$.

  * $(1)$

If $\mu\notin\cup_{i\geq 2}\Gamma_{i}$ then

$\operatorname{HF}_{S/I}(\mu)=\chi(\mu)$

where
$\chi((a,b))=\sum_{i}(-1)^{i}\binom{n+m}{i}\binom{n+a-id}{n}\binom{m+b-ie}{m}$.

  * $(2)$

If $\mu\notin\Gamma_{1}$ then

$\operatorname{HF}_{S/I}(\mu)=\operatorname{HF}_{V}(\mu)$

  * $(3)$

If $\mu\notin\Gamma_{0}$ then

$\operatorname{HF}_{V}(\mu)+\operatorname{HF}_{V}(\sigma-\mu)=\deg(V)=\binom{n+m}{n}d^{n}e^{m}.$

In particular, if
$\mu\in(nd-n,(n+m)e-m)+\operatorname{\mathbb{N}}^{2}\cup((n+m)d-n,me-m)+\operatorname{\mathbb{N}}^{2}$
then $\operatorname{HF}_{V}(\mu)=\deg(V)$ and
$\mu\in\operatorname{reg}_{B}(V)$.

  * $(4)$

If $\mu\in\Gamma_{0}\setminus\\{\Gamma_{-1}\cup\Gamma_{1}\\}$ then

$\operatorname{HF}_{V}(\mu)+\operatorname{HF}_{V}(\sigma-\mu)=\binom{n+m}{n}(d^{n}e^{m}-\epsilon(\mu)),$

where $\epsilon(\mu)=\operatorname{HF}_{S}^{\prime}(\mu-
v_{n})+\operatorname{HF}_{S}^{\prime\prime}(\mu-w_{n})$ as in the Definition
4.2.

  * $(5)$

If $\mu\in\cup_{i\geq 1}\Gamma_{i}\setminus\\{\Gamma_{0}\\}$ then

$\operatorname{HF}_{V}(\mu)=\binom{n+m}{n}d^{n}e^{m}-\chi(\sigma-\mu).$

In particular, if $\mu\in\Gamma_{1}\setminus\\{\Gamma_{0}\cup\Gamma_{2}\\}$
then

$\dim(I_{V}/I)_{\mu}=\binom{n+m}{n+1}\operatorname{HF}_{S}^{\prime}(\mu-
v_{n+1})+\binom{n+m}{m+1}\operatorname{HF}_{S}^{\prime\prime}(\mu-w_{m+1}).$

###### Proof.

Denote the saturation of $I$ with respect to $B$ by $J$. In this case
$\operatorname{HF}_{V}(\mu)=\dim(S/J)_{\mu}$ for all $\mu\in\mathbb{Z}^{2}$.
In the proof we use these two simple fact that
$\mu\in\Gamma_{i-1}\cup\Gamma_{i+1}$ yields $\mu\in\Gamma_{i}$ and if
$\mu\in\Gamma_{i}$ then $\sigma-\mu\in\Gamma_{-i}$. By Proposition 3.2 the
Hilbert polynomial of $V$ is $D:=\binom{n+m}{n}d^{n}e^{m}$ which in this case
is equal to the $\deg(V)$. By Serre Grothendieck formula,

(4.1) $\operatorname{HF}_{S/J}(\mu)+\dim(H^{1}_{B}(S/J)_{\mu})=D$

$(1)$ If $\mu\notin\cup_{i\geq 2}\Gamma_{i}$, by Lemma 3.7
$(H_{i}(\operatorname{K}(\operatorname{\mathbf{f}},S))_{\mu}=0$ for all $i\geq
1$. Therefore by Equation 3.2, $(S/I)_{\mu}=\chi(\mu)$.
$(2)$ If $\mu\notin\Gamma_{1}$, by Lemma 3.7, $(H^{0}_{B}(S/I))_{\mu}=0$
therefore $(S/I)_{\mu}=(S/J)_{\mu}$.
$(3)$ First note that $\mu,\sigma-\mu\notin\Gamma_{0}$. We claim that
$\mu\notin\Gamma_{-1}$ or $\sigma-\mu\notin\Gamma_{-1}$. Suppose not, then
$\mu\in\Gamma_{-1}\cap\Gamma_{1}$ which yields $\mu\in\Gamma_{0}$ which is a
contradiction. So assume $\mu\notin\Gamma_{-1}$. By Corollary 3.8,

$\dim(H^{1}_{B}(S/I)_{\mu})=\operatorname{HF}_{S/I}(\sigma-\mu).$

Since $\sigma-\mu\notin\Gamma_{1}$, by part $(2)$ and Equation 3.1

$\binom{n+m}{n}d^{n}e^{m}-\operatorname{HF}_{V}(\mu)\operatorname{HF}_{V}(\sigma-\mu).$

The same argument works if we assume $\sigma-\mu\notin\Gamma_{-1}$.
$(4)$ By part $(1)$ $\operatorname{HF}_{S/I}(\mu)=\operatorname{HF}_{V}(\mu)$.
By considering the double complex as in the proof of Theorem 3.5 and Lemma
3.7,

$\displaystyle\left(H_{n}(H^{n+1}_{B_{1}}\operatorname{K}(\operatorname{\mathbf{f}},S))\right)\mu=(H^{n+1}_{B_{1}}(K_{n}(\operatorname{\mathbf{f}},S))_{\mu},\,\,\text{and}$
$\displaystyle\left(H_{m}(H^{m+1}_{B_{2}}\operatorname{K}(\operatorname{\mathbf{f}},S))\right)_{\mu}=\left(H^{m+1}_{B_{2}}(K_{m}(\operatorname{\mathbf{f}},S)\right)_{\mu}.$

By the abutment of the spectral sequence

$\left(H^{1}_{B}(S/J)\right)_{\mu}=\operatorname{HF}_{S/I}(\sigma-\mu)+\dim(H^{n+1}_{B_{1}}(K_{n}(\operatorname{\mathbf{f}},S))_{\mu}+.\left(H^{m+1}_{B_{2}}(K_{m}(\operatorname{\mathbf{f}},S)\right)_{\mu}.$

Since $\sigma-\mu\notin\Gamma_{1}$, by parts $(1)$ and $(2)$,
$\operatorname{HF}_{S/I}(\mu)=\operatorname{HF}_{V}(\mu)$ and by the
definition,
$\dim(H^{n+1}_{B_{1}}(K_{n}(\operatorname{\mathbf{f}},S))_{\mu}+\left(H^{m+1}_{B_{2}}(K_{m}(\operatorname{\mathbf{f}},S)\right)_{\mu}=\binom{n+m}{n}\epsilon(\mu)$.
The assertion follows by
$\left(H^{1}_{B}(S/J)\right)_{\mu}=\binom{n+m}{n}d^{n}e^{m}-\operatorname{HF}_{V}(\mu)$.
$(5)$ By part $(3)$,
$\operatorname{HF}_{V}(\mu)=\binom{n+m}{n}d^{n}e^{m}-\operatorname{HF}_{V}(\sigma-\mu)$.
Because $\mu\notin\Gamma_{-1}$ ($\mu\in\Gamma_{-1}$ yields $\mu\in\Gamma_{0}$)
therefore $\sigma-\mu\notin\Gamma_{1}$, which by part $(2)$, yields
$\operatorname{HF}_{V}(\sigma-\mu)=\operatorname{HF}_{S/I}(\sigma-\mu)$. On
the other hand, since $\mu\notin\cup_{i\geq 2}\Gamma_{i}$, by part $(1)$,
$\operatorname{HF}_{S/I}(\sigma-\mu)=\chi(\sigma-\mu)$. ∎

Let $S=[x_{0},x_{1},x_{2},y_{0},y_{1},y_{2}]$ and $\deg(x_{i})=(1,0)$ and
$\deg(y_{i})=(0,1)$. Let $I=(f_{1},\dots,f_{4})$ where $\deg(f_{i})=(2,2)$.
Suppose scheme $V$ defined by $I$ is complete intersection. The following
picture demonstrate the regions $\Gamma_{i}$ in the Theorem 4.3.

$-1.$$1.$$2.$$3.$$4.$$5.$$6.$$-1.$$1.$$2.$$3.$$4.$$5.$$6.$$0$$\sigma$$M$

$\Gamma_{-1}$ is the red region, $\Gamma_{0}$ is the blue region and
$\Gamma_{1}$ is the green region. By Theorem 4.3, Hilbert function of $V$ at
the bidegrees except the intersection of blue and green are independent from
the choices of $f_{i}$’s and they can be computed via $\chi$ and $\epsilon$
defined in the Theorem 4.3. On the other hand, the rest do depends upon the
$f_{i}$’s. Here we computed two example via computer system Macaulay2 [5].

######  Example 4.4.

Let $S=k[x_{0},x_{1},x_{2},y_{0},y_{1},y_{2}]$,
$I=(x_{0}^{2}y_{0}^{2},x_{1}^{2}y_{1}^{2},x_{2}^{2}y_{2}^{2},(x_{0}+x_{1}+x_{2})^{2}(y_{0}+y_{1}+y_{2})^{2})$
and $V$ be the complete intersection scheme of points defined by $I$. For
$(0,0)\leq\mu\leq(7,7)$ the bigraded Hilbert function
$\operatorname{HF}_{V}(\mu)$ is

$\begin{matrix}{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}24}&{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}72}&{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}96}&{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}96}&96&96&96&96\\\
{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}24}&{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}72}&{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}96}&{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}96}&96&96&96&96\\\
{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}21}&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}63}&86&90&93&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}95}&96&96\\\
{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}15}&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}45}&66&78&87&93&96&96\\\
10&30&48&64&78&90&{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}96}&{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}96}\\\
6&18&32&48&66&86&{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}96}&{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}96}\\\
3&9&18&30&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}45}&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}63}&{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}72}&{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}72}\\\
1&3&6&10&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}15}&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}21}&{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}24}&{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}24}\end{matrix}.$

Where blue corresponds to the bidegrees in $\Gamma_{0}$, green corresponds to
$\Gamma_{1}$ and orange to their intersections. Also red indicate the
$\operatorname{HF}_{V}(\sigma)$. For $\mu\notin\Gamma_{0}$ by Theorem 4.3
parts $(1),(2)$ and $(3)$ one can compute the bigraded Hilbert function and
for blue points by part $(4)$. For the orange points, Theorem 4.3 does not say
anything. In addition, any points except the orange ones only depend on the
degree of the generator of $I$, which in this case is $(2,2)$.

Asking random command in Macaulay2 and constructing $I^{\prime}$ with $4$
generators of bidegree $(2,2)$. Let $V^{\prime}$ be the complete intersection
scheme of points defined by $I$. For $(0,0)\leq\mu\leq(13,7)$ the bigraded
Hilbert function $\operatorname{HF}_{V^{\prime}}(\mu)$ is

$\textstyle{33}$$\textstyle{84}$$\textstyle{96}$$\textstyle{96}$$\textstyle{96}$$\textstyle{96}$$\textstyle{96}$$\textstyle{96}$$\textstyle{96}$$\textstyle{96}$$\textstyle{96}$$\textstyle{96}$$\textstyle{96}$$\textstyle{96}$$\textstyle{27}$$\textstyle{81}$$\textstyle{96}$$\textstyle{96}$$\textstyle{96}$$\textstyle{96}$$\textstyle{96}$$\textstyle{96}$$\textstyle{96}$$\textstyle{96}$$\textstyle{96}$$\textstyle{96}$$\textstyle{96}$$\textstyle{96}$$\textstyle{21}$$\textstyle{63}$$\textstyle{86}$$\textstyle{90}$$\textstyle{93}$$\textstyle{95}$$\textstyle{96}$$\textstyle{96}$$\textstyle{96}$$\textstyle{96}$$\textstyle{96}$$\textstyle{96}$$\textstyle{96}$$\textstyle{96}$$\textstyle{15}$$\textstyle{45}$$\textstyle{66}$$\textstyle{78}$$\textstyle{87}$$\textstyle{93}$$\textstyle{96}$$\textstyle{96}$$\textstyle{96}$$\textstyle{96}$$\textstyle{96}$$\textstyle{96}$$\textstyle{96}$$\textstyle{96}$$\textstyle{10}$$\textstyle{30}$$\textstyle{48}$$\textstyle{64}$$\textstyle{78}$$\textstyle{90}$$\textstyle{96}$$\textstyle{96}$$\textstyle{96}$$\textstyle{96}$$\textstyle{96}$$\textstyle{96}$$\textstyle{96}$$\textstyle{96}$$\textstyle{6}$$\textstyle{18}$$\textstyle{32}$$\textstyle{48}$$\textstyle{66}$$\textstyle{86}$$\textstyle{96}$$\textstyle{96}$$\textstyle{96}$$\textstyle{96}$$\textstyle{96}$$\textstyle{96}$$\textstyle{96}$$\textstyle{96}$$\textstyle{3}$$\textstyle{9}$$\textstyle{18}$$\textstyle{30}$$\textstyle{45}$$\textstyle{63}$$\textstyle{81}$$\textstyle{84}$$\textstyle{96}$$\textstyle{96}$$\textstyle{96}$$\textstyle{96}$$\textstyle{96}$$\textstyle{96}$$\textstyle{1}$$\textstyle{3}$$\textstyle{6}$$\textstyle{10}$$\textstyle{15}$$\textstyle{21}$$\textstyle{28}$$\textstyle{36}$$\textstyle{45}$$\textstyle{55}$$\textstyle{66}$$\textstyle{78}$$\textstyle{91}$$\textstyle{96}$

The only differences are in the orange spots. In addition, in this case
$\operatorname{HF}_{V^{\prime}}(0,13)=\operatorname{HF}_{V^{\prime}}(13,0)=96$
which means the natural projection of $V^{\prime}$ to each $\mathbb{P}^{n}$
and $\mathbb{P}^{m}$ is one to one.

######  Theorem 4.5.

Let $S=k[x_{0},\dots,x_{n},y_{0},\dots,y_{m}]$ be a bigraded polynomial ring
where $\deg(x_{i})=(1,0)$ and $\deg(y_{i})=(0,1)$. Assume $V$ be a complete
intersection scheme of points defined by $I=(f_{1},\dots,f_{n+m})$ with
$\deg(f_{i})=(d,e)$. Let $\pi_{1}:V\rightarrow\mathbb{P}^{n}$ and
$\pi_{2}:V\rightarrow\mathbb{P}^{m}$ be the natural projections. Then

$\displaystyle\max\\{\operatorname{reg}(\pi_{1}^{-1}(p))\,\,\text{for}\,\,p\in\mathbb{P}^{n}\\}$
$\displaystyle=\min\\{a\in\mathbb{N}|(a,b)\in\operatorname{reg}_{B}(V)\,\,\text{for
some }\,b\\}$
$\displaystyle\max\\{\operatorname{reg}(\pi_{2}^{-1}(p))\,\,\text{for}\,\,p\in\mathbb{P}^{m}\\}$
$\displaystyle=\min\\{b\in\mathbb{N}|(a,b)\in\operatorname{reg}_{B}(V)\,\,\text{for
some }\,a\\}.$

###### Proof.

Adopt Notation 2.1 and let $J$ be the saturation of $I$ with respect to $B$.
Since $S/J$ is $B$-saturated, it is also $B_{1}$ and $B_{2}$ saturated. Let
$\mu\gg 0$ and $\nu\gg 0$, therefore
$H^{1}_{\operatorname{\mathfrak{m}}}(S/J)_{\mu,\nu}=H^{2}_{\operatorname{\mathfrak{m}}}(S/J)_{\mu,\nu}=0$.
Suppose $\mu\gg 0$, Hence by Mayer-Vietoris exact sequence

$H^{1}_{B_{1}}(S/J)_{\mu,\nu}\oplus
H^{1}_{B_{2}}(S/J)_{\mu,\nu}=H^{1}_{B}(S/J)_{\mu,\nu}.$

Since $S/J$ is $B_{1}$-saturated, for $\mu\gg 0$,
$H^{1}_{B_{1}}(S/J)_{\mu,\nu}=0$ which means

$H^{1}_{B_{2}}(S/J)_{\mu,\nu}=H^{1}_{B}(S/J)_{\mu,\nu}.$

We state the proof for the first equality, the argument for the second one is
the same. There exist $(a_{1},b_{1}),(a_{2},b_{2})\in\mathbb{N}^{2}$ such that
$H^{1}_{\operatorname{\mathfrak{m}}}(S/J)_{\mu}=H^{2}_{\operatorname{\mathfrak{m}}}(S/J)_{\nu}=0$
for all $\mu\geq(a_{1},b_{1})$ and $\nu\geq(a_{2},b_{2})$. Also for any
$a\in\mathbb{N}$, there exists $b_{a}$ such that for all $\mu\geq b_{a}$,
$(H^{1}_{B_{2}}(S/J))_{(a,\mu)}=0$. Set

$b:=\max_{a\in\mathbb{N}}\\{b_{1},b_{2},b_{a}\\}.$

The Mayer-Vietoris short exact sequence yields
$H^{1}_{B}(S/J)_{(\mu,\nu)}=H^{1}_{B_{1}}(S/J)_{(\mu,\nu)}$ if $\nu>b$.
Therefore

$\displaystyle\min\\{a\in\mathbb{N}|(a,b)\in\operatorname{reg}_{B}(V)\,\,\text{for
some
}\,b\\}=\min\\{\mu\in\mathbb{N}|H^{1}_{B_{1}}(S/J)_{\mu,\nu}=0\,\,\text{for}\,\,\nu>b\\}$
$\displaystyle=\min\\{\mu\in\mathbb{N}|\exists\,\,\mathfrak{p}\neq(y_{0},y_{1},y_{2})\,;\,H^{1}_{B_{1}}(S/J\otimes_{k[y_{0},y_{1},y_{2}]}k[y_{0},y_{1},y_{2}]_{\mathfrak{p}})_{\mu,\nu}=0\,\,\text{for
}\,\,\nu>b\\}$

∎

##  5\. Generic complete intersection points in
$\mathbb{P}^{n}\times\mathbb{P}^{m}$

In this subsection we adopt the Notation 2.1 as well. We focus on the case
where ideal $I$ is generated by the generic forms. In this regard, we first
need to provide an analogous version of Bertini’s theorem in our case.

######  Proposition 5.1.

Let $k$ be a field of characteristics zero and
$V\subset\mathbb{P}_{k}^{n}\times\mathbb{P}_{k}^{m}$ be a reduced scheme and
$(d,e)\in\mathbb{N}^{2}$ with $d,e\neq 0$. Let $f_{U}$ be a form of bidegree
$(d,e)$ with indeterminate coefficients $U_{\alpha,\beta}$. There exists a non
empty open set $\Omega\subset\operatorname{Spec}(k[U_{\alpha,\beta}])$ such
that for any $\mathfrak{p}\in\Omega$, the corresponding form
$f_{\mathfrak{p}}$ is such that

$V\cap
Z(f_{\mathfrak{p}})\subseteq\mathbb{P}^{n}_{k_{\mathfrak{p}}}\times\mathbb{P}^{m}_{k_{\mathfrak{p}}},$

where $k_{\mathfrak{p}}$ is the residue field of
$k[U_{\alpha,\beta}]_{\mathfrak{p}}$, is reduced of dimension equal $\dim
V-1$, unless the $\dim V=0$ in which case $V\cap
Z(f_{\mathfrak{p}})=\emptyset$.

###### Proof.

Consider the Segre-Veronese map
$\Psi:\mathbb{P}^{n}\times\mathbb{P}^{m}\rightarrow\mathbb{P}^{N}$ with
$N=\binom{n+d}{d}\times\binom{m+e}{e}-1$. Let $V^{\prime}=\Psi(V)$. Under the
map $\Psi$, $f_{U}$ is mapped to a linear form $\ell_{U}$ in $N+1$ variables
and this correspondence is one to one. By Bertini theorem (see [3, Corollary
3.4.9]) there exists $\Omega$ as claimed such that for
$\mathfrak{p}\in\Omega$, $V^{\prime}\cap Z(\ell_{\mathfrak{p}})\cong V\cap
Z(f_{\mathfrak{p}})$ is a reduced scheme with the asserted dimension. ∎

######  Definition 5.2.

Forms of bidegree $(d,e)$ are in one to one correspondence with closed points
in $\mathbb{A}_{k}^{\dim_{k}S_{(d,e)}}$.

######  Remark 5.3.

Notice that $\Omega\subset\operatorname{spec}k[U_{\alpha,\beta}]\setminus
Z(F)$ for some form $F\neq 0$. In particular, for $c=(c_{\alpha,\beta})\in
k^{N}$ such that $F(c)\neq 0$, Proposition 5.1 with $\mathfrak{p}$ generated
by the elements $u_{\alpha,\beta}-c_{\alpha,\beta}$, shows that $Z(f_{c})\cap
V$ is reduced.

######  Theorem 5.4.

Let $S=k[x_{0},\dots,x_{n},y_{0},\dots,y_{m}]$ be a bigraded polynomial ring
over a field $k$ of characteristic zero where $\deg(x_{i})=(1,0)$ and
$\deg(y_{i})=(0,1)$. Let $(d_{i},e_{i})\geq(1,1)$ for $i=1,\dots,n+m$ be
bidegrees. Then, there exists a non-empty open set

$\Omega\subset\mathbb{A}_{k}^{N}:=\prod_{i=1}^{n+m}\mathbb{A}_{k}^{|S_{(d_{i},e_{i})}|}$

such that, for any $(f_{1},\dots,f_{n+m})$ corresponding to a point in
$\Omega$,

  * $(1)$

$Z:=\operatorname{proj}\left(S/(f_{1},\dots,f_{n+m})\right)$ is reduced of
dimension zero.

  * $(2)$

The natural projections of $Z$ to the factors $\mathbb{P}^{n}$ and
$\mathbb{P}^{m}$ are isomorphism.

###### Proof.

Let
$V:=Z(g_{1},\dots,g_{n+m})\subset\mathbb{A}_{k}^{N}\times\mathbb{P}_{k}^{n}\times\mathbb{P}_{k}^{m}$
where $\mathbb{A}_{k}^{N}=\operatorname{spec}k[U_{i,\alpha,\beta}]$ for $1\leq
i\leq n+m$, $|\alpha|=d_{i}$, $|\beta|=e_{i}$ and $g_{i}:=\sum
U_{i,\alpha,\beta}\textbf{x}^{\alpha}\textbf{y}^{\beta}$. Set
$\pi:\mathbb{A}_{k}^{N}\times\mathbb{P}_{k}^{n}\times\mathbb{P}_{k}^{m}\rightarrow\mathbb{P}_{k}^{n}\times\mathbb{P}_{k}^{m}$
be the natural projection, then $V$ is a vector bundle over
$\mathbb{P}_{k}^{n}\times\mathbb{P}_{k}^{m}$ via $\pi$. For any point
$p\in\mathbb{P}_{k}^{n}\times\mathbb{P}_{k}^{m}$ the fiber of $p$ is a linear
space of dimension $N-n-m$. Hence $V$ is geometrically irreducible scheme and
$\dim(V)=n+m+(N-n-m)=N$.

$(1)$ Set
$W:=(g_{1},\dots,g_{n+m},\operatorname{Jac}^{n+m}(g_{1},\dots,g_{n+m}))\subset
V$ where $\operatorname{Jac}^{n+m}$ is the Jacobian of order $n+m$. By using
Proposition 5.1 inductively, there exist a point $u\in\mathbb{A}_{k}^{N}$ and
$(f_{1},\dots,f_{n+m})$ corresponding to $u$ such that
$\operatorname{Jac}^{n+m}(f_{1},\dots,f_{n+m})\neq 0$ which yields
$W\subsetneq V$ and $\dim(W)<\dim(V)=N$. Consider the natural projection
$p:\mathbb{A}_{k}^{N}\times\mathbb{P}_{k}^{n}\times\mathbb{P}_{k}^{m}\rightarrow\mathbb{A}_{k}^{N}$.
Since $\dim(W)<N$, $p(W)$ is a non empty closed subset of
$\mathbb{A}_{k}^{N}$. Set
$\Omega_{0}:=\left(\mathbb{A}_{k}^{N}\setminus\pi(W)\right)$. Therefore for
any $(f_{1},\dots,f_{n+m})$ corresponding to a point in $\Omega_{0}$,
$\operatorname{proj}\left(S/(f_{1},\dots,f_{n+m})\right)$ is geometrically
reduced and is of dimension zero.

$(2)$ Set
$p_{1}:\mathbb{A}_{k}^{N}\times\mathbb{P}_{k}^{n}\times\mathbb{P}_{k}^{m}\rightarrow\mathbb{A}_{k}^{N}\times\mathbb{P}_{k}^{n}$
and
$q_{2}:\mathbb{A}_{k}^{N}\times\mathbb{P}_{k}^{n}\rightarrow\mathbb{A}_{k}^{N}$
the natural projections. First we show that it is enough to show that the
restriction $p_{1}^{\prime}:V\rightarrow p_{1}(V)$ is birational. Indeed if
$p_{1}^{\prime}$ is birational then there exists a closed subset
$Z_{1}\subsetneq p_{1}(V)$ such that setting
$W_{1}:=(p_{1}^{\prime})^{-1}(Z_{1})$ then $p_{1}^{\prime}|_{V\setminus
W_{1}}:V\setminus W_{1}\rightarrow p_{1}(V)\setminus Z_{1}$ is an isomorphism.
As $\dim(Z_{1})<N$, $q_{2}(Z_{1})\subsetneq\mathbb{A}_{k}^{N}$. Set
$\Omega_{1}:=\mathbb{A}_{k}^{N}\setminus\left(q_{2}(Z_{1})\cup\pi(W)\right)$.
For any point $u\in\Omega_{1}$ and $(f_{1},\dots,f_{n+m})$ corresponding to
$u$,

$q_{2}^{-1}(u)=p_{1}(u,Z(f_{1},\dots,f_{n+m}))\cong\left(u,Z(f_{1},\dots,f_{n+m})\right)$

showing that the natural projection of $Z$ onto $\mathbb{P}^{n}$ is an
isomorphism. By replacing $p_{1}$ and $q_{2}$ by
$p_{2}:\mathbb{A}_{k}^{N}\times\mathbb{P}_{k}^{n}\times\mathbb{P}_{k}^{m}\rightarrow\mathbb{A}_{k}^{N}\times\mathbb{P}_{k}^{m}$
and
$q_{1}:\mathbb{A}_{k}^{N}\times\mathbb{P}_{k}^{m}\rightarrow\mathbb{A}_{k}^{N}$,
along the same lines it shows the existence of $Z_{2}\in p_{2}(V)$ and
$\Omega_{2}=\mathbb{A}_{k}^{N}\setminus\left(q_{1}(Z_{2})\cup\pi(W)\right)$
such that for $u\in\Omega_{2}$ and $(f_{1},\dots,f_{n+m})$ corresponding to
$u$,

$q_{1}^{-1}(u)=p_{2}(u,Z(f_{1},\dots,f_{n+m}))\cong\left(u,Z(f_{1},\dots,f_{n+m})\right).$

Consider the following diagram

$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\subset}$$\scriptstyle{p_{1}^{\prime}}$$\textstyle{\mathbb{A}_{k}^{N}\times\mathbb{P}_{k}^{n}\times\mathbb{P}_{k}^{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{1}}$$\textstyle{V_{1}:=p_{1}(V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\subset}$$\scriptstyle{q_{2}|_{V_{1}}}$$\textstyle{\mathbb{A}_{k}^{N}\times\mathbb{P}_{k}^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{2}}$$\textstyle{\mathbb{A}_{k}^{N}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{=}$$\textstyle{\mathbb{A}_{k}^{N}.}$

We now show that $p_{1}^{\prime}$ is birational.

Let $I:=(g_{1},\dots,g_{n+m)}$ and $J:=I:B^{\infty}$. By part $(1)$, $V_{1}$
is a geometrically irreducible scheme. Hence $J$ and $I_{V_{1}}=J\cap
k[\textbf{x},\textbf{U}]$ are prime and remain prime under any extension of
$k$.

As $V_{1}$ is generically smooth, there exists $G\in I_{V_{1}}$ such that
$D:=\dfrac{\partial G}{\partial U_{i,\alpha,\beta}}\notin I_{V_{1}}$ for some
$i,\alpha$, and $\beta$ unless $G\in k[\textbf{y}]$ which is impossible:
indeed $I_{V_{1}}\cap k[\textbf{y}]=(0)$ because the projection
$V\rightarrow\mathbb{P}^{m}$ is onto.

Set $Z_{1}:=Z\left(D\right)\subset V_{1}$. By [7, Lemma 4.6.1],

$\textbf{x}^{\alpha}\textbf{y}^{\beta}\dfrac{\partial G}{\partial
U_{i,\alpha^{\prime},\beta^{\prime}}}-\textbf{x}^{\alpha^{\prime}}\textbf{y}^{\beta^{\prime}}\dfrac{\partial
G}{\partial U_{i,\alpha,\beta}}\in I_{V}$

for any choices of $\alpha^{\prime}$ and $\beta^{\prime}$. Let
$\textbf{y}^{\beta}=y_{p}\textbf{y}^{\beta_{0}}$ for some $i$. For any $0\leq
j\leq m$ put $\alpha^{\prime}:=\alpha$ and $\beta^{\prime}:=y_{j}\beta_{0}$.
As $\textbf{x}^{\alpha}$ and $\textbf{y}^{\beta_{0}}$ are not in $I_{V}$ it
follows that

$y_{p}\dfrac{\partial G}{\partial
U_{i,\alpha^{\prime},\beta^{\prime}}}-y_{j}\dfrac{\partial G}{\partial
U_{i,\alpha,\beta}}\in I_{V}$

for all $0\leq j\leq m$. Therefore by localizing at $D$, we get an isomorphism

$\phi:\dfrac{\left(k[\textbf{U},\textbf{x}]_{D}[\textbf{y}]\right)}{I_{V}}\overset{\cong}{\longrightarrow}\dfrac{\left(k[\textbf{U},\textbf{x}]_{D}[y_{p}]\right)}{I_{V_{1}}}.$

Therefore the natural maps

$\left(\dfrac{k[\textbf{U},\textbf{x}]}{I_{V_{1}}}\right)_{D}\overset{\iota}{\hookrightarrow}\dfrac{\left(k[\textbf{U},\textbf{x}]_{D}[\textbf{y}]\right)}{I_{V}}\overset{\phi}{\longrightarrow}\dfrac{\left(k[\textbf{U},\textbf{x}]_{D}[y_{p}]\right)}{I_{V_{1}}}$

are such that $\phi\circ\iota$ induces the identity from
$\operatorname{proj}\left(\left(k[\textbf{U},\textbf{x}]_{D}[y_{p}]\right)/I_{V_{1}}\right)$
to
$\operatorname{spec}\left(\left(k[\textbf{U},\textbf{x}]_{D}\right)/I_{V_{1}}\right)$.
Hence $\phi$ provides the inverse $\varphi$

$V_{1}\setminus
Z_{1}\overset{\varphi}{\longrightarrow}V\setminus\left(Z_{1}\times\mathbb{P}_{k}^{m}\right)\overset{p_{1}^{\prime}}{\longrightarrow}V_{1}\setminus
Z_{1}.$

∎

######  Theorem 5.5.

Let $S=k[x_{0},\dots,x_{n},y_{0},\dots,y_{m}]$ be a bigraded polynomial ring
over a field $k$ of characteristics zero. If $I$ is generated by $n+m$ generic
forms of bidegree $(d,e)$, then the scheme $V$ defined by $I$ is a set of
reduced points and

$|\\{\mu\in\mathbb{N}^{2}|\operatorname{HF}_{V}(\mu)\neq\deg(V)\\}|<\infty,$

which means the natural projections are one-to-one.

###### Proof.

By Theorem 5.4 part $(1)$, $V$ is a set of reduced points and by part $(2)$,
$\operatorname{HF}_{V}(0,i)=\operatorname{HF}_{V}(j,0)=\deg(V)$ for any
$i,j\gg 0$. ∎

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