Mixed multiplicities and projective degrees of rational maps
aa r X i v : . [ m a t h . A C ] A p r MIXED MULTIPLICITIES AND PROJECTIVE DEGREES OFRATIONAL MAPS
YAIRON CID-RUIZ
Abstract.
We consider the notion of mixed multiplicities for multigraded modulesby using Hilbert series, and this is later applied to study the projective degrees ofrational maps. We use a general framework to determine the projective degrees of arational map via a computation of the multiplicity of the saturated special fiber ring.As specific applications, we provide explicit formulas for all the projective degrees ofrational maps determined by perfect ideals of height two or by Gorenstein ideals ofheight three. Introduction
Let B be a standard multigraded algebra B = L ν ∈ N r [ B ] ν over an Artinian localring A = [ B ] (0 ,..., . Let M be a finitely generated Z r -graded B -module. It is knownthat the Hilbert function ν ∈ Z r length A (cid:0) [ M ] ν (cid:1) ∈ N of M coincides with a polynomial P M ( X ) = P M ( X , . . . , X r ) for ν ≫ (0 , . . . , ∈ N r (see [15, Theorem 4.1]). This polynomial P M ( X ) is referred to as the Hilbert polynomial of M , and its total degree is equal to the dimension d ++ = dim (cid:0) Supp ++ ( M ) (cid:1) of therelevant support Supp ++ ( M ) of M . Let N ⊂ B be the multigraded irrelevant ideal N = L ν > ,...,ν r > [ B ] ν . Then, the relevant support Supp ++ ( M ) is given by Supp ++ ( M ) = { p ∈ Supp( M ) | p is N r -graded and p N } . If we write P M ( X ) = X n ,...,n r ≥ e ( n , . . . , n r ) (cid:18) X + n n (cid:19) · · · (cid:18) X r + n r n r (cid:19) , then 0 ≤ e ( n , . . . , n r ) ∈ Z for all n + . . . + n r = d ++ . For n ∈ N r with | n | = n + · · · + n r = d ++ , the non-negative integer e ( n , . . . , n r ) is called the mixed multiplicityof M of type n = ( n , . . . , n r ) and it is denoted as e ( n ; M ) = e ( n , . . . , n r ; M ).The concept of mixed multiplicities provides the right generalization of multiplicities(or degrees) to a multigraded setting, and its study goes back to seminal work byvan der Waerden [28]. These invariants have been further studied and developed byBhattacharya [1], by Katz, Mandal and Verma [20], by Herrmann, Hyry, Ribbe andTang [15], and by Trung [27]. For more details, see the survey paper [25] and thereferences therein.Motivated by the notion of multidegree defined by Miller and Sturmfels in [24, § Hilbert series . Contrary to the single-graded case, in a multigraded setting theapproaches with Hilbert polynomials or Hilbert series may yield different results. Thiscomes from the fact that the Hilbert polynomial can only read irreducible components
Mathematics Subject Classification.
Primary 13H15, 14E05; Secondary 13A30, 13D02.
Key words and phrases. mixed multiplicities, projective degrees, rational maps, Hilbert polynomials,Hilbert series, Rees algebra, saturated special fiber ring. which are relevant in the multigraded geometric sense. To be more precise, let d =dim( M ) and denote the Hilbert series of M byHilb M ( t ) = Hilb M ( t , . . . , t r ) = X ν ∈ Z r length A ([ M ] ν ) t ν · · · t ν r r . In our first main result we provide a different notion of mixed multiplicities (definedin terms of Hilbert series) and we relate it to the above notion of mixed multiplicities(defined in terms of Hilbert polynomials). For the moment, note that dim (cid:0) M / H N ( M ) (cid:1) = r + d ++ (see Lemma 3.3). We show that the mixed multiplicities e ( n ; M ) can be expressedin terms of the new ones applied to the module M / H N ( M ), i.e., after modding out thetorsion with respect to N . Definition-Theorem A (Theorem 2.2, Lemma 2.3, Definition 2.4, Theorem 3.4) . Let B be a standard multigraded algebra B = L ν ∈ N r [ B ] ν over an Artinian local ring. Let M be a finitely generated Z r -graded B -module. Let d = dim( M ) and d ++ = dim (cid:0) Supp ++ ( M ) (cid:1) .(I) [A structural result for Hilbert series] For each n = ( n , . . . , n r ) ∈ N r with | n | = d , there exists a Laurent polynomial Q n ( t ) = Q ( n ,...,n r ) ( t, . . . , t ) ∈ Z [ t , t − ] = Z [ t , . . . , t r , t − , . . . , t − r ] with Q n ( ) = Q ( n ,...,n r ) (1 , . . . , ≥ , such that Hilb M ( t ) = X | n | = d Q n ( t )(1 − t ) n · · · (1 − t r ) n r , and the following statements hold:(a) There is at least one n ∈ N r with | n | = d such that Q n ( ) > .(b) The values of Q n ( ) are uniquely determined by the module M .(II) [Definition] For each n = ( n , . . . , n r ) ∈ Z r with n i ≥ − and | n | = d − r , we de-fine the mixed multiplicity of M of type n = ( n , . . . , n r ) in terms ofHilbert series as e n ( M ) = e ( n ,...,n r ) ( M ) = Q n + ( ) , where Q n + ( t ) = Q ( n +1 ,...,n r +1) ( t, . . . , t ) ∈ Z [ t , t − ] are Laurent polynomialsobtained in part ( I ) .(III) [Relation between the two notions of mixed multiplicities] For each n ∈ N r with | n | = d ++ , we have the following equality e ( n ; M ) = e n (cid:16) M / H N ( M ) (cid:17) . One advantage of our approach with Hilbert series is that we can read certain mixedmultiplicities which cannot be read using the Hilbert polynomial approach. For instance,if M has a minimal prime of maximal dimension containing the irrelevant ideal N , thenthe contribution of that minimal prime is summed up in some e n ( M ) but not in any e ( n ; M ).After defining the new mixed multiplicities, we prove several general results aboutthese invariants (see Lemma 2.7, Theorem 2.8 and Theorem 3.10). We also provide adifferent proof for the existence of the Hilbert polynomial in a multigraded setting (seeTheorem 3.4). Similarly to [27], we use filter-regular elements to substitute the notionof general elements .In the second half of the paper, we consider the multidegrees of multiprojectiveschemes as mixed multiplicities (see Definition 4.3). In particular, our main focus will IXED MULTIPLICITIES AND PROJECTIVE DEGREES OF RATIONAL MAPS 3 be on the projective degrees of rational maps. The projective degrees of rational mapsare fundamental and classical invariants in Algebraic Geometry. For more details onthe subject, the reader is referred to [14, Example 19.4] and [9, § § k be a field and R bethe polynomial ring R = k [ x , . . . , x d ]. Let n ≥ d and(1) F : P d k = Proj( R ) P n k be a rational map defined by n + 1 homogeneous elements { f , . . . , f n } ⊂ R of the samedegree δ > I ⊂ R be the homogeneous ideal I = ( f , . . . , f n ). The projectivedegrees of F are defined as the multidegrees of the graph of F (see Definition 5.2), andthey are denoted as d i ( F ) for 0 ≤ i ≤ d . In classical geometrical terms, d i ( F ) equals thenumber of points in the intersection of the graph Γ ⊂ P d k × k P n k of F with the product H × k K ⊂ P d k × k P n k , where H ⊂ P d k and K ⊂ P n k are general subspaces of dimension d − i and n − d + i , respectively.Our main tool for the computation of the projective degrees d i ( F ) will be to exploitthe saturated special fiber ring [5] (see Theorem 5.4).As specific applications, we compute all the projective degrees for rational mapsdetermined by perfect ideals of height two or by Gorenstein ideals of height three. Inboth cases we assume the condition G d +1 , that is, µ ( I p ) ≤ dim( R p ) for all p ∈ V ( I ) ⊂ Spec( R ) such that ht( p ) < d + 1. However, it should be noted that the condition G d +1 isalways satisfied by generic perfect ideals of height two and by generic Gorenstein idealsof height three.In the conditions expressed in the theorem below, we use the fact that the minimalresolution of any perfect ideal of height two is described by the Hilbert-Burch theorem(see, e.g., [10, Theorem 20.15]). Theorem B (Theorem 5.7) . With the notations above, assume the following conditions:(i) I is perfect of height two with Hilbert-Burch resolution of the form → n M i =1 R ( − δ − µ i ) ϕ −→ R ( − δ ) n +1 → I → . (ii) I satisfies the condition G d +1 .Then, the projective degrees of the rational map F : P d k P n k in (1) are given by d i ( F ) = e d − i ( µ , µ , . . . , µ n ) where e d − i ( µ , µ , . . . , µ n ) denotes the elementary symmetric polynomial e d − i ( µ , µ , . . . , µ n ) = X ≤ j Theorem C (Theorem 5.8) . With the notations above, assume the following conditions:(i) I is a Gorenstein ideal of height three.(ii) Every non-zero entry of an alternating minimal presentation matrix of I has degree D ≥ .(iii) I satisfies the condition G d +1 . YAIRON CID-RUIZ Then, the projective degrees of the rational map F : P d k P n k in (1) are given by d i ( F ) = ( D d − i P ⌊ n − d + i ⌋ k =0 (cid:0) n − − kd − i − (cid:1) if 0 ≤ i ≤ d − δ d − i if d − ≤ i ≤ d. The basic outline of this paper is as follows. In Section 2, we introduce the notion ofmixed multiplicities in terms of Hilbert series. In Section 3, we relate this notion withthe usual mixed multiplicities in terms of Hilbert polynomials. In Section 4, we considerthe multidegrees of multiprojective schemes and we prove van der Waerden’s originalresult [28] in a multigraded setting. In Section 5, we concentrate on the projectivedegrees of rational maps and we obtain the formulas of Theorem B and Theorem C.2. Mixed multiplicities of multigraded modules via Hilbert series During this section, we study and develop a definition of mixed multiplicities thatdepends on Hilbert series. The results in this section can be seen as a natural continua-tion of [24, § ν ∈ Z r , we define its weight as | ν | = ν + · · · + ν r . If µ, ν ∈ Z r are two multi-indexes, we write µ ≥ ν whenever µ i ≥ ν i ,and µ > ν whenever µ i > ν i . For 1 ≤ i ≤ r , let e i be the i -th elementary vector e i = (0 , . . . , , . . . , ∈ N r and ∈ N r be the vectors = (0 , . . . , 0) and = (1 , . . . , 1) of r copies of 0 and 1, respectively.The following setup is fixed throughout this section. Setup 2.1. Let ( A, n , k ) be an Artinian local ring with maximal ideal n and residuefield k = A/ n . Let B be a finitely generated standard N r -graded algebra over A , thatis, [ B ] = A and B is finitely generated over A by elements of degree e i with 1 ≤ i ≤ r .For any Z r -graded B -module M , its graded part of degree ν ∈ Z r is denoted by [ M ] ν .Unless specified otherwise, a graded B -module M will always means an Z r -graded B -module, that is, M = L ν ∈ Z r [ M ] ν . If ν ∈ Z r and x ∈ [ B ] ν , we say that x is homogeneousof degree ν and its total degree is | ν | . Let M ⊂ B be the graded ideal M := L ν = [ B ] ν ,and for 1 ≤ i ≤ r let M i ⊂ B be the graded ideal M i := L ν ≥ e i [ B ] ν . In this multigradedsetting the irrelevant ideal is defined as the graded ideal N := M ∩ · · · ∩ M r = M ν> [ B ] ν . Similarly to the single-graded case, we can define a multiprojective scheme from B .The multiprojective scheme MultiProj( B ) is given by the set of all graded prime idealsin B which do not contain N , that is,MultiProj( B ) := (cid:8) p ∈ Spec( B ) | p is graded and p N (cid:9) , and its scheme structure is obtained by using multi-homogeneous localizations (see,e.g., [18, § B ) are given by V ++ ( J ) := V ( J ) ∩ MultiProj( B ) = { p ∈ MultiProj( B ) | p ⊇ J } for J ⊂ B a graded ideal. For a finitelygenerated graded B -module M , set Supp ++ ( M ) to be the closed subset Supp ++ ( M ) :=Supp( M ) ∩ MultiProj( B ) = { p ∈ MultiProj( B ) | M p = 0 } = V ++ (Ann( M )). Given a IXED MULTIPLICITIES AND PROJECTIVE DEGREES OF RATIONAL MAPS 5 graded B -module M , we denote as M gr the single-graded module given as M gr := M n ∈ Z M ν ∈ Z r | ν | = n [ M ] ν . We have that M gr is naturally a graded module over the single-graded A -algebra B gr .For any n ∈ Z r , the terms t n and ( − t ) n represent the elements t n · · · t n r r and(1 − t ) n · · · (1 − t r ) n r , respectively. Note that any graded part [ M ] ν of a finitely generatedgraded B -module M is a finitely generated module over the Artinian local ring A , andso a module of finite length. The length of a finitely generated A -module E is denotedas length A ( E ). The Hilbert series of a finitely generated graded B -module M is denotedas Hilb M ( t ) and given by Hilb M ( t ) := X ν ∈ Z r length A (cid:0) [ M ] ν (cid:1) t ν . For any ν ∈ Z r , M ( ν ) denotes the shifted graded B -module given as [ M ( ν )] µ = [ M ] ν + µ .Also, we have that Hilb M ( − ν ) ( t ) = t ν Hilb M ( t ).The following theorem gives a structural result for the Hilbert series of a finitelygenerated graded B -module. Theorem 2.2. Assume Setup 2.1. Let M be a finitely generated graded B -module. Let d = dim( M ) . Then, for each n ∈ N r with | n | = d , there exists a Laurent polynomial Q n ( t ) ∈ Z [ t , t − ] with Q n ( ) ≥ , such that Hilb M ( t ) = X | n | = d Q n ( t )( − t ) n , and the following statements hold:(i) There is at least one n ∈ N r with | n | = d such that Q n ( ) > .(ii) If Q n ( t ) = 0 and Q n ( ) = 0 , then Q n ( t ) is divisible by (1 − t i ) for some ≤ i ≤ r such that n i = 0 .Proof. We proceed by induction on d .For the statement in part ( ii ), we will actually prove that, if Q n ( t ) = 0 and Q n ( ) = 0,then n ≥ e and (1 − t ) divides Q n ( t ).There exists a finite filtration0 = M ⊂ M ⊂ · · · ⊂ M k = M of M such that M l / M l − ∼ = ( B / p l ) ( − ν l ) where p l ⊂ B is a graded prime ideal withdimension dim( B / p l ) ≤ d and ν l ∈ Z r . The short exact sequences0 → M l − → M l → ( B / p l ) ( − ν l ) → M ( t ) = k X l =1 t ν l Hilb B / p l ( t ) . Let d l = dim ( B / p l ) and suppose for the moment that we have shown thatHilb B / p l ( t ) = X | n | = d l Q ( l ) n ( t )( − t ) n , YAIRON CID-RUIZ where Q ( l ) n ( t ) ∈ Z [ t , t − ] satisfy the same conditions of parts ( i ) and ( ii ) (with i = 1).By an abuse of notation, for any m we set Q ( l ) m ( t ) = 0. Then, we would get theequation Hilb M ( t ) = k X l =1 t ν l (1 − t ) d − d l (1 − t ) d − d l Hilb B / p l ( t )= X | n | = d P kl =1 t ν l (1 − t ) d − d l Q ( l ) n − ( d − d l ) e ( t )( − t ) n . Therefore, it is enough to consider the case M = B / p where p ⊂ B is a gradedprime ideal. Suppose that d = dim ( B / p ) = 0. Since A is an Artinian local ring, itfollows that p is equal to the unique maximal graded ideal n + M . Hence, it follows thatHilb B / p ( t ) = 1, and the result is clear.Suppose that d = dim ( B / p ) > 0. Then, we may choose a homogeneous element0 = x ∈ B / p of degree deg( x ) = e i where 1 ≤ i ≤ r . From the short exact sequence0 → ( B / p ) ( − e i ) x −→ B / p → B / ( x, p ) → B / ( x, p ) ( t ) = Hilb B / p ( t ) − t i Hilb B / p ( t ) = (1 − t i )Hilb B / p ( t ) . Since dim ( B / ( x, p )) = d − 1, from the induction hypothesis we may assume thatHilb B / ( x, p ) ( t ) = X | n | = d − Q n ( t )(1 − t ) n · · · (1 − t r ) n r , where Q n ( t ) ∈ Z [ t , t − ] satisfy the same conditions of parts ( i ) and ( ii ) (with i = 1).Thus, the equation Hilb B / p ( t ) = − t i Hilb B / ( x, p ) ( t ) implies thatHilb B / p ( t ) = X | n | = d − Q n ( t )(1 − t ) n · · · (1 − t i ) n i +1 · · · (1 − t r ) n r , and so the statement of the theorem is obtained. (cid:3) Although the decomposition of the Hilbert series Hilb M ( t ) given in Theorem 2.2 isnot necessarily unique, we can easily see from the simple lemma below that the valuesof Q n ( ) are uniquely determined by the module M . Lemma 2.3. Suppose that X | n | = d Q n ( t )( − t ) n = X | n | = d Q ′ n ( t )( − t ) n where n ∈ N r and Q n ( t ) , Q ′ n ( t ) ∈ Z [ t , t − ] . Then, we have that Q n ( ) = Q ′ n ( ) for all n ∈ N r with | n | = d .Proof. Take m ∈ N r such that P n ( t ) = t m Q n ( t ) ∈ Z [ t ] and P ′ n ( t ) = t m Q ′ n ( t ) ∈ Z [ t ],i.e., P n ( t ) and P ′ n ( t ) become polynomials in t . Multiplying both sides of the aboveequation by ( − t ) d t m (where d = ( d, d, . . . , d ) ∈ N r ) gives us that X | n | = d P n ( t )( − t ) d − n = X | n | = d P ′ n ( t )( − t ) d − n . IXED MULTIPLICITIES AND PROJECTIVE DEGREES OF RATIONAL MAPS 7 Then, the substitution t − t yields X | n | = d P n ( − t ) t d − n = X | n | = d P ′ n ( − t ) t d − n . Since both sums above are restricted to the multi-indexes n ∈ N r with | n | = d , bycomparing the terms of the smallest possible degree | d − n | = ( r − d , we obtain that Q n ( ) = P n ( ) = P ′ n ( ) = Q ′ n ( ), and so the result follows. (cid:3) We are now ready to define the following notion of mixed multiplicities in a multi-graded setting. Definition 2.4. Assume Setup 2.1. Let M be a finitely generated graded B -module.Let d = dim( M ). From Theorem 2.2 choose any decompositionHilb M ( t ) = X n ∈ N r | n | = d Q n ( t )( − t ) n . For any n = ( n , . . . , n r ) ∈ Z r with n ≥ − = ( − , . . . , − ∈ Z r and | n + | ≥ d , the mixed multiplicity of M of type n defined in terms of Hilbert series is given by e n ( M ) := ( Q n + ( ) if | n + | = d | n + | > d. From Lemma 2.3, the mixed multiplicities e n ( M ) are uniquely determined by M . Remark 2.5. In the above definition we have chosen to enumerate the mixed multiplic-ities for multi-indexes with n ≥ − . We made this choice so that we can relate e n ( M )with the usual definition in terms of Hilbert polynomials for n ≥ (see Theorem 3.4below).For instance, let K be a field and S = K [ x , . . . , x n , y , . . . , y m ] be a bigraded polyno-mial ring with deg( x i ) = (1 , 0) and deg( y i ) = (0 , S are given by P S ( ν , ν ) = (cid:18) ν + n − n − (cid:19)(cid:18) ν + m − m − (cid:19) and Hilb S ( t , t ) = 1(1 − t ) n (1 − t ) m , respectively. Therefore, from both definitions, we obtain that e n − ,m − ( S ) = e ( n − , m − S ) = 1 and that e i,j ( S ) = e ( i, j ; S ) = 0 for i, j ≥ , i + j = n + m − Remark 2.6. In Definition 2.4 we allow the flexibility of having | n + | ≥ d so thatthe function e n ( • ) becomes additive in the full subcategory of finitely generated graded B -modules with dimension at most | n + | (for the same setting in the single-gradedcase, see, e.g., [3, Corllary 4.7.7]).Next we derive some basic properties of the mixed multiplicities e n ( • ). Lemma 2.7. Let M be a finitely generated graded B -module with dim( B ) = d . Let n ∈ Z r such that n ≥ − and | n + | ≥ d . Then, the following statements hold:(i) For any ν ∈ Z , we have that e n ( M ( − ν )) = e n ( M ) .(ii) (additivity) Let → M ′ → M → M ′′ → be a short exact sequence of finitelygenerated graded B -modules. Then e n ( M ) = e n ( M ′ ) + e n ( M ′′ ) . YAIRON CID-RUIZ (iii) (associativity formula) e n ( M ) = X p ∈ Supp( M )dim( B / p )= d length B p (cid:0) M p (cid:1) e n ( B / p ) . Proof. ( i ) It follows from the fact that Hilb M ( − ν ) ( t ) = t ν Hilb M ( t ).( ii ) SinceHilb M ( t ) = (1 − t ) d − dim( M ′ ) (1 − t ) d − dim( M ′ ) Hilb M ′ ( t ) + (1 − t ) d − dim( M ′′ ) (1 − t ) d − dim( M ′′ ) Hilb M ′′ ( t ) , the result is obtained from Theorem 2.2 and Lemma 2.3.( iii ) Take a finite filtration0 = M ⊂ M ⊂ · · · ⊂ M n = M of M such that M l / M l − ∼ = ( B / p l ) ( − ν l ) where p l ⊂ B is a graded prime ideal withdimension dim( B / p l ) ≤ d and ν l ∈ Z r . From parts ( i ) and ( ii ), it follows that e n ( M ) = P nl =1 e n ( B / p l ). The inequality dim ( B / p l ) ≤ d − e n ( B / p l ) = 0. Onthe other hand, for any p ∈ Supp( M ) with dim ( B / M ) = d , the cardinality of the set { l | p = p l } ⊆ { , , . . . , n } is equal to length B p (cid:0) M p (cid:1) . So, the result follows. (cid:3) The following theorem is a generalization of the results of [15, Theorem 4.3] and[20, Theorem 4.1]. Additionally, it characterizes when multiplicities of negative typeare positive, that is, when e n ( M ) > n and some finitely generated graded B -module M . Theorem 2.8. Assume Setup 2.1. Let M be a finitely generated graded B -module with dim( M ) = d . Then, the following statements hold:(i) We have the equality e (cid:0) M gr (cid:1) = X n ≥− | n + | = d e n ( M ) . (ii) { p ∈ Supp ( M ) | dim( B / p ) = d } ∩ V ( N ) = ∅ if and only if e n ( M ) > for some n .Proof. ( i ) The Hilbert series Hilb M gr ( t ) of the single-graded module M gr can be (uniquely)written as Hilb M gr ( t ) = Q ( t )(1 − t ) d for some Q ( t ) ∈ Z [ t, t − ] (see, e.g., [3, § M ( t ) by making the substitutions t i t for 1 ≤ i ≤ r . Hence, takinga decomposition of Hilb M ( t ) from Theorem 2.2 gives the equation Q ( t )(1 − t ) d = P n ≥− | n + | = d Q n + ( t, . . . , t )(1 − t ) d . Therefore, the equality e (cid:0) M gr (cid:1) = X n ≥− | n + | = d e n ( M )follows from the fact that e ( M gr ) = Q (1) and e n ( M ) = Q n + ( ).( ii ) Suppose that e m ( M ) > − ≤ m ∈ Z r such that | m + | = d and that m i = − ≤ i ≤ r ; and fix such m and i . From Lemma 2.7( iii ), there exists a IXED MULTIPLICITIES AND PROJECTIVE DEGREES OF RATIONAL MAPS 9 prime p ∈ Supp( M ) with dim ( B / p ) = d such that e m ( B / p ) > 0. By using Theorem 2.2take a decomposition Hilb B / p ( t ) = X n ≥− | n + | = d Q n + ( t )( − t ) n + . Since e m ( B / p ) > 0, it follows that Q m + ( t ) = 0. Let t ′ = ( t , . . . , t i − , t i +1 , . . . , t r )and write Q m + ( t ) = P cj =0 t ji G j ( t ′ ) where G j ( t ′ ) ∈ Z [ t ′ ].For any F ( t ) = P ν ≥ f ν t ν ∈ Z [[ t ]] power series we use the notation (cid:10) F ( t ) (cid:11) ic := X ν ≥ ν i ≤ c f ν t ν ∈ Z [[ t ]] . Clearly, the function (cid:10) • (cid:11) ic : Z [[ t ]] → Z [[ t ]] is additive.From the following isomorphism B / (cid:0) p , M c +1 i (cid:1) ∼ = M ν ≥ ν i ≤ c [ B / p ] ν , we then obtain thatHilb B / ( p , M c +1 i )( t ) = (cid:10) Hilb B / p ( t ) (cid:11) ic = X n ≥− | n + | = d D Q n + ( t )( − t ) n + E ic . Since D Q m + ( t )( − t ) m + E ic = Q m + ( t )( − t ) m + , Lemma 2.3 and Theorem 2.2 imply thatdim (cid:0) B / (cid:0) p , M c +1 i (cid:1)(cid:1) = dim ( B / p ) = d. But then we get the containment p ⊇ M i ⊇ N because p is a prime ideal. Therefore, itfollows that { p ∈ Supp ( M ) | dim( B / p ) = d } ∩ V ( N ) = ∅ .For the reverse implication, assume that Z = { p ∈ Supp ( M ) | dim( B / p ) = d } ∩ V ( N ) = ∅ , and choose p ∈ Z . Since p ⊇ N , [ B / p ] ν = 0 for ν > and so it follows that adecomposition from Theorem 2.2 can be written asHilb B / p ( t ) = X n ∈ Σ Q n + ( t )( − t ) n + where Σ = { n ∈ Z r | n ≥ − , | n + | = d and n i = − ≤ i ≤ r } . Therefore,Lemma 2.7( iii ) implies that e n ( M ) > − ≤ n ∈ Z r such that | n + | = d andthat n i = − ≤ i ≤ r .So, the result of the theorem follows. (cid:3) Relation with mixed multiplicities via Hilbert polynomials In this section we relate the mixed multiplicities introduced in Definition 2.4 withthe usual mixed multiplicities defined in terms of Hilbert polynomials. Here we alsostudy how mixed multiplicities behave after taking quotients by filter-regular sequences.Throughout this section we continue using all the notations and conventions of theprevious section.The following theorem shows that in a multigraded setting we can define a multi-graded Hilbert polynomial, which provides the usual approach for defining mixed mul-tiplicities. Below in Theorem 3.4 we obtain a different proof of this result. Theorem 3.1 ([15, Theorem 4.1]) . Assume Setup 2.1. Let M be a finitely generatedgraded B -module. Then, there exists a polynomial P M ( X ) = P M ( X , . . . , X r ) ∈ Q [ X ] = Q [ X , . . . , X r ] which can be written as P M ( X ) = X n ,...,n r ≥ e ( n , . . . , n r ) (cid:18) X + n n (cid:19) · · · (cid:18) X r + n r n r (cid:19) , where e ( n , . . . , n r ) ∈ Z , and that satisfies the following conditions:(i) The total degree of P M ( X ) is equal to dim (cid:0) Supp ++ ( M ) (cid:1) .(ii) e ( n , . . . , n r ) ≥ for any n + · · · + n r = dim (cid:0) Supp ++ ( M ) (cid:1) .(iii) P M ( ν ) = length A ([ M ] ν ) for all ν ≫ . Motivated by the previous theorem, we have the following definition which goes backto the work of van der Waerden ([28]). Definition 3.2. Assume Setup 2.1. Let M be a finitely generated graded B -module.Let d ++ = dim (cid:0) Supp ++ ( M ) (cid:1) . Let P M ( X ) = P M ( X , . . . , X r ) be the polynomial P M ( X ) = X n ,...,n r ≥ e ( n , . . . , n r ) (cid:18) X + n n (cid:19) · · · (cid:18) X r + n r n r (cid:19) obtained according to Theorem 3.1. For any n = ( n , . . . , n r ) ∈ N r with | n | ≥ d ++ , the mixed multiplicity of M of type n defined in terms of Hilbert polynomials is given by e ( n ; M ) := ( e ( n , . . . , n r ) if | n | = d ++ | n | > d ++ . The next simple lemma will be needed to relate Definition 2.4 and Definition 3.2. Lemma 3.3. Let M be a finitely generated graded B -module. Then, the following state-ments hold:(i) [ M ] ν = (cid:2) M / H N ( M ) (cid:3) ν for all ν ≫ .(ii) Ass (cid:0) M / H N ( M ) (cid:1) = Ass( M ) \ V ( N ) .(iii) dim (cid:0) Supp ++ ( M ) (cid:1) = dim (cid:0) M / H N ( M ) (cid:1) − r .Proof. ( i ) Follows from the fact that N k · H N ( M ) = 0 for some k ≥ ii ) It is well-known (see, e.g., [10, Proposition 3.13]).( iii ) By using part ( ii ) and Supp ++ ( M ) = Supp ++ (cid:0) M / H N ( M ) (cid:1) , we getdim (cid:0) Supp ++ ( M ) (cid:1) = sup (cid:8) dim (cid:0) MultiProj ( B / p ) (cid:1) | p ∈ Supp ++ ( M ) (cid:9) and dim (cid:0) M / H N ( M ) (cid:1) = sup (cid:8) dim ( B / p ) | p ∈ Supp ++ ( M ) (cid:9) . Finally, for any p N it is known that dim (MultiProj( B / p )) = dim( B / p ) − r (see,e.g., [15, Lemma 1.1], [18, Lemma 1.2]). If Supp ++ ( M ) = ∅ , then by convention we havedim (cid:0) Supp ++ ( M ) (cid:1) = −∞ and dim (cid:0) M / H N ( M ) (cid:1) = −∞ . (cid:3) The following theorem shows that the mixed multiplicities in terms of Hilbert poly-nomials can be expressed as the ones in terms of Hilbert series after modding out thetorsion with respect to N . Additionally, we provide a different proof of Theorem 3.1. Theorem 3.4. Assume Setup 2.1. Let M be a finitely generated graded B -module. Let d ++ = dim (cid:0) Supp ++ ( M ) (cid:1) . Then, the following statements hold: IXED MULTIPLICITIES AND PROJECTIVE DEGREES OF RATIONAL MAPS 11 (i) For ν ≫ , the function length A ([ M ] ν ) becomes a polynomial in Q [ X ] of totaldegree d ++ which can be written in the form X | n | = d ++ e n (cid:16) M / H N ( M ) (cid:17) n ! X n + ( terms of total degree < d ++ ) . (ii) For each n ∈ N r with | n | ≥ d ++ we have the equality e ( n ; M ) = e n (cid:16) M / H N ( M ) (cid:17) . Proof. ( i ) For simplicity of notation set M = M / H N ( M ). Note that Lemma 3.3( iii ) yields d ++ = dim( M ) − r . From Theorem 2.2 take a decompositionHilb M ( t ) = X | n | = d ++ n ≥− Q n + ( t )( − t ) n + . None of the hypotheses or conclusions change if we consider instead the shifted module M ( − ν ) for any ν ≥ ; therefore, without any loss of generality we may assume that Q n + ( t ) ∈ Z [ t ] are polynomials in t . Let F ( t ) = Hilb M ( t ) − X | n | = d ++ n ≥ Q n + ( )( − t ) n + = X | n | = d ++ n ≥− P n + ( t )( − t ) n + where P n + ( t ) = ( Q n + ( t ) − Q n + ( ) if n ≥ Q n + ( t ) if n . From Lemma 3.3( ii ) and Theorem 2.8( ii ) we obtain that e n ( M ) > n ≥ .Then, Theorem 2.2( ii ) implies that, for any n , P n + ( t ) = Q n + ( t ) and ( − t ) n + are both divisible by some (1 − t i ). For n ≥ , since P n + ( ) = 0, we can write P n + ( t ) = P | α | > p α n + ( − t ) α where p α n + ∈ Z . By summing up, we conclude that F ( t ) can be written in the following form F ( t ) = X | n | Let 1 ≤ i ≤ r and M be a finitely generated graded B -module. Ahomogeneous element is said to be filter-regular on M if z p for all associated primes p ∈ Ass( M ) of M such that p N . A sequence of homogeneous elements z , . . . , z m ∈ B is said to be filter-regular on M if z j is a filter-regular element on M / ( z , . . . , z j − ) M forall 1 ≤ j ≤ m . Lemma 3.6. Let ≤ i ≤ r , z ∈ [ B ] e i and M be a finitely generated graded B -module.Then, the following statements are equivalent:(i) (cid:2) (0 : M z ) (cid:3) ν = 0 for ν ∈ Z r with ν ≫ .(ii) N k · (0 : M z ) = 0 for some k > .(iii) Supp ((0 : M z )) ⊆ V ( N ) .(iv) z is filter-regular on M .Proof. The equivalence ( i ) ⇔ ( ii ) is clear. For the equivalence ( ii ) ⇔ ( iii ), note thatSupp((0 : M z )) = V (Ann((0 : M z ))) ⊆ V ( N ) is equivalent to N ⊆ p Ann ((0 : M z )).( iii ) ⇒ ( iv ) Suppose that Supp ((0 : M z )) ⊆ V ( N ) and let p ∈ Ass( M ) such that p N . Then, there is an injection R/ p ֒ → M , and if z ∈ p then we would get thecontradiction 0 = R p / p R p = (cid:0) R p / p R p zR p (cid:1) ֒ → (cid:0) M p zR p (cid:1) = (0 : M z ) p . Therefore z p .( iv ) ⇒ ( iii ) Suppose z is filter-regular on M . Since the minimal primes of Ass ((0 : M z ))and Supp ((0 : M z )) coincide, it is enough to show that Ass ((0 : M z )) ⊆ V ( N ). Take p ∈ Ass( M ) such that p N , then we obtain (0 : M z ) p = (cid:0) M p zR p (cid:1) = (cid:0) M p R p (cid:1) = 0because z p . Therefore Ass ((0 : M z )) ⊆ V ( N ). (cid:3) The following lemma shows that under the assumption that the residue field k = A/ n is infinite we can always find filter-regular elements. Lemma 3.7. Let ≤ i ≤ r , M be a finitely generated graded B -module and suppose thatthe residue field k = A/ n of A is infinite. Let V be the finite dimensional k -vector space V = [ B ] e i ⊗ A k , choose a basis for V and consider the Zariski topology on V ∼ = k dim k ( V ) .Denote by π the canonical map π : [ B ] e i → V = [ B ] e i ⊗ A k . Then U = { w ∈ V | w = π ( z ) for some z ∈ [ B ] e i filter-regular on M } is a dense open subset of V .Proof. Let Ass( M ) \ V ( N ) = { p , . . . , p m } . Since p j M i , then Nakayama’s lemmaimplies that V j = π (cid:0) p j ∩ [ B ] e i (cid:1) is a proper linear subspace of V . Since n is nilpotent, p j ⊃ n and in particular p j ⊃ n [ B ] e i . Then, for any z ∈ [ B ] e i we have that z is filter-regular on M if and only if π ( z ) V ∪ · · · ∪ V m . Therefore, using that k is an infinitefield, the result follows and we obtain U = V \ ( V ∪ · · · ∪ V m ). (cid:3) IXED MULTIPLICITIES AND PROJECTIVE DEGREES OF RATIONAL MAPS 13 Remark 3.8. As customary, the assumption on the infiniteness of k = A/ n can beachieved by making the faithfully flat base change B ⊗ A B where B = A [ x ] n A [ x ] and x is an indeterminate. Note that the residue field of B is k ( x ) and that length A ([ M ] ν ) =length B ([ M ⊗ A B ] ν ) for all ν ∈ Z r and M a finitely generated graded B -module.The next proposition shows that mixed multiplicities behave nicely when taking quo-tient by a filter-regular element. Lemma 3.9. Let ≤ i ≤ r , M be a finitely generated graded B -module with d ++ =dim (cid:0) Supp ++ ( M ) (cid:1) and z ∈ [ B ] e i be a filter-regular element on M . Then, we obtain that dim (cid:0) Supp ++ ( M /z M ) (cid:1) ≤ d ++ − and e ( n ; M ) = e ( n − e i ; M /z M ) for all e i ≤ n ∈ N r such that | n | ≥ d ++ .Proof. The four-term exact sequence0 → (0 : M z ) ( − e i ) → M ( − e i ) z −→ M → M /z M → , Theorem 3.1 (or Theorem 3.4) and Lemma 3.6 give the equation P M /z M ( X ) = P M ( X ) − P M ( X − e i ) . It is easy to check that P M ( X ) − P M ( X − e i ) is a polynomial of total degree smaller orequal than d ++ − n − e i is equal to e ( n ; M )( n − e i )! . So, theresult follows from Theorem 3.1 (or Theorem 3.4). (cid:3) Finally, we now extend [27, Theorem 2.4] to a multigraded setting. It reduces thecomputation of mixed multiplicities to the computation of a single-graded module. Theorem 3.10. Assume Setup 2.1. Let M be a finitely generated graded B -module.Let d ++ = dim (cid:0) Supp ++ ( M ) (cid:1) and n ∈ N r with | n | = d ++ . For ≤ i ≤ r , let z i = z i, . . . , z i,n i ∈ [ B ] e i . Suppose that z , . . . , z r is a filter-regular sequence on M . Set E tobe the single-graded module E = M / ( z , . . . , z r ) M H N (cid:16) M / ( z , . . . , z r ) M (cid:17) gr . Then, the following equation holds e ( n ; M ) = ( e ( E ) if dim( E ) = r otherwise . Proof. Applying Lemma 3.9 successively it follows that e ( n ; M ) = e ( ; M / ( z , . . . , z r ) M ).Therefore, from Theorem 3.4(ii) and Theorem 2.8 we obtain the result. (cid:3) Multidegrees of multiprojective schemes In this section, we study the degrees of multiprojective schemes via the use of mixedmultiplicities. The main objective here is to obtain a direct generalization of van derWaerden’s result [28] in a multigraded setting. The results exposed in this short sectionare probably well-known and part of the folklore, but, for the sake of completeness,we include a very short account that depends directly on the previous sections. Per-haps worthy of mentioning, our approach here is completely based upon the use offilter-regular elements (as introduced in Definition 3.5). The following setup is usedthroughout this section. Setup 4.1. Let k be a field, A be the standard multigraded polynomial ring A = k [ x , , . . . , x ,d ] ⊗ k · · · ⊗ k k [ x r, , . . . , x r,d r ]and P be the corresponding multiprojective space P = MultiProj( A ) = P d k × k · · · × k P d r k . First, we recall the notion of degree for zero-dimensional schemes over k (see, e.g.,[11, § II.3.2]). Definition 4.2. Let Y be a k -scheme of finite type with dim( Y ) = 0. The degree of Y relative to k is given by deg k ( Y ) := X y ∈ Y [ k ( y ) : k ] length ( O Y ,y ) , where k ( y ) denotes the residue field of the local ring O Y ,y .Next we define the multidegrees of closed subschemes of P in terms of mixed multi-plicities. Definition 4.3. Let X ⊂ P be a closed subscheme of P defined as X = MultiProj ( A / J )where J ⊂ A is a graded ideal. Let n ∈ N r with | n | ≥ dim( X ). The multidegree of X oftype n with respect to P is given bydeg n P ( X ) := e (cid:18) n ; A J (cid:19) . Note that, equivalently, the multidegrees of a closed subscheme of P can be definedeasily in terms of Chow rings. Remark 4.4. The Chow ring of P = P d k × k · · · × k P d r k is given by A ∗ ( P ) = Z [ ξ , . . . , ξ r ] (cid:16) ξ d +11 , . . . , ξ d r +1 r (cid:17) where ξ i represents the class of the inverse image of a hyperplane of P d i k under thecanonical projection π i : P → P d i k . If X ⊂ P is a closed subscheme of P of dimension d = dim( X ), then the class of the cycle associated to X coincides with[ X ] = X ≤ n i ≤ d i | n | = d deg n P ( X ) ξ d − n · · · ξ d r − n r r ∈ A ∗ ( P ) . Definition 4.5. For a closed subscheme X = MultiProj ( A / J ) ⊂ P , we say that H ⊂ P is a filter-regular hyperplane on X if H is given by H = V ++ ( h ) where h ∈ [ A ] e i , for some1 ≤ i ≤ r , is a filter-regular element on A / J . Similarly, we say that H , . . . , H m ⊂ P isa filter-regular sequence of hyperplanes on X if H k is a filter-regular hyperplane on theclosed subscheme X ∩ H ∩ · · · ∩ H k − for all 1 ≤ k ≤ m .We say that H ⊂ P is a hyperplane in the i -th component of P if H = V ++ ( h ) forsome h ∈ [ A ] e i . Remark 4.6. (i) A property P is said to be satisfied by a general hyperplane in the i -th component of P , if there exists a dense open subset U of [ A ] e i with the Zariskitopology such that every hyperplane in U satisfies the property P .(ii) If we fix a closed subscheme X ⊂ P and assume that k is an infinite field, thenLemma 3.7 implies that a sequence H , . . . , H m ⊂ P of general hyperplanes will be afilter-regular sequence of hyperplanes on X . IXED MULTIPLICITIES AND PROJECTIVE DEGREES OF RATIONAL MAPS 15 Since it could be of interest, we do not assume that the field k is infinite and weexpress the following result in terms of filter-regular hyperplanes. Theorem 4.7. Assume Setup 4.1. Let X ⊂ P be a closed subscheme of P . Let n ∈ N r with | n | = dim( X ) . For ≤ i ≤ r , let H i, , . . . , H i,n i ⊂ P be a sequence of hyperplanesin the i -th component of P . Suppose that H , , . . . , H ,n , . . . , H i, , . . . , H i,n i , . . . , H r, , . . . , H r,n r ⊂ P is a filter-regular sequence of hyperplanes on X . Then, the following equality holds deg n P ( X ) = deg k X ∩ \ ≤ i ≤ r ≤ j ≤ n i H i,j . Proof. Suppose that X = MultiProj( B ) with B = A / J . Let h i,j ∈ [ A ] e i such that H i,j = V ++ ( h i,j ) ⊂ P . The closed subscheme Y = X ∩ \ ≤ i ≤ r ≤ j ≤ n i H i,j can be expressed as Y = MultiProj (cid:0) B / ( h , , . . . , h r,n r ) B (cid:1) .By using Lemma 3.9 and Theorem 3.1 (or Theorem 3.4), it follows that either dim( Y ) =0 or Y = ∅ and thatdeg n P ( X ) = e ( n ; B ) = e (cid:0) ; B / ( h , , . . . , h r,n r ) B (cid:1) . From Serre’s Vanishing Theorem (see [21, Lemma 4.2] for a multigraded setting) we getthat e (cid:0) , B / ( h , , . . . , h r,n r ) B (cid:1) = dim k (cid:0) [ B / ( h , , . . . , h r,n r ) B ] m (cid:1) = dim k (cid:0) H ( Y , O Y ( m )) (cid:1) for m ≫ . Since dim( Y ) = 0, we obtain that Y ∼ = ` y ∈ Y Spec( O Y ,y ) and thatH ( Y , O Y ( m )) = H ( Y , O Y ) for any m (see, e.g., [12, Proposition 5.11]). By summingup, we have deg n P ( X ) = dim k (cid:0) H ( Y , O Y ) (cid:1) = X y ∈ Y [ k ( y ) : k ] length ( O Y ,y ) . So, the result follows. (cid:3) Projective degrees of rational maps During this section, we concentrate on the projective degrees of a rational map. Thesenumbers are defined as the multidegrees of the graph of a rational map. As applications,we provide explicit formulas for the projective degrees of rational maps determined byperfect ideals of height two or by Gorenstein ideals of height three. The following setupwill be used throughout this section. Setup 5.1. Let k be a field, R be the polynomial ring R = k [ x , . . . , x d ] and m ⊂ R bethe graded irrelevant ideal m = ( x , . . . , x d ). Let n ≥ d and F : P d k = Proj( R ) P n k be a rational map defined by n + 1 homogeneous elements f = { f , . . . , f n } ⊂ R of thesame degree δ > 0. Let I ⊂ R be the homogeneous ideal I = ( f , . . . , f n ) and P be thebiprojective space P = P d k × k P n k . Let Y ⊆ P n k and Γ ⊆ P be the closed subschemesgiven as the closures of the image and the graph of F , respectively. Definition 5.2. For 0 ≤ i ≤ d , the i -th projective degree of F : P d k P n k is given by d i ( F ) := deg i,d − i P (Γ)(see Definition 4.3).Equivalently, the projective degrees of a rational map can be defined as in Remark 4.4(see, e.g., [14, Example 19.4] and [9, § Y and Γ in algebraic terms as follows (for more details, see, e.g.,[8, § A be the bigraded polynomial ring A = R [ y , . . . , y n ] where bideg( x i ) =(1 , 0) and bideg( y i ) = (0 , R ( I ) = L ∞ q =0 I q t q ⊂ R [ t ] gives thebihomogeneous coordinate ring of Γ and can be presented as a quotient of A via thecanonical R -epimorphism Ψ : A ։ R ( I ) ⊂ R [ t ] y i f i t. The standard graded k -algebra S = k [ f , . . . , f n ] = L ∞ q =0 [ I q ] qδ gives the homogeneouscoordinate ring of Y and we have the canonical epimorphism k [ y , . . . , y n ] ։ S, y i f i .In geometrical terms, we obtain the closed immersions Γ = BiProj( R ( I )) ֒ → P =BiProj( A ) and Y = Proj( S ) ֒ → P n k = Proj( k [ y , . . . , y n ]).Our main tool for the computation of the projective degrees of a rational map willbe the saturated special fiber ring. Definition 5.3 ([5]) . The saturated special fiber ring of I is given by the graded k -algebra e F ( I ) := ∞ M q =0 (cid:2)(cid:0) I q : m ∞ (cid:1)(cid:3) qδ . A very important feature of e F ( I ) is that it is a finitely generated S -module and thatits multiplicity is equal to e (cid:16)e F ( I ) (cid:17) = deg( F ) deg P n k ( Y ) (i.e., the product of the degreesof the map F and its image Y ; see [5, Theorem 2.4]). Although it is well-known that d ( F ) = deg( F ) deg P n k ( Y ), in the following theorem we provide a direct proof of theequality d ( F ) = e (cid:16)e F ( I ) (cid:17) . Theorem 5.4. Assume Setup 5.1. If F : P d k P n k is a generically finite map, then d ( F ) = e (cid:16)e F ( I ) (cid:17) . Proof. For notational purposes set b = ( y , . . . , y n ), N = m ∩ b and M = m + b . TheMayer-Vietoris sequence (see, e.g., [2, Theorem 3.2.3]) yields the exact sequenceH i M ( R ( I )) → H i m ( R ( I )) ⊕ H i b ( R ( I )) → H i N ( R ( I )) → H i +1 M ( R ( I ))for all i ≥ 0. Since (cid:2) H i M ( R ( I )) (cid:3) (0 ,j ) = 0 and (cid:2) H i b ( R ( I )) (cid:3) (0 ,j ) = 0 for all j ≫ 0, it followsthat(2) (cid:2) H i m ( R ( I )) (cid:3) (0 ,j ) ∼ = (cid:2) H i N ( R ( I )) (cid:3) (0 ,j ) for all i ≥ j ≫ X = Proj R -gr ( R ( I )) be the projective scheme obtained by considering R ( I ) assingle-graded with the grading of R (i.e., by setting deg( x i ) = 1 and deg( y i ) = 0). Then, e F ( I ) is also given by e F ( I ) ∼ = H ( X, O X ) IXED MULTIPLICITIES AND PROJECTIVE DEGREES OF RATIONAL MAPS 17 (see [5, Lemma 2.8]).We have following relations between sheaf and local cohomologies (see, e.g., [18,Corollary 1.5], [10, Appendix A4.1])(3) 0 → (cid:2) H N ( R ( I )) (cid:3) (0 ,j ) → [ R ( I )] (0 ,j ) → H (cid:0) Γ , O Γ (0 , j ) (cid:1) → (cid:2) H N ( R ( I )) (cid:3) (0 ,j ) → → (cid:2) H m ( R ( I )) (cid:3) (0 ,j ) → [ R ( I )] (0 ,j ) → (cid:2) H (cid:0) X, O X (cid:1)(cid:3) j → (cid:2) H m ( R ( I )) (cid:3) (0 ,j ) → . Combining (2), (3) and (4) we obtain that(5) he F ( I ) i j ∼ = (cid:2) H ( X, O X ) (cid:3) j ∼ = H (cid:0) Γ , O Γ (0 , j ) (cid:1) for j ≫ R ( I ) is given by P R ( I ) ( u, v ) = d X i =0 d i ( F ) i !( d − i )! u i v d − i + (terms of total degree < d ) . By using the bigraded version of the Grothendieck-Serre formula (see, e.g., [21, Lemma4.3],[19, Theorem 2.4]), we obtain that P R ( I ) (0 , j ) = X i ≥ ( − i dim k (cid:16) H i (Γ , O Γ (0 , j )) (cid:17) for all j . Then, (2), [18, Corollary 1.5] and (5) imply that P R ( I ) (0 , j ) = dim k (cid:18)he F ( I ) i j (cid:19) + X i ≥ ( − i dim k (cid:16)(cid:2) H i +1 m ( R ( I )) (cid:3) (0 ,j ) (cid:17) for j ≫ 0. From [8, Corollary 4.5] (also, see [5, Proposition 3.1]), we have that (cid:2) H i m ( R ( I )) (cid:3) (0 , ∗ ) is a finitely generated graded S -module withdim (cid:16)(cid:2) H i m ( R ( I )) (cid:3) (0 , ∗ ) (cid:17) ≤ d + 1 − i. Since S ֒ → e F ( I ) is an integral extension and F is generically finite, it follows thatdim (cid:16)e F ( I ) (cid:17) = dim( S ) = d + 1. Therefore, for j ≫ 0, dim k (cid:18)he F ( I ) i j (cid:19) becomes apolynomial of degree d whose leading coefficient coincides with the leading coefficientof P R ( I ) (0 , v ). This implies that d ( F ) = e (cid:16)e F ( I ) (cid:17) , and so we are done. (cid:3) We recall a condition that will assumed in the next subsection. We say that I satisfiesthe condition G d +1 when µ ( I p ) ≤ dim( R p ) for all p ∈ Spec( R ) such that ht( p ) < d + 1 . Remark 5.5. In terms of Fitting ideals, I satisfies the condition G d +1 if and only ifht(Fitt i ( I )) > i for all 1 ≤ i < d + 1. Proof. It follows from [10, Proposition 20.6]. (cid:3) The proposition below contains some reductions to be used later. Proposition 5.6. Assume Setup 5.1 and suppose that k is an infinite field. There existelements h , . . . , h d ∈ [ R ] such that, if we set S i = R/ ( h , . . . , h i ) R and J i = IS i for ≤ i ≤ d , then the following statements hold:(i) d i ( F ) = e (cid:0) , d − i ; R S i ( J i ) (cid:1) .(ii) If ht( I ) = c , then d i ( F ) = δ d − i for all d − c + 1 ≤ i ≤ d .(iii) If R/I is Cohen-Macaulay with minimal graded free resolution F • : 0 → F c → · · · → F → F → R/I → , then, for all ≤ j ≤ d − c , S j /J j is Cohen-Macaulay with minimal graded freeresolution F • ⊗ R S j : 0 → F c ⊗ R S j → · · · → F ⊗ R S j → F ⊗ R S j → S j /J j → . Additionally, if I satisfies the condition G d +1 , then J j satisfies the condition G d +1 − j for all ≤ j ≤ d − c .Proof. Set L i ⊂ R to be the ideal L i = Fitt i ( I ) for 1 ≤ i < d + 1. By using Lemma 3.7we can find a sequence h , . . . , h d ∈ [ R ] = [ R ( I )] (1 , which is filter-regular on R ( I ), ongr I ( R ) = R ( I ) ⊗ R ( R/I ) = ∞ M q =0 I q /I q +1 , on R/I , and on R/L i for all i .( i ) Applying − ⊗ R S i to the inclusion R ( I ) ֒ → R [ t ] yields a natural map s : R ( I ) ⊗ R S i ։ R S i ( J i ) ⊂ S i [ t ] . For p ∈ Spec( R ) \ V ( I ), localizing the surjection s : R ( I ) ⊗ R S i ։ R S i ( J i ) at R \ p ,we easily see that it becomes an isomorphism. It then follows that some power of I annihilates Ker( s ), that is, I l · Ker( s ) = 0 for some l > 0. We have that dim (gr I ( R )) =dim( R ) = d + 1 and dim ( R ( I )) = dim( R ) + 1 = d + 2 (see, e.g., [17, § (cid:0) Supp ++ (Ker( s )) (cid:1) ≤ dim (cid:0) Supp ++ (cid:0) ( R ( I ) ⊗ R S i ) ⊗ R ( R/I ) (cid:1)(cid:1) = dim (cid:0) Supp ++ (gr I ( R ) ⊗ R S i ) (cid:1) ≤ dim(gr I ( R )) − − i = d − − i and dim (cid:0) Supp ++ ( R ( I ) ⊗ R S i ) (cid:1) = dim (cid:0) R ( I ) (cid:1) − − i = d − i. Hence, from the short exact sequence 0 → Ker( s ) → R ( I ) ⊗ R S i → R S i ( J i ) → e (0 , d − i ; R ( I ) ⊗ R S i ) = e (0 , d − i ; R S i ( J i )) . By using Lemma 3.9 successively we obtain d i ( F ) = e (cid:16) i, d − i ; R ( I ) (cid:17) = e (cid:0) , d − i ; R ( I ) ⊗ R S i (cid:1) . So, the result follows.( ii ) The condition of h , . . . , h d being a filter-regular sequence on R/I yields that J i is an m S i -primary ideal for d − c + 1 ≤ i ≤ d . It then follows that d i ( F ) = e (cid:0) , d − i ; R S i ( J i ) (cid:1) = δ d − i (see, e.g., [22, Observation 3.2]).( iii ) Since pd( R/I ) = c , the Auslander-Buchsbaum formula implies that depth( R/I ) = d − c . When R/I is Cohen-Macaulay and k is infinite, we can assure that h , . . . , h d − c is a regular sequence on R and on R/I (see, e.g., [3, Proposition 1.5.12]). Then, using IXED MULTIPLICITIES AND PROJECTIVE DEGREES OF RATIONAL MAPS 19 that h , . . . , h d − c is a regular sequence on R and on R/I , for 1 ≤ j ≤ d − c , it followsthat S j /J j ∼ = R/ ( I, h , . . . , h j ) is Cohen-Macaulay and thatH l ( F • ⊗ R S j ) ∼ = Tor Rl ( R/I, S j ) ∼ = H l (cid:0) K • ( h , . . . , h j ; R/I ) (cid:1) = 0for l ≥ K • ( h , . . . , h j ; R/I ) denotes the Koszul complex). Thus, F • ⊗ R S j is theminimal graded free resolution of S j /J j .For 1 ≤ j ≤ d − c , since F ⊗ R S j → F ⊗ R S j → J j → J j ,we get that Fitt i ( J j ) = L i S j . Since h , . . . , h d − c is a filter-regular sequence on R/L i ,the assumption of the condition G d +1 yields that ht( L i S j ) = min { ht( L i ) , dim( S j ) } ≥ min { i + 1 , dim( S j ) } for all 1 ≤ j ≤ d − c . This implies that J j satisfies G d +1 − j for all1 ≤ j ≤ d − c .So, we are done. (cid:3) Certain families of rational maps. In this short subsection we compute allthe projective degrees of the rational map F : P d k P n k when I is a perfect ideal ofheight two or a Gorenstein ideal of height three, under the assumption of the condition G d +1 . The results of this subsection are easy consequences of the previous developmentstogether with [6, Theorem A] and [7, Theorem A]. It should be noted that the condition G d +1 is always satisfied by generic perfect ideals of height two and by generic Gorensteinideals of height three. Theorem 5.7. Assume Setup 5.1 with the following conditions:(i) I is perfect of height two with Hilbert-Burch resolution of the form → n M i =1 R ( − δ − µ i ) ϕ −→ R ( − δ ) n +1 → I → . (ii) I satisfies the condition G d +1 .Then, the projective degrees of F : P d k P n k are given by d i ( F ) = e d − i ( µ , µ , . . . , µ n ) where e d − i ( µ , µ , . . . , µ n ) denotes the elementary symmetric polynomial e d − i ( µ , µ , . . . , µ n ) = X ≤ j We can assume that k is an infinite field. From Proposition 5.6, we can find h , . . . , h d such that, if we set S i = R/ ( h , . . . , h i ) and J i = IS i , then d i ( F ) = e (cid:0) , d − i ; R S i ( J i ) (cid:1) and, for all 1 ≤ j ≤ d − J j is a perfect ideal of ideal height two thatsatisfies G d +1 − j with syzygies of degrees µ , µ , . . . , µ n .For 0 ≤ j ≤ d − 2, note that d j ( F ) = e (cid:0) , d − j ; R S i ( J j ) (cid:1) is equal to the 0-th projectivedegree of a rational map determined by the minimal generators of J j , then Theorem 5.4and [6, Theorem A] yield that d j ( F ) = e (cid:16)e F S j ( J j ) (cid:17) = e d − j ( µ , . . . , µ n ) . On the other hand, the case d − ≤ j ≤ d follows directly from Proposition 5.6( ii ).So, we are done. (cid:3) Theorem 5.8. Assume Setup 5.1 with the following conditions:(i) I is a Gorenstein ideal of height three.(ii) Every non-zero entry of an alternating minimal presentation matrix of I has degree D ≥ . (iii) I satisfies the condition G d +1 .Then, the projective degrees of F : P d k P n k are given by d i ( F ) = ( D d − i P ⌊ n − d + i ⌋ k =0 (cid:0) n − − kd − i − (cid:1) if 0 ≤ i ≤ d − δ d − i if d − ≤ i ≤ d. Proof. The proof follows verbatim to the one of Theorem 5.7 but now using [7, TheoremA] (instead of using [6, Theorem A]). More explicitly, for 0 ≤ j ≤ d − 3, by substitutingin the formula of [7, Theorem A] we obtain that d j ( F ) = D ( d − j +1) − (cid:4) ( n +1) − ( d − j +1)2 (cid:5)X k =0 (cid:18) ( n + 1) − − k ( d − j + 1) − (cid:19) . Again, the case d − ≤ j ≤ d follows directly from Proposition 5.6( ii ). 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