FIBERS OF RATIONAL MAPS AND REES ALGEBRAS OF THEIR BASE IDEALS

DOI: 10.26459/hueuni-jns.v129i1B.5349 5

FIBERS OF RATIONAL MAPS AND REES ALGEBRAS OF THEIR BASE IDEALS

Tran Quang Hoa¹, Ho Vu Ngoc Phuong^2*

1 University of Education, Hue University, 34 Le Loi St., Hue, Vietnam

2 University of Sciences, Hue University, 77 Nguyen Hue St., Hue, Vietnam

* Correspondence to Ho Vu Ngoc Phuong <hvnphuong@husc.edu.vn>

(Received: 03 August 2019; Accepted: 18 March 2020)

Abstract. We consider a rational map : _k^m→ _kⁿ that is a parameterization of an m-dimensional variety. Our main goal is to study the (m−1)-dimensional fibers of  in relation to the m-th local cohomology modules of the Rees algebra of its base ideal.

Keywords: approximation complexes, base ideals, fibers of rational maps, parameterizations, Rees algebras

1 Introduction

Let k be a field and : _k^m→ _kⁿ be a rational map.

Such a map  is defined by homogeneous polynomials

0, , _n,

f f of the same degree d, in a standard graded polynomial ring R= [k X₀, ,X_m], such that

( ,0 , ) = 1.

gcd f f_n The ideal I of R generated by these polynomials is called the base ideal of . The scheme :=Proj R I( / ) _k^m is called the base locus of . Let

= [ ,0 , _n]

B k T T be the homogeneous coordinate ring of

n.

k The map  corresponds to the k-algebra homomorphism :B→R, which sends each Ti _to

i.

f Then, the kernel of this homomorphism defines the closed image of . In other words, after degree renormalization, k f[ ₀, ,f_n] B Ker/ ( ) is the homogeneous coordinate ring of . The minimal set of generators of Ker( ) is called its implicit equations and the implicitization problem is to find these implicit equations.

The implicitization problem has been of increasing interest to commutative algebraists and algebraic geometers due to its applications in Computer Aided Geometric Design as explained by Cox [1].

We blow up the base locus of  and obtain the following commutative diagram:

The variety ^ is the blow-up of _k^m along , and it is also the Zariski closure of the graph of

 in _k^m _kⁿ. Moreover,  is the geometric version of the Rees algebra of I, i.e.,

( ) = .

Proj _As is the graded domain defining , the projection  ₂( ) = is defined by the graded domain k T[ ,₀ ,T_n], and we

6

can thus obtain the implicit equations of from the defining equations of .

Besides the computation of implicit representations of parameterizations, in geometric modeling it is of vital importance to have a detailed knowledge of the geometry of the object and of the parametric representation one is working with.

The question of how many times is the same point being painted (i.e., corresponds to distinct values of parameter) depends not only on the variety itself but also on the parameterization. It is of interest for applications to determine the singularities of the parameterizations. The main goal of this paper is to study the fibers of parameterizations in relation to the Rees and symmetric algebras of their base ideals. More precisely, we set

:= 2 : .

  _|  → _kⁿ

For every closed point y _kⁿ, we will denote its residue field by k y( ). If k is assumed to be algebraically closed, then k y( ) k. The fiber of  at y _kⁿ is the subscheme

1

( ) = ( ( )) ( ) .

⁻ y Proj _Bk y  _{k y}^m _k^m Suppose that m ≥ 2, and  is generically finite onto its image. Then, the set

1

1= { ⁻ ( ) = 1}

−  ⁿ| dim −

m y k y m

consists of only a finite number of points in _kⁿ. For each y _m−1, the fiber of  at y is an

(m−1)-dimensional subscheme of _k^m, and thus the unmixed component of maximal dimension is defined by a homogeneous polynomial h_yR. One of the interesting problems is to establish an upper bound for

1

( )

 −



y m ^{deg y}

h in terms of d. This problem was studied in [2, 3].

The paper is organized as follows. In Section 2, we study the structure of _m−1. Some results in

this section were proved in [2]. The main result of this section is Theorem 2.5 that gives an upper bound for

1

( )

 −



y m ^{deg y}

h by the initial degree of certain symbolic powers of its base ideal. This is a generalization of [3, Proposition 1] where the first author only proved this result for parameterizations of surfaces : _k²→ _k³ under the assumption that the base locus is locally a complete intersection. More precisely, we have the following.

Theorem If there exists an integer s such that

= (( ) ) <

 indeg I^{s sat} sd, then

1

( )  < .

 −



^{deg y} 

y m

h sd

In particular, if indeg I( ^sat) <d , then

1

( ) < .

 −



y ^{deg y} m

h d

In Section 3, we study the part of graded m in X_i of the m-th local cohomology modules of the Rees algebra with respect to the homogeneous maximal ideal m= (X₀, ,X_m)

( , ) 0

= ^m( )_{− m} =_s ^m( )^s _{sd m−} .

N H_m H_m I

The main result of this section is the following.

Theorem (Theorem 3.2) We have that N is a finitely generated B-module satisfying

(i) Supp_B( ) =N _m−1 and dim( ) = 1.N (ii)

1

( ) 1

( ) = .

 −

 + − 

 

 

 



^deg

deg ^y

y m

h m

N m

In the last section, we treat the case of parameterization : _k² → _k³ of surfaces. We establish a bound for the Castelnuovo-Mumford regularity and the degree of the B-module

2

0 2

=_s ( )^s _sd− ,

N H_m I

see Corollary 4.2 and Proposition 4.3.

DOI: 10.26459/hueuni-jns.v129i1B.5349 7 Proposition Assume =Proj R I( / ) is

locally a complete intersection. Then

( ) ( ) 2 ,

3

 + 

   

 

and deg n

reg N n N

where n=dimkH¹_m( / )R I d₋₂. Moreover, if indeg I( ^sat) =d, then

( 3) 2 3.

 d d− + d n

2 Fibers of rational maps

^: _k^m→ _kⁿ

Let n m 2 be integers and R= [k X₀, ,X_m] be the standard graded polynomial ring over an algebraically closed field k. Denote the homogeneous maximal ideal of R by m= (X₀, ,X_m). Suppose we are given an integer d ≥ 1 and n + 1 homogeneous polynomials f₀, ,f_nR_d, not all zero. We may further assume that gcd( ,f₀ ,f_n) = 1, replacing the



f si by their quotient by the greatest common divisor of f₀, ,f_n if needed; hence, the ideal I of R generated by these polynomials is of codimension at least two. Set :=Proj R I( / ) _k^m:=Proj R( ) and consider the rational map

: _k^m− → _kⁿ ( 0( ) : : _n( ))

x f x f x

whose closed image is the subvariety in _kⁿ. In this paper, we always assume that  is generically finite onto its image, or equivalently that the closed image is of dimension m. In this case, we say that  is a parameterization of the

m-dimensional variety .

Let  _{0 k}^m _kⁿ be the graph of

: _k^m \ → _kⁿ and  be the Zariski closure of

0. We have the following diagram

where the maps ₁ and ₂ are the canonical projections. One has

2 0 2

= ( ) = ( ),

where the bar denotes the Zariski closure.

Furthermore,  is the irreducible subscheme of



m n

k k defined by the Rees algebra :=Rees_R( ) =I _s0I^s.

Denote the homogeneous coordinate ring of

n

k by B:= [k T₀, ,T_n]. Set

:= _k = [ ,0 , _n]

S R B R T T

with the grading deg(X_i) = (1, 0) _and ( ) = (0,1)

degT_j for all i = 0,…,m and j=0,...,n. The natural bi-graded morphism of k-algebras

0 0

: = ( ) = ( )

 S→ _s I d ^s _s I sd^s

i i

T f

is onto and corresponds to the embedding

  _k^m _kⁿ.

Let P be the kernel of  . Then, it is a bi- homogeneous ideal of S, and the part of degree one of P in T_i, denoted by P₁=P_{( ,1)} , is the module of syzygies of the I

0 0+ + _{n n} 1 0 0+ + _{n n}= 0.

a T a T P a f a f

Set :=Sym_R( )I for the symmetric algebra of I. The natural bi-graded epimorphisms

1 1

/ ( ) : / ( ) /

→ and →

S S P S P S P

correspond to the embeddings of schemes

  V _k^m _kⁿ, where V is the projective scheme defined by .

8

Let be the kernel of , one has the following exact sequence

0→ → → →0.

Notice that the module is supported in because I is locally trivial outside .

As the construction of symmetric and Rees algebras commutes with localization, and both algebras are the quotient of a polynomial extension of the base ring by the Koszul syzygies on a minimal set of generators in the case of a complete intersection ideal, it follows that  and V coincide on ( _k^m \X) _kⁿ, where X is the (possibly empty) set of points where is not locally a complete intersection.

Now we set :=_2|: → _kⁿ. For every closed point y _kⁿ, we will denote its residue field by k y( ), that is, k y( ) =B_p/pB_p, where p is the defining prime ideal of y. As k is algebraically closed, k y( ) k. The fiber of  at

 _kⁿ

y is the subscheme

1

( ) = ( ( )) ( ) .

⁻ y Proj _Bk y  _{k y}^m _k^m

Let 0 m, we define

= {y _kⁿ | dim⁻1( ) = }y  _kⁿ.

Our goal is to study the structure of . Firstly, we have the following.

Lemma 2.1 [2, Lemma 3.1] Let : _k^m→ _kⁿ be a parameterization of m-dimensional variety and

 be the closure of the graph of . Consider the projection : → _kⁿ. Then

+  .

dim m

Furthermore, this inequality is strict for any l > 0. As a consequence,  has no m-dimensional fibers and only has a finite number of (m−1)_- dimensional fibers.

The fibers of  are defined by the specialization of the Rees algebra. However, Rees algebras are difficult to study. Fortunately, the symmetric algebra of I is easier to understand than , and the fibers of  are closely related to the fibers of

:= 2 : .

^'  _|V V→ _kⁿ

Recall that for any closed point y _kⁿ, the fiber of ^' at y is the subscheme

1

( ) = ( ( )) ( ) .

^'− y Proj _Bk y  _{k y}^m _k^m The next result gives a relation between fibers of  and ^' – recall that X is the (possible empty) set of points where is not locally a complete intersection.

Lemma 2.2 [2, Lemma 3.2] The fibers of  and ^' agree outside X, hence they have the same

(m−1)-dimensional fibers.

The next result is a generalization of [4, Lemma 10] that gives the structure of the unmixed part of a (m−1) -dimensional fiber of . Note that our result does not need the assumption that is locally a complete intersection as in [4], thanks to Lemma 2.2. Recall that the saturation of an ideal J of R is defined by J^sat:=J:_R ( )m ^_.

Lemma 2.3 [2, Lemma 3.3] Assume such that p_i = 1. Then, the unmixed part of the fiber

1( )

⁻ y is defined by

0 0

=gcd( − , , − ).

y i n n i

h f p f f p f

Furthermore, if f_j−p f_{j i} =h g_{y j} for all j ≠ i, then

0 1 1

= ( )_i + _y( , , _i− , _i+, , _n) and ^sat ( ,_{i y}).

I f h g g g g I f h

Remark 2.4 The above lemma shows that the (m−1)-dimensional fibers of  can only occur when

  as V f h( ,_{i y}). It also shows that

DOI: 10.26459/hueuni-jns.v129i1B.5349 9 ( ) ( ),

deg _y deg

d h

if there is a (m – 1)-dimensional fiber with unmixed part given by h_y. As a consequence, deg(h_y) <d _{for any}

1.

 _m−

y

By Lemma 2.1,  only has a finite number of (m−1)-dimensional fibers. The following gives an upper bound for this number in terms of the initial degree of certain symbolic powers of its base ideal. Recall that the initial degree of a graded R- module M is defined by

( ) :=inf{  | _n0}

indeg M n M

with convention that sup += .

Theorem 2.5 If there exists an integer s1 such that  =indeg I(( )^{s sat}) <sd, then

1

( )  < .

 −



^{deg y} 

y m

h sd

In particular, if indeg I( ^sat) <d, then

1

( ) < .

 −



^{deg y}

y m

h d

Proof. As _m−1 is finite, by Lemma 2.3, there exists a homogeneous polynomial fI of degree

d such that, for any y _m−1,

= ( )+ _y( 1_y, , _ny) and ^sat( , _y)

I f h g g I f h

for some g_1y, ,g_ny _{R. Since} ( ,f h_y) is a complete intersection ideal, it follows from [5, Appendix 6, Lemma 5] that ( ,f h_y)^s is unmixed, hence saturated for every integer s ≥ 1. Therefore, for all y _m−1,

1

( ) (( ) ) (( , ) ) = ( , )

= ( , , , ).

s sat sat s sat s sat s

y y

s s s

y y

I I f h f h

f f ⁻h h

 

Now, let 0 F ( )I^{s sat} such that deg(F) = v

< sd, then h_y is a divisor of F. Moreover, if yy^' in _m−1, then gcd( _y, _') = 1.

h hy We deduce that

 −1



^y|

y m

h F

which gives

1

( ) ( ) =< .

 −



^{deg y} ^deg

y m

h F sd

Remark 2.6 In the case where : _k²→ _k³ is a parameterization of surfaces. In [3], the first author showed that if is locally a complete intersection of dimension zero, then

1

4 = 3,

( )

1 4.

 2



     − 



^{deg y if}_if

y

d

h d

d d

Example 2.7 Consider the parameterization

2 3

: _k → _k of surface given by

0 0 1 0 2 0 2 0 2

1 0 1 1 2 1 2 1 2

2 0 2 0 2 0 2 0 2

3 1 2 1 2 1 2 1 2

= ( )( )( 2 )

= ( )( )( 2 )

= ( )( )( 2 )

= ( )( )( 2 ).

− + −

− + −

− + −

− + −

f X X X X X X X X

f X X X X X X X X

f X X X X X X X X

f X X X X X X X X

Using Macaulay2 [6], it is easy to see that

= ^sat

I I _and indeg I(( ²)^sat)= 8 < 2.5 = 10. Furthermore, I admits a free resolution

where matrix M is given by

2 1 2 1 2 1 2

2 0 2 0 2 0 2

1 0

0 ( )( )( 2 )

0 ( )( )( 2 )

0 0 .

0 0

− − + −

 

 − − − + − 

 

 

 

 

 

X X X X X X X

X X X X X X X

X X

Thus, we obtain ₁= { ,p p p p p p p p_{1 2}, ₃, ₄, ₅, ₆, ₇, ₈} with

10

1 0 2 1

1 2

3 3 0 2 4 4 0 2

5 0 2 6 1 2

5 6

7 7 1 2 8 8 1 2

= (0 : 0 : 0 :1) = = (0 : 0 :1: 0) =

= (0 :1: 0 :1) = = (0 : 1: 0 :1) =

= (0 : 2 : 0 :1) = 2 = (1: 0 :1: 0) =

= ( 1: 0 :1: 0) = = (2 : 0 :1: 0) = 2 .

h X h X

h X X h X X

h X X h X X

h X X h X X

− − +

− −

− + −

p p

p p

p p

p p

p p

p p

p p

p p

Consequently, we have ²

1

( ) = 8 = (( ) ).



 ^{deg y sat} y

h indeg I

3 Local cohomology of Rees algebras of the base ideal of parameterizations

Let : _k^m→ _kⁿ be a parameterization of m-dimensional variety. Let R= [k X₀, ,X_m] and

= [ ,0 , _n]

B k T T be the homogeneous coordinate ring of _k^m and _kⁿ, respectively. For every closed point y _kⁿ, the fiber of  at y is the subscheme

1

( ) = ( ( )) ( )

⁻ y Proj _Bk y  _{k y}^m _k^m and we are interested in studying the set

1

1= { ⁻ ( ) = 1}.

−  ⁿ|dim −

m y k y m

We now consider the B-module

( , ) 0

= ( ) = ( ) ,

 ^m  _s ^{m s} +_sd

M H_m H_m I

where m= (X₀, ,X_m) is the homogeneous maximal ideal of R. By [7, Theorem 2.1], M_ is a finitely generated B-module for all  _{. The} following result gives a relation between the support of M_ and _m−1. For each

= ( )

 _kⁿ

y Proj B , we can see y as a homogeneous prime ideal of B.

Proposition 3.1 One has

1

( _) = {  ₋1| deg(⁻ ( )) + + 1}.

B m

Supp M y y m

Proof. As k is algebraically closed, we have

1

( ) = ( ( )) ( ) .

⁻ y Proj _Bk y  _{k y}^m _k^m

Therefore, the homogeneous coordinate ring of ₂⁻¹( )y is

( ) / ,

_Bk y R J

where J is a satured ideal of R depending on .

y Let y _m−1_{. As} dim⁻¹( ) =y m−1, one has

(  ( )) = / = .

dim _Bk y dimR J m

Since dimR = m + 1, there exists a homogeneous polynomial f of degree d_f such that J= ( )f J^', with codim J( ) 2. Notice that f is exactly the defining equation of unmixed part of ⁻¹( ).y Consider the exact sequence

0→( ) /f J→R J/ →R/ ( )f →0

which deduces the exact sequence in cohomology

1

0 = (( ) / ) ( / )

( / ( )) (( ) / ) = 0,

m m

m m

H f J H R J

H R f H ⁺ f J

→

→ →

m m

m m

since codim J( )  2, hence ( ) /f J is of dimension at most m−1. It follows from the above exact sequence that

(  ( )) ( / ) ( / ( )).

m m m

H_m Bk y H_m R J H_m R f (3.1) We consider the following exact sequence

which implies the exact sequence in cohomology

In degree  , one has the following exact sequence

DOI: 10.26459/hueuni-jns.v129i1B.5349 11 (3.2)

On the other hand,

1 1 1 1

0 0

( ) ( ) [ , , ],

+ − − −

m

m m

H_m R X X k X X

hence

1

1 1

( )_ ( )_ := ( _ , ).

+ 

+ + − − −

m

m R m

H_m R R Homgr R k

It follows that H_m^m+1( ) = 0R _ for all

> 1

 ^{− −}m and H_m^m+1( )R _ 0 for any ^{ − −}m 1. It follows from (3.2) that

1

( / ( )) =

0 > 1

( / ( )) 0 1.

if if

m

f

df m f

H R f

d m

R f d m







 

− + +

 − −

   − −



m

(3.3)

By definition, M_ is a graded B-module and Supp_B(M_)Proj B( ). Now let pProj B( ), we have

( _) _ 0

Supp_B M M _BB_p p

( / ) 0

M_BB_p_B B p 

( , )

( )_{ } ( ) 0

H_m^m _Bk p 

( , )

( ( ))_{ } 0.

H_m^m _Bk p 

In particular, H_m^m( _Bk( ))p 0, _hence

(  ( )) =

dim _Bk p m which shows that p _m−1. It follows from (3.1) and (3.3) that

(⁻1( )) + + 1.

deg p m

In particular, if =−m, then the finitely generated B-module

=_s0 ^m( )^s _{sd m−}

N H_m I

satisfies Supp_B( ) =N _m−1 by Proposition 3.1.

Furthermore, we have the following.

Theorem 3.2 Let N be the finitely generated B-module as above. Then

(i) dim( ) = 1.N (ii)

1

( ) 1

( ) = .

 −

+ −

 

 

 

 



^deg

deg ^y

y m

h m

N m

Proof. Let y= (p₀:p₁: :p_n) _m−1. Without loss of generality, we can assume that

0= 1.

p Hence,

1 1 0 0

= (T−p T, ,T_n−p T_n )B p

is the defining ideal of y. For any fB, we have

1 1 1 0 0 0

= ( − )+ + _n( _n− _n )+ forsome  [ ].

f g T p T g T p T v v k T

It follows that f +p=v+p. This implies that B/p k T[ ₀]. Therefore,

( / ) = 1  ₋1

dim B p for anyp _m

and thus,

( )

( ) = ( / ) = 1

dim max dim

SuppB N

N B

p

p

which shows (i). We now prove for item (ii). It was known that

( ) = ( ) =dimk =dimk ^m( )^s ₋

N N s sd m

HP s HF s N H_m I

for all s 0 , where HP_N and HF_N is the Hilbert polynomial and the Hilbert function ofN, respectively. As dimN= 1, the Hilbert polynomial of N is constant, which is equal to deg( ).N On the other hand,

( / )=1

( ) =



( ). ( / ).

dim

deg _B deg

B

N length N_p B

p p

p

We proved that B/p k T[ ₀], hence ( / ) = 1

dimB p and deg( / ) = 1B p for the defining ideal p _of y _m−1. Therefore,

1

( ) = ( ).

 −



deg _B

y m

N length N_p

p

As N^p is an Artinian B^p-module and ( / ) = ( [ 0]) = 1

dimk B ps dimk k T s for any s0, one has

12

( ) =dimk(  )

B B

length N_p N B_p

p

=



^dimk( ^B )^s

s

N B_p

( , )

=



^dimk( ^m( )^B )₋^{m s}

s

H_m B_p

( , )

=



^dimk( ^m( )^B )₋^{m s}.^dimk( / )^s

s

H_m B_p B p

( , )

=



^{dimk m}( ^B ^B / )₋^{m s}

s

H_m B_p B p

( , )

=



^{dimk m}( ^B ( ))₋^{m s}

s

H_m k p

(3.1)

= dimkH_m^m( / ( ))R f ₋m (3.3)

= dimk( / ( ))d ₋1

R f f

=dimk dR f₋1 since deg( ) =f df

= + −1.

 

 

 

df m m

It follows that

1

( ) 1

( ) = .

 −

+ −

 

 

 

 



^deg

deg ^y

y m

h m

N

m

4 Parameterization

: _k² → _k³

of surfaces

In this section, we consider a parameterization

2 3

: _k → _k of surface defined by four homogeneous polynomials f0, ,f3R= [k X0,X X1, 2] of the same degree d such that gcd( ,f₀ ,f₃) = 1. Denote the homogeneous maximal ideal of R by

0 1 2

= (X ,X X, ).

m From now on we assume that is

locally a complete intersection. Under this hypothesis, the module is supported in mS, hence Hⁱ_m( ) = 0 for any i1. The exact sequence

0→ → → →0

deduces that

( ) ( ), 1.

i i

H_m H_m i

Let B= [ ,k T₀ ,T₃] be the homogeneous coordinate ring of _k³. It follows from Theorem 3.2 that the finitely generated B-module

2 2 2

0 2 ( 2, ) ( 2, )

:=_s ( )^s _sd− = ( )_{− } ( )_{− }

N H_m I H_m H_m

satisfying dim( ) = 1N and

3 1

( ) = 1= {  | dim⁻ ( ) = 1}.

B k

Supp N y y

Furthermore,

2 2 1

( ) 1

= ( ) = ( )

2 ⁻



+

 

 

 

 



^{deg y deg dimk s sd}

y

h N H_m I

for s ≥ reg(N) + 1, where reg(N) is the Castelnuove- Mumford regularity of N. Thus, it is useful to establish the bounds for deg(N) and reg(N).

Let K_•:=K_•( ; )f R and Z_•:=Z_•( ; )f R be the Koszul complex and the module of cycles associated to the sequence f:= f₀, ,f₃ with coefficients in R, respectively. Since the ideal

= ( )f

I is homogeneous, these modules inherit a natural structure of graded R-modules. Let _• be the approximation complex associated to I. The approximation complexes were introduced by Herzog, Simis and Vasconcelos in [8] to study the Rees and symmetric algebras of ideals. By definition

0 3

= [ ] [ , , ](− )

q Z qdq RR T T q

for all q= 0, ,3 with deg(X_i) = (1, 0) and ( ) = (0,1)

degT_i . This complex is of the form ( _•)

where v a a a a₁( ₀, ₁, ₂, ₃) =a T_{0 0}+ +a T_{3 3} . As is locally a complete intersection, the complex ( _•)

DOI: 10.26459/hueuni-jns.v129i1B.5349 13 is acyclic and is a resolution of H₀( _•) , see

[9, Theorem 4].

Proposition 4.1 Assume is locally a complete intersection. Then N admits a finite representation of free B-modules

( 2)− ^m→ −( 1)ⁿ→ →0,

B B N

where

1

=dimk ( / )d₋2

n H_m R I and m=dimkH_m³(Z_{2 2}) d₋₂. Proof. We consider the two spectral sequences associated to the double complex C_m^•( _•), where

( )

C_m• M denotes the Čech complex on M

relatively to the ideal m.Since ( _•) is acyclic, one of them abuts at step two with:

2

( ) = 0

= =

0 0.

 

 



for for

p

h p h p

q q

H q

E E

q

m

The other one gives at step one:

1 = ( ) = ( )[ ] [ 0, , 3]( )

= ( )[ ] ( ).

 −

 −

v p p p

q q q R

p

q k

E H H Z qd R T T q

H Z qd B q

m m

m

By [9, Lemma 1], H_m^p(Z_q) = 0 for p= 0,1 and by definition, Z₃ R[ 4 ]− d and Z₀=R. Therefore, the first page of the vertical spectral sequence has only two nonzero lines

In bi-degree ( 2, )−  _{, we have}

3 3

0 2 2

( )₋ _k = ( )₋ _k = 0.

H_m Z B H_m R B Therefore, we

obtain the complex (C_•) of free B-modules

Notice that

3 2 1

1 2 2 2

=dimk ( )d₋ =dimk ( )d₋ =dimk ( / )d₋ .

n H_m Z H_m I H_m R I

It remains to show that H C₁( _•) =N. It is easy to see that

=2 = = 3 = 2, = 1.

^vE_q^{p v}E_q^p unless p q or p q Therefore,

3 2

2 1

=2 =2

= = ( ) = ,

 

− −



^{v qp v}



^{h qp}

p q p q

E E H_m E

in other words,

2 2

1( _•) = ( )( 2, )_{− } = ( )( 2, )_{− } = .

H C H_m H_m N

We now establish a bound for the Castelnuove-Mumford regularity and the degree

of B-module N in terms of n=dimkH¹_m( / )R I d₋₂

as follows.

Corollary 4.2 Suppose is locally a complete intersection. Then

( ) ( ) 2 .

3 +

 

   

 

and deg n

reg N n N

Proof. As dimB= 4, hence codim N( ) = 3 and by Proposition 4.1, N admits a finite representation

( 2)− ^m→ −( 1)ⁿ→ →0.

B B N

The corollary follows from [10, Corollaries 2.4 and 3.4].

Theorem 2.5 shows that if indeg I( ^sat) <d , then

1

( ) < .



 ^{deg y} y

h d

Hence, the delicate case is when the ideal I satisfies indeg I( ^sat) =indeg I( ) = .d In this case, the first author in [3] established an upper bound for

1

=dimk ( / )d₋2

n H_m R I in terms of d as follows.

14

Proposition 4.3 Assume is locally a complete intersection and indeg I( ^sat) =d. Then

1 2

( 1) ( ) 2 3

2d d+ deg d − d+ and

1 ( 3)

= ( ) ( 1) 3.

2 2

 deg − − d d− +

d n d d

References

1. Cox D. What is the role of algebra in applied mathematics?. Notices of the AMS.2005;52(10):1193- 1198.

2. Chardin M, Cutkosky SD, Tran QH. Fibers of rational maps and jacobian matrices. Journal of Algebra. 2019.

3. Tran QH. Bound for the number of one-dimensional fibers of a projective morphism. Journal of Algebra.

2018;494:220-236.

4. Botbol N, Busé L, Chardin M. Fitting ideals and multiple points of surface parameterizations.

Journal of Algebra. 2014;420:486-508.

5. Zariski O, Samuel P. Commutative algebra, Vol II.

Berlin: Springer; 1960.

6. Daniel G, Michael S, David E. Macaulay2. Version 1.16. Illinois: National Science Foundation; 2020.

7. Chardin M. Powers of ideals and the cohomology of stalks and fibers of morphisms. Algebra & Number Theory. 2013;7(1):1-18.

8. Herzog J, Simis A, Vasconcelos W. Approximation complexes of blowing-up rings. Journal of Algebra.

1982;74(2):466-493.

9. Busé L, Chardin M. Implicitizing rational hypersurfaces using approximation complexes.

Journal of Symbolic Computation. 2005;40(4-5):

1150-1168.

10. Chardin M, Fall AL, Nagel U. Bounds for the Castelnuovo–Mumford regularity of modules.

Mathematische Zeitschrift. 2007;258(1):69-80.