Multigraded regularity of complete intersections
aa r X i v : . [ m a t h . A C ] D ec MULTIGRADED REGULARITY OF COMPLETEINTERSECTIONS
MARC CHARDIN AND NAVID NEMATI
Abstract. V is a complete intersection scheme in a multiprojective space if itcan be defined by an ideal I with as many generators as codim( V ). We inves-tigate the multigraded regularity of complete intersections scheme in P n × P m .We explicitly compute many values of the Hilbert functions of 0-dimensionalcomplete 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 upperbound for the multigraded regularity of 0-dimensional complete intersections. Introduction
The theory of syzygies offers a microscope for looking at systems of equations.Castelnuovo-Mumford regularity is an important invariant in commutative algebraand algebraic geometry which is strongly related to syzygies. D. Eisenbud and S.Goto in [2] showed that Castelnuovo-Mumford regularity can be obtained from theminimal free resolution. The minimal free resolution was first introduced by Hilbertto study Hilbert function. It is one of the finest invariant that we can associatewith a finitely generated graded module M over a polynomial ring. Castelnuovo-Mumford regularity measures the maximum degree of the syzygies and provides aquantitative version of Serre vanishing theorem for the associated sheaf. In par-ticular, it bounds the largest degree of the minimal generators and the smallesttwist for which the sheaf is generated by its global sections. It has been used as ameasure for the complexity of computational problems in algebraic geometry andcommutative algebra. The two most frequent definitions of Z -graded Castelnuovo-Mumford regularity are the one in terms of graded Betti numbers and the one usinglocal cohomology.An extension of Castelnuovo-Mumford regularity for the multigraded setting ina special case was first introduced by Hoffman and Wang in [6]. Later by Maclaganand Smith in [8], and Botbol and Chardin in [1] in a more general setting. The mainmotivation for studying regularity over multigraded polynomial rings was from toricgeometry. Maclagan and Smith [8] developed a multigraded theory of regularity ofsheaves on a simplicial toric variety X with an algebraic variant defined in termsof the vanishing of graded pieces of H iB ( M ), the i -th local cohomology module of M . Here B is the irrelevant ideal of the homogeneous coordinate ring of X .The conceptual difficulty of computing multigraded regularity lies in the simplefact that bounded subsets of Z r need not have single maximal or minimal ele-ments. This makes it very hard to capture the vanishing or non-vanishing of the Mathematics Subject Classification.
Key words and phrases.
Multigraded regularity, Hilbert function, multiprojective space, com-plete intersection. multigraded pieces of local cohomology. One can define a complete intersectionin projective space as a subscheme defined by as many forms as its codimension.In this case, it corresponds to complete intersection homogeneous ideals; thereforemany homological invariants are determined by the degrees of the forms. One natu-ral question that may arise is to ask the same question for the complete intersectionschemes in a product of projective spaces.Proposition 6.7 in [8] implies that the multigraded graded regularity a 0-dimensionalschemes is the same as the stabilization region of the Hilbert functions. Understand-ing the Hilbert functions and the minimal free resolutions of the coordinate ringsof points in multiprojective space is included among the list of open problems incommutative algebra found in the survey article of Peeva-Stillman[9].Based on the above motivations, our main concern in this article is to studymultigraded regularity and bigraded Hilbert function complete intersection pointsin P n × P m . In Section 2, we start by setting our notation and define multigradedregularity. We show that if the ring is bigraded, then one can consider an alternativeway to define the multigraded regularity (see Theorem 2.5).In section 3, we study the multigraded regularity of general complete intersectionschemes in P n × P m . A scheme V ⊂ P n × P m is complete intersection if it can bedefined by as many forms as its codimension. Our motivation is to show whichhomological invariants are determined by the degrees of these forms.In Section 4, we focus on the case of complete intersection points in P n × P m .The main result of this section is Theorem 4.3. For a complete intersection schemeof point V ⊆ P n × P m defined by f , . . . , f n + m , many values of its bigraded Hilbertfunction are independent from the choices of f i ’s and they only depend upon theirbidegrees. Moreover, in some regions, there is a duality among these values. Example A. (Example 4.4) Let S = k [ x , x , x , y , y , y ] , and I = ( f , . . . , f ) isgenerated by bigraded forms of bidegree (2 , such that proj( S/I ) = V is completeintersection scheme of points . For any µ / ∈ (2 ,
6) + ( − N , N ) ∪ (6 ,
2) + ( N , − N ) ,the Hilbert function of V at µ is independent from the choices of f i ’s and can becomputed explicitly via Theorem 4.3. In Section 5, we study generic complete intersection points in P n × P m . Themain Theorem of this section is the following Theorem A. (Theorem 5.5) Let S = k [ x , . . . , x n , y , . . . , y m ] be a bigraded poly-nomial ring over a field k of characteristics zero. If I is generated by n + m genericforms of bidegree ( d, e ) , then the scheme V defined by I is a set of reduced pointsand |{ µ ∈ N | HF V ( µ ) = deg( V ) }| < ∞ , which means the natural projections are one-to-one. 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 , . . . , x n , y , . . . , y m ] be a bigraded polynomial ring where deg( x i ) =(1 , and deg( y i ) = (0 , . Let B = ( x , . . . , x n ) , and B = ( y , . . . , y m ) be theirrelevant ideal of P n and P m . Define B := B · B the irrelevant ideals of P n × P m ,and m = B + B 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].
ULTIGRADED REGULARITY OF COMPLETE INTERSECTIONS 3
Definition 2.2.
Let M be a graded S -module. The support of the module M is Supp( M ) := { γ | M γ = 0 } . Definition 2.3. (Multigraded regularity) Let C ⊆ m be a finitely generated bigradedideal of S . Define for a γ ∈ Z , M is ( C, γ ) -regular if γ + N ∩ ∪ i Supp( H iC ( M )) + F i − = ∅ , where F i := { ( i − , , ( i − , , . . . , (0 , i − } for i > and F := { } , F − = −F ,and F i = 0 for i < − . C -regularity of M is reg C ( M ) := { γ ∈ Z r | M is ( C, γ ) -regular } . In particular, if C = B , B -regularity of M , reg B ( M ) , is called the multigradedregularity of M . Our approach for computing multigraded regularity relies on Mayer-Vietoris se-quence to relate the various cohomology modules.
Remark 2.4.
Let S = k [ x , . . . , x m , y , . . . , y n ] be a bigraded polynomial ring. De-fine B , B , B and m as in Notation 2.1. Then we have the following complexbetween local cohomology modules: · · · → H i − B ( M ) → H i m ( M ) → H iB ( M ) ⊕ H iB ( M ) → H iB ( M ) → · · · The following Theorem provide an alternative way to define multigraded regu-larity in the bigraded setting by knowing m , B , and B -regularity. Theorem 2.5.
Let S = k [ x , . . . , x m , y , . . . , y n ] be a bigraded polynomial ring and M is a graded S -module. Adopt Notations 2.1, then reg B ( M ) = reg m ( M ) ∩ reg B ( M ) ∩ reg B ( M ) . Proof.
By analyzing the spectral sequences correspond to ˇCech-Koszul double com-plex C • B K • ( x , y ; M ) we getSupp(Tor Si ( M, k )) ⊆ ∪ ij =0 Supp( H jB ( M )) + E i + j where E i denote the set of twists of the summands in the i -th step of the minimalfree resolution of B ; see [1, Section 4.2] for more details regarding the relationbetween B -regularity and the support of Tor modules. By [1, Corollary 3.12], forevery i we haveSupp( H i m ( M )) ⊆ ∪ Supp(Tor Sm + n − i ( M, k )) + Supp( H m + n m ( S )) ⊆ ∪ Supp(Tor Sm + n − i ( M, k )) − ( m, n ) − N ⊆ ∪ ∪ ij =0 Supp( H jB ( M )) + E m + n − ( i − j ) − ( m, n ) − N ⊆ ∪ ij =0 Supp( H jB ( M )) + E i − i − N . Therefore Supp( H i m ( M )) + F i − ⊆ ∪ ij =0 Supp( H jB ( M )) + F j − − N , which means(2.1) reg B ( M ) ⊆ reg m ( M ) . Suppose µ ∈ reg B ( M ). By the definition, (cid:0) µ − F i − + N (cid:1) ∩ Supp H iB ( M ) = ∅ for every i . By 2.1, (cid:0) µ − F i − + N (cid:1) ∩ Supp H i m ( M ) = ∅ . By using Mayer-Vietoris MARC CHARDIN AND NAVID NEMATI exact sequence one gets (cid:0) µ − F i − + N (cid:1) ∩ (cid:0) Supp H iB ( M ) ∩ Supp H iB ( M ) (cid:1) = ∅ ,that implies reg B ( M ) ⊆ reg m ( M ) ∩ reg B ( M ) ∩ reg B ( M ) . Suppose µ ∈ reg m ( M ) ∩ reg B ( M ) ∩ reg B ( M ). By the definition, for every i , (cid:0) µ − F i − + N (cid:1) ∩ (cid:0) Supp H iB ( M ) ∩ Supp H iB ( M ) (cid:1) = ∅ , and (cid:0) µ − F i + N (cid:1) ∩ Supp H i +1 m ( M ) = ∅ . Since (cid:0) µ − F i − + N (cid:1) ⊂ (cid:0) µ − F i + N (cid:1) , (cid:0) µ − F i − + N (cid:1) ∩ Supp H i +1 m ( M ) = ∅ . The assertion follows from Mayer-Vietoris exact sequence andthe definition of reg B ( M ). (cid:3) Complete intersections in P n × P m We start this section by stating the definition of complete intersection scheme ina product of projective that we are using in this article.
Definition 3.1.
A subscheme V ⊆ P n × P m , is a complete intersection if V =proj( S/I ) where I is generated by codim( V ) bihomogeneous elements. Let V be a complete intersection scheme in a projective space, then there existideal 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 entirelydetermined by the degrees of the generators. Let V = proj( S/I ) ⊂ P n × P m bea complete intersection scheme, where I is generated by codim( V ) forms. In thiscase, I is not complete intersection unless n, m = 1 in which case I is not thedefining ideal of V (it is not B -saturated). Although the similar argument wouldnot work in this case, but our goal is to show that many homological invariants areonly depend upon the degrees of the generators of I . Proposition 3.2. [4, Example 8.4.2]
Let S = k [ x , . . . , x n , y , . . . , y m ] be a bigradedpolynomial ring where deg( x i ) = (1 , and deg( y i ) = (0 , . Let I = ( f , . . . , 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 ) = X d i · · · d i n · e j · · · e j m , where the sum is over all permutations ( d i , . . . , d i n , e j , . . . , e j m ) of (1 , . . . , n + m ) with i < i < · · · < i n and j < j < · · · < j m . In particular if d i = d and e i = e for all i , then deg( V ) = (cid:18) n + mn (cid:19) d n e m . Proposition 3.3.
Let I = ( f , . . . , f r ) be a bigraded ideal defining a completeintersection of codimension r and let deg( f i ) = d i . For µ ∈ ∩ i reg B ( H i ( K ( f , S )) , HF S/I ( µ ) = P ( µ ) , where P is a polynomial that only depends upon d , . . . , d r .Proof. By the Serre Grothendieck formula [1, Proposition 4.27](3.1) HP S/I ( µ ) = HF S/I ( µ ) + X i ( − i dim H iB ( S/I ) µ . ULTIGRADED REGULARITY OF COMPLETE INTERSECTIONS 5
Hence, if µ ∈ reg B ( S/I ) then HP S/I ( µ ) = HF S/I ( µ ). Set(3.2) χ ( µ ) := X i ( − i dim( H i ( K ( f , S )) µ = X i ( − i dim( K i ) µ . Note that the second equality shows that χ is a function that only depends upon thedegrees. Since I is complete intersection, H i ( K ( f , S ) = H B ( H i ( K ( f , S )). Hence,Supp( H i ( K ( f , S )) ∩ reg B ( H i ( K ( f , S )) = ∅ . Therefore, for µ ∈ ∩ i> reg B ( H i ( K ( f , S )), χ ( µ ) = HF S/I ( µ ) . Hence, for µ ∈ ∩ i reg B ( H i ( K ( f , S )), HP S/I ( µ ) = HF S/I ( µ ) = χ ( µ ) . Setting P := HP S/I the first equality shows that HF
S/I ( µ ) = P ( µ ) and the secondequality shows this function only depends upon the degrees. (cid:3) Proposition 3.4.
With the Notation 2.1, for any bigraded free S -module M , H n + m +1 B ( M ) ∼ = H n + m +2 m ( M ) . Furthermore, If n = m then H n +1 B ( M ) ∼ = H n +1 B ( M ) ⊕ H n +1 B ( M ) , else if n < m then, H n +1 B ( M ) ∼ = H n +1 B ( M ) and H m +1 B ( M ) ∼ = H m +1 B ( M ) . Proof.
Note that H aB ( M ) = H bB ( M ) = H c m ( M ) = 0 if a = n + 1, b = m + 1 and c = n + m + 2. The Mayer-Vietoris exact sequence · · · → H i m ( M ) → H iB ( M ) ⊕ H iB ( M ) → H iB ( M ) → H i +1 m ( M ) → · · · gives the results. (cid:3) Theorem 3.5.
Let S = k [ x , . . . , x n , y , . . . , y m ] be a bigraded polynomial ringwhere deg( x i ) = (1 , and deg( y i ) = (0 , . Define B = ( x , . . . , x n ) · ( y , . . . , y m ) the irrelevant ideal of P n × P m . Assume I = ( f , . . . , f r ) with deg( f i ) = ( d i , e i ) and V be the scheme defined by I . If codim( V ) = r then • Supp( H i ( K ( f , S )) = Supp( V n +1+ i ) ∪ Supp( W m +1+ i ) for i > • Supp( H iB ( S/I )) = Supp( V n +1 − i ) ∪ Supp( W m +1 − i ) for ≤ i ≤ dim V, • Supp( H dim V +1 B ( S/I )) = { ( µ, ν ) | H dim V ( V, O V ( µ, ν )) = 0 }⊆ Supp( V r − m ) ∪ Supp( W r − n ) ∪ { σ − Supp (
S/I ) } , where V i (resp. W i ) is a subquotient of H n +1 B (K i ( f , S )) (resp. H m +1 B (K i ( f , S )) ),and σ := P i ( d i , e i ) − ( n + 1 , m + 1) .Proof. Consider the double complex C • B ( K • ( f , S )) and suppose n ≤ m . If we starttaking homologies vertically, by 3.4 in the third page we have:0 0 · · · V r V r − · · · V V ... ... ... ... ... W r W r − · · · W W ... ... ... ... ... M · · · , MARC CHARDIN AND NAVID NEMATI where M ∼ = H r ( H n + m +1 B ( K ( f , S )) ∼ = H r (cid:0) H n + m +2 m ( K ( f , S )) (cid:1) ∼ = H r (cid:0) ( K ( f , S ) ( σ )) ⋆ (cid:1) ∼ = ( S/I ) ⋆ ( σ ) . If we start taking homology horizontally, in the second page we have · · · H γ · · · H H B ( S/I ) · · · H B ( S/I ) · · · H dim( V )+1 B ( S/I )... ... ... ... ... 0 , where γ := max { , r − n − , r − m − } . The result is obtained by comparing thetwo abutments. (cid:3) Definition 3.6.
Let d = ( d, e ) ∈ N and r ∈ N , define v i := i · d − ( n + 1 , ,w i := i · d − (0 , m + 1) ,σ := r · d − ( n + 1 , m + 1) . Lemma 3.7.
Let I = ( f , . . . , f r ) ⊆ S and deg( f i ) = d = ( d, e ) . Suppose V is thescheme defined by I , if codim( V ) = r then Supp( H i ( K ( f , S )) ⊆ v n +1+ i + ( − N , N ) ∪ w m +1+ i + ( N , − N ) for i > , Supp( H iB ( S/I )) ⊆ v n +1 − i + ( − N , N ) ∪ w m +1 − i + ( N , − N ) for ≤ i ≤ dim V. In addition, the inclusions are sharp.Proof.
With the proof of Theorem 3.5, in the second page of the spectral sequencewe have for p < n + m + 1 v E ∞ p,q = v E p,q = H q ( H n +1 B K ( f , S )) if p = n + 1 ,H q ( H n +1 B K ( f , S )) if p = m + 1 , . Combining with the Theorem 3.5 shows the inclusions. The sharpness follows fromthe fact that for any q , v q (resp. w q ) is in the support of H n +1 B K q ( f , S ) (resp. H m +1 B K q ( f , S )) and it is not in the support of H n +1 B K q − ( f , S ) and H n +1 B K q +1 ( f , S )(resp. H m +1 B K q − ( f , S ) and H m +1 B K q − ( f , S )). (cid:3) Corollary 3.8.
Let I = ( f , . . . , f r ) ⊆ S and deg( f i ) = d = ( d, e ) and V be thescheme defined by I . If codim( V ) = r , then (1) if µ / ∈ v i +( − N , N ) ∪ w j +( N , − N ) for i = r − m, r − m − and j = r − n, r − n − then H dim( V )+1 B ( S/I )) µ = ( S/I ) ⋆σ − µ . ULTIGRADED REGULARITY OF COMPLETE INTERSECTIONS 7
Proof.
Note thatSupp (cid:0) H n +1 B ( K i ( f , S )) (cid:1) = v i +( − N , N ) and Supp H m +1 B ( K i ( f , S )) = w i + ( N , + N ) for all i . Consider the double complex C • B ( K • ( f , S )) and suppose n ≤ m as in the proof of Theorem 3.5. Since Supp( V i ) ∩ Supp( W j ) = ∅ , there willbe no nonzero map in the spectral sequence among them. In this case, in the n +2-thpage we have an induced map ψ : M → W r − n − , and in the m + 2-th page we havean induced map ψ : ker ψ → V r − m − . If µ / ∈ v r − m + ( − N , N ) ∪ w r − n + ( N , − N ), H rB ( S/I ) µ ∼ = (ker ψ ) µ . In addition, if µ / ∈ v r − m − + ( − N , N ) ∪ w r − n − + ( N , − N ) H dim( V )+1 B ( S/I ) µ ∼ = (ker ψ ) µ = M µ = ( S/I ) ⋆σ − µ . (cid:3) Complete intersection points in P n × P m In this section we apply the results in the previous section to the case of completeintersection points in P n × P m . In the rest of this section, I = ( f , . . . , f n + m ) and V be the complete intersection scheme of points defined by I with d := deg( f i ) =( d, e ). Definition 4.1.
Set Γ i := Supp( H n +1 B (K n + i ( f , S ))) ∪ Supp( H m +1 B (K m + i ( f , S ))) .Notice that Γ i = ∅ if and only if i ≥ max { n, m } . If i ≤ m then Supp( H n +1 B (K n + i ( f , S ))) = v n + i + ( − N , N ) , and similarly for Supp( H m +1 B (K m + i ( f , S ))) if i ≤ n . Definition 4.2.
For a function F : Z → Z define F ⋆ ( a, b ) := F ( − a, − b ) ,F ′ ( a, b ) := F ( − a, b ) , and F ′′ ( a, b ) := F ( a, − b ) . Theorem 4.3.
Let S = k [ x , . . . , x n , y , . . . , y m ] be a bigraded polynomial ringwhere deg( x i ) = (1 , and deg( y i ) = (0 , . Assume V be a complete intersectionscheme of points defined by I = ( f , . . . , f n + m ) with deg( f i ) = ( d, e ) . (1) If µ / ∈ ∪ i ≥ Γ i then HF S/I ( µ ) = χ ( µ ) where χ (( a, b )) = P i ( − i (cid:0) n + mi (cid:1)(cid:0) n + a − idn (cid:1)(cid:0) m + b − iem (cid:1) . (2) If µ / ∈ Γ then HF S/I ( µ ) = HF V ( µ )(3) If µ / ∈ Γ then HF V ( µ ) + HF V ( σ − µ ) = deg( V ) = (cid:18) n + mn (cid:19) d n e m . In particular, if µ ∈ ( nd − n, ( n + m ) e − m )+ N ∪ (( n + m ) d − n, me − m )+ N then HF V ( µ ) = deg( V ) and µ ∈ reg B ( V ) . (4) If µ ∈ Γ \ { Γ − ∪ Γ } then HF V ( µ ) + HF V ( σ − µ ) = (cid:18) n + mn (cid:19) ( d n e m − ǫ ( µ )) , where ǫ ( µ ) = HF ′ S ( µ − v n ) + HF ′′ S ( µ − w n ) as in the Definition 4.2. MARC CHARDIN AND NAVID NEMATI (5) If µ ∈ ∪ i ≥ Γ i \ { Γ } then HF V ( µ ) = (cid:18) n + mn (cid:19) d n e m − χ ( σ − µ ) . In particular, if µ ∈ Γ \ { Γ ∪ Γ } then dim( I V /I ) µ = (cid:18) n + mn + 1 (cid:19) HF ′ S ( µ − v n +1 ) + (cid:18) n + mm + 1 (cid:19) HF ′′ S ( µ − w m +1 ) . Proof.
Denote the saturation of I with respect to B by J . In this case HF V ( µ ) =dim( S/J ) µ for all µ ∈ Z . In the proof we use these two simple fact that µ ∈ Γ i − ∪ Γ i +1 yields µ ∈ Γ i and if µ ∈ Γ i then σ − µ ∈ Γ − i . By Proposition 3.2the Hilbert polynomial of V is D := (cid:0) n + mn (cid:1) d n e m which in this case is equal to thedeg( V ). By Serre Grothendieck formula,(4.1) HF S/J ( µ ) + dim( H B ( S/J ) µ ) = D (1) If µ / ∈ ∪ i ≥ Γ i , by Lemma 3.7 ( H i (K( f , S )) µ = 0 for all i ≥
1. Therefore byEquation 3.2, (
S/I ) µ = χ ( µ ).(2) If µ / ∈ Γ , by Lemma 3.7, ( H B ( S/I )) µ = 0 therefore ( S/I ) µ = ( S/J ) µ .(3) First note that µ, σ − µ / ∈ Γ . We claim that µ / ∈ Γ − or σ − µ / ∈ Γ − . Supposenot, then µ ∈ Γ − ∩ Γ which yields µ ∈ Γ which is a contradiction. So assume µ / ∈ Γ − . By Corollary 3.8,dim( H B ( S/I ) µ ) = HF S/I ( σ − µ ) . Since σ − µ / ∈ Γ , by part (2) and Equation 3.1 (cid:18) n + mn (cid:19) d n e m − HF V ( µ ) HF V ( σ − µ ) . The same argument works if we assume σ − µ / ∈ Γ − .(4) By part (1) HF S/I ( µ ) = HF V ( µ ). By considering the double complex as in theproof of Theorem 3.5 and Lemma 3.7, (cid:0) H n ( H n +1 B K( f , S )) (cid:1) µ = ( H n +1 B ( K n ( f , S )) µ , and (cid:0) H m ( H m +1 B K( f , S )) (cid:1) µ = (cid:0) H m +1 B ( K m ( f , S ) (cid:1) µ . By the abutment of the spectral sequence (cid:0) H B ( S/J ) (cid:1) µ = HF S/I ( σ − µ ) + dim( H n +1 B ( K n ( f , S )) µ + . (cid:0) H m +1 B ( K m ( f , S ) (cid:1) µ . Since σ − µ / ∈ Γ , by parts (1) and (2), HF S/I ( µ ) = HF V ( µ ) and by the definition,dim( H n +1 B ( K n ( f , S )) µ + (cid:0) H m +1 B ( K m ( f , S ) (cid:1) µ = (cid:0) n + mn (cid:1) ǫ ( µ ). The assertion follows by (cid:0) H B ( S/J ) (cid:1) µ = (cid:0) n + mn (cid:1) d n e m − HF V ( µ ).(5) By part (3), HF V ( µ ) = (cid:0) n + mn (cid:1) d n e m − HF V ( σ − µ ). Because µ / ∈ Γ − ( µ ∈ Γ − yields µ ∈ Γ ) therefore σ − µ / ∈ Γ , which by part (2), yields HF V ( σ − µ ) =HF S/I ( σ − µ ). On the other hand, since µ / ∈ ∪ i ≥ Γ i , by part (1), HF S/I ( σ − µ ) = χ ( σ − µ ). (cid:3) Let S = [ x , x , x , y , y , y ] and deg( x i ) = (1 ,
0) and deg( y i ) = (0 , I = ( f , . . . , f ) where deg( f i ) = (2 , V defined by I is completeintersection. The following picture demonstrate the regions Γ i in the Theorem 4.3. ULTIGRADED REGULARITY OF COMPLETE INTERSECTIONS 9 − . . . . . . . − . . . . . . . σM Γ − is the red region, Γ is the blue region and Γ is the green region. By Theorem4.3, Hilbert function of V at the bidegrees except the intersection of blue and greenare independent from the choices of f i ’s and they can be computed via χ and ǫ 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 , x , x , y , y , y ] , I = ( x y , x y , x y , ( x + x + x ) ( y + y + y ) ) and V be the complete intersection scheme of points defined by I . For (0 , ≤ µ ≤ (7 , the bigraded Hilbert function HF V ( µ ) is
24 72 96 96 96 96 96 9624 72 96 96 96 96 96 9621 63 86 90 93 95 96 9615 45 66 78 87 93 96 9610 30 48 64 78 90 96 966 18 32 48 66 86 96 963 9 18 30 45 63 72 721 3 6 10 15 21 24 24 . Where blue corresponds to the bidegrees in Γ , green corresponds to Γ and orangeto their intersections. Also red indicate the HF V ( σ ) . For µ / ∈ Γ by Theorem4.3 parts (1) , (2) and (3) one can compute the bigraded Hilbert function and forblue 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 thegenerator of I , which in this case is (2 , .Asking random command in Macaulay2 and constructing I ′ with generators ofbidegree (2 , . Let V ′ be the complete intersection scheme of points defined by I . For (0 , ≤ µ ≤ (13 , the bigraded Hilbert function HF V ′ ( µ ) is
33 84 96 96 96 96 96 96 96 96 96 96 96 9627 81 96 96 96 96 96 96 96 96 96 96 96 9621 63 86 90 93 95 96 96 96 96 96 96 96 9615 45 66 78 87 93 96 96 96 96 96 96 96 9610 30 48 64 78 90 96 96 96 96 96 96 96 966 18 32 48 66 86 96 96 96 96 96 96 96 963 9 18 30 45 63 81 84 96 96 96 96 96 961 3 6 10 15 21 28 36 45 55 66 78 91 96
The only differences are in the orange spots. In addition, in this case HF V ′ (0 ,
13) =HF V ′ (13 ,
0) = 96 which means the natural projection of V ′ to each P n and P m isone to one. Theorem 4.5.
Let S = k [ x , . . . , x n , y , . . . , y m ] be a bigraded polynomial ringwhere deg( x i ) = (1 , and deg( y i ) = (0 , . Assume V be a complete intersectionscheme of points defined by I = ( f , . . . , f n + m ) with deg( f i ) = ( d, e ) . Let π : V → P n and π : V → P m be the natural projections. Then max { reg( π − ( p )) for p ∈ P n } = min { a ∈ N | ( a, b ) ∈ reg B ( V ) for some b } max { reg( π − ( p )) for p ∈ P m } = min { b ∈ N | ( a, b ) ∈ reg B ( V ) 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 and B saturated. Let µ ≫ ν ≫
0, therefore H m ( S/J ) µ,ν = H m ( S/J ) µ,ν = 0. Suppose µ ≫
0, Hence by Mayer-Vietoris exactsequence H B ( S/J ) µ,ν ⊕ H B ( S/J ) µ,ν = H B ( S/J ) µ,ν . Since
S/J is B -saturated, for µ ≫ H B ( S/J ) µ,ν = 0 which means H B ( S/J ) µ,ν = H B ( S/J ) µ,ν . We state the proof for the first equality, the argument for the second one is thesame. There exist ( a , b ) , ( a , b ) ∈ N such that H m ( S/J ) µ = H m ( S/J ) ν = 0 forall µ ≥ ( a , b ) and ν ≥ ( a , b ). Also for any a ∈ N , there exists b a such that forall µ ≥ b a , ( H B ( S/J )) ( a,µ ) = 0. Set b := max a ∈ N { b , b , b a } . The Mayer-Vietoris short exact sequence yields H B ( S/J ) ( µ,ν ) = H B ( S/J ) ( µ,ν ) if ν > b . Thereforemin { a ∈ N | ( a, b ) ∈ reg B ( V ) for some b } = min { µ ∈ N | H B ( S/J ) µ,ν = 0 for ν > b } = min { µ ∈ N |∃ p = ( y , y , y ) ; H B ( S/J ⊗ k [ y ,y ,y ] k [ y , y , y ] p ) µ,ν = 0 for ν > b } (cid:3) ULTIGRADED REGULARITY OF COMPLETE INTERSECTIONS 11 Generic complete intersection points in P n × P m In this subsection we adopt the Notation 2.1 as well. We focus on the case whereideal I is generated by the generic forms. In this regard, we first need to providean analogous version of Bertini’s theorem in our case. Proposition 5.1.
Let k be a field of characteristics zero and V ⊂ P nk × P mk bea reduced scheme and ( d, e ) ∈ N with d, e = 0 . Let f U be a form of bidegree ( d, e ) with indeterminate coefficients U α,β . There exists a non empty open set Ω ⊂ Spec( k [ U α,β ]) such that for any p ∈ Ω , the corresponding form f p is such that V ∩ Z ( f p ) ⊆ P nk p × P mk p , where k p is the residue field of k [ U α,β ] p , is reduced of dimension equal dim V − ,unless the dim V = 0 in which case V ∩ Z ( f p ) = ∅ .Proof. Consider the Segre-Veronese map Ψ : P n × P m → P N with N = (cid:0) n + dd (cid:1) × (cid:0) m + ee (cid:1) −
1. Let V ′ = Ψ( V ). Under the map Ψ, f U is mapped to a linear form ℓ U in N + 1 variables and this correspondence is one to one. By Bertini theorem (see[3, Corollary 3.4.9]) there exists Ω as claimed such that for p ∈ Ω, V ′ ∩ Z ( ℓ p ) ∼ = V ∩ Z ( f p ) is a reduced scheme with the asserted dimension. (cid:3) Definition 5.2.
Forms of bidegree ( d, e ) are in one to one correspondence withclosed points in A dim k S ( d,e ) k . Remark 5.3.
Notice that Ω ⊂ spec k [ U α,β ] \ Z ( F ) for some form F = 0 . Inparticular, for c = ( c α,β ) ∈ k N such that F ( c ) = 0 , Proposition 5.1 with p generatedby the elements u α,β − c α,β , shows that Z ( f c ) ∩ V is reduced. Theorem 5.4.
Let S = k [ x , . . . , x n , y , . . . , y m ] be a bigraded polynomial ringover a field k of characteristic zero where deg( x i ) = (1 , and deg( y i ) = (0 , . Let ( d i , e i ) ≥ (1 , for i = 1 , . . . , n + m be bidegrees. Then, there exists a non-emptyopen set Ω ⊂ A Nk := n + m Y i =1 A | S ( di,ei ) | k such that, for any ( f , . . . , f n + m ) corresponding to a point in Ω , (1) Z := proj ( S/ ( f , . . . , f n + m )) is reduced of dimension zero. (2) The natural projections of Z to the factors P n and P m are isomorphism.Proof. Let V := Z ( g , . . . , g n + m ) ⊂ A Nk × P nk × P mk where A Nk = spec k [ U i,α,β ] for1 ≤ i ≤ n + m , | α | = d i , | β | = e i and g i := P U i,α,β x α y β . Set π : A Nk × P nk × P mk → P nk × P mk be the natural projection, then V is a vector bundle over P nk × P mk via π .For any point p ∈ P nk × P mk 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 , . . . , g n + m , Jac n + m ( g , . . . , g n + m )) ⊂ V where Jac n + m is theJacobian of order n + m . By using Proposition 5.1 inductively, there exist a point u ∈ A Nk and ( f , . . . , f n + m ) corresponding to u such that Jac n + m ( f , . . . , f n + m ) = 0which yields W ( V and dim( W ) < dim( V ) = N . Consider the natural projection p : A Nk × P nk × P mk → A Nk . Since dim( W ) < N , p ( W ) is a non empty closed subsetof A Nk . Set Ω := (cid:0) A Nk \ π ( W ) (cid:1) . Therefore for any ( f , . . . , f n + m ) corresponding toa point in Ω , proj ( S/ ( f , . . . , f n + m )) is geometrically reduced and is of dimension zero.(2) Set p : A Nk × P nk × P mk → A Nk × P nk and q : A Nk × P nk → A Nk the naturalprojections. First we show that it is enough to show that the restriction p ′ : V → p ( V ) is birational. Indeed if p ′ is birational then there exists a closed subset Z ( p ( V ) such that setting W := ( p ′ ) − ( Z ) then p ′ | V \ W : V \ W → p ( V ) \ Z isan isomorphism. As dim( Z ) < N , q ( Z ) ( A Nk . Set Ω := A Nk \ ( q ( Z ) ∪ π ( W )).For any point u ∈ Ω and ( f , . . . , f n + m ) corresponding to u , q − ( u ) = p ( u, Z ( f , . . . , f n + m )) ∼ = ( u, Z ( f , . . . , f n + m ))showing that the natural projection of Z onto P n is an isomorphism. By replacing p and q by p : A Nk × P nk × P mk → A Nk × P mk and q : A Nk × P mk → A Nk , along thesame lines it shows the existence of Z ∈ p ( V ) and Ω = A Nk \ ( q ( Z ) ∪ π ( W ))such that for u ∈ Ω and ( f , . . . , f n + m ) corresponding to u , q − ( u ) = p ( u, Z ( f , . . . , f n + m )) ∼ = ( u, Z ( f , . . . , f n + m )) . Consider the following diagram V ⊂ / / p ′ (cid:15) (cid:15) A Nk × P nk × P mkp (cid:15) (cid:15) V := p ( V ) ⊂ / / q | V (cid:15) (cid:15) A Nk × P nkq (cid:15) (cid:15) A Nk = / / A Nk . We now show that p ′ is birational.Let I := ( g , . . . , g n + m ) and J := I : B ∞ . By part (1), V is a geometricallyirreducible scheme. Hence J and I V = J ∩ k [ x , U ] are prime and remain primeunder any extension of k .As V is generically smooth, there exists G ∈ I V such that D := ∂G∂U i,α,β / ∈ I V for some i, α , and β unless G ∈ k [ y ] which is impossible: indeed I V ∩ k [ y ] = (0)because the projection V → P m is onto.Set Z := Z ( D ) ⊂ V . By [7, Lemma 4.6.1], x α y β ∂G∂U i,α ′ ,β ′ − x α ′ y β ′ ∂G∂U i,α,β ∈ I V for any choices of α ′ and β ′ . Let y β = y p y β for some i . For any 0 ≤ j ≤ m put α ′ := α and β ′ := y j β . As x α and y β are not in I V it follows that y p ∂G∂U i,α ′ ,β ′ − y j ∂G∂U i,α,β ∈ I V for all 0 ≤ j ≤ m . Therefore by localizing at D , we get an isomorphism φ : ( k [ U , x ] D [ y ]) I V ∼ = −→ ( k [ U , x ] D [ y p ]) I V . ULTIGRADED REGULARITY OF COMPLETE INTERSECTIONS 13
Therefore the natural maps (cid:18) k [ U , x ] I V (cid:19) D ι ֒ → ( k [ U , x ] D [ y ]) I V φ −→ ( k [ U , x ] D [ y p ]) I V are such that φ ◦ ι induces the identity from proj (( k [ U , x ] D [ y p ]) /I V ) to spec (( k [ U , x ] D ) /I V ).Hence φ provides the inverse ϕV \ Z ϕ −→ V \ ( Z × P mk ) p ′ −→ V \ Z . (cid:3) Theorem 5.5.
Let S = k [ x , . . . , x n , y , . . . , y m ] be a bigraded polynomial ring overa 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 |{ µ ∈ N | HF V ( µ ) = deg( V ) }| < ∞ , 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),HF V (0 , i ) = HF V ( j,
0) = deg( V ) for any i, j ≫ (cid:3) References [1]
Botbol, N., and Chardin, M.
Castelnuovo Mumford regularity with respect to multigradedideals.
Journal of Algebra 474 (mar 2017), 361–392.[2]
Eisenbud, D., and Goto, S.
Linear free resolutions and minimal multiplicity.
Journal ofAlgebra 88 , 1 (may 1984), 89–133.[3]
Flenner, H., O’Carroll, L., and Vogel, W.
Joins and intersections . Springer Monographsin Mathematics. Springer-Verlag, Berlin, 1999.[4]
Fulton, W.
Intersection theory , second ed., vol. 2 of
Ergebnisse der Mathematik und ihrerGrenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematicsand Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics] . Springer-Verlag,Berlin, 1998.[5]
Grayson, D. R., and Stillman, M. E.
Macaulay2, a software system for research in algebraicgeometry. Available at https://faculty.math.illinois.edu/Macaulay2/ .[6]
Hoffman, J. W., and Wang, H. H.
Castelnuovo–Mumford regularity in biprojective spaces.
Advances in Geometry 4 , 4 (jan 2004).[7]
Jouanolou, J.-P.
Le formalisme du r´esultant.
Adv. Math. 90 , 2 (1991), 117–263.[8]
Maclagan, D., and Smith, G. G.
Multigraded Castelnuovo-Mumford regularity.
J. ReineAngew. Math 14 , 1 (2005), 137–164.[9]
Peeva, I., and Stillman, M.
Open problems on syzygies and hilbert functions.
J. Commut.Algebra 1 , 1 (03 2009), 159–195.
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