Degree of Rational Maps versus Syzygies
aa r X i v : . [ m a t h . A C ] J u l Degree of Rational Maps versus Syzygies
M. Chardin S. H. Hassanzadeh A. Simis ∗ Abstract
One proves a far-reaching upper bound for the degree of a generically finite rationalmap between projective varieties over a base field of arbitrary characteristic. Thebound is expressed as a product of certain degrees that appear naturally by consideringthe Rees algebra (blowup) of the base ideal defining the map. Several special casesare obtained as consequences, some of which cover and extend previous results in theliterature.
Introduction
Let X ⊆ P nk = Proj( k [ x , · · · , x n ]) be a reduced and irreducible closed projective variety,where k is a field of arbitrary characteristic. Let Φ : X P mk = Proj( k [ y , · · · , y m ]) be a generically finite rational map. The purpose of this work is to give bounds forthe degree deg(Φ) , a numerical invariant that can be identified as the field degree ofthe extension of the corresponding function fields of X and the (closure of the) imageof Φ . It is common in algebraic geometry to deal with this invariant in terms of Chernclasses. In commutative algebra, it can be addressed in several other ways, includingreformulations in terms of certain multiplicities. Along this line, pretty much of the theoryof rational/birational maps meets some important algebraic constructions and invariants.In this work the main discussion will be carried within this perspective; in particular, theobtained bounds are expressed in terms of these algebraic invariants.The most basic fact at the outset is that Φ is represented by a set of homogeneouselements { f , · · · , f m } ⊂ R of a certain degree d – the so to say coordinates of Φ – where R stands for the homogeneous coordinate ring of the given embedding X ⊆ P nk . Such arepresentation is unique up to proportionality of the coordinates with multipliers in the ∗ Partially supported by a CNPq grant (302298/2014-2). Mathematics Subject Classification 2020 (MSC2020). Primary 13A30, 13D02, 14E05 Secondary 13A02,13H10, 13H15, 14A10. R . The ideal I := ( f , · · · , f m ) ⊂ R is often called a base ideal of Φ . Itis this ideal, along with the various algebras associated to it, that will encompass most ofthe work. This approach has been shown quite useful in the previous literature (see, e.g.,[BJ], [Si], [SiVi], [DHS], [KPU]).The main result of this paper is Theorem 3.3, along with its various consequences andspecial cases, some of which have been proved before. It gives upper bounds for deg(Φ) and rests on a close study of the Rees algebra (blowup) R R ( I ) of the base ideal I andits main features. In particular, the essential statement draws upon the structure of thedefining degrees of certain subideals of the defining ideal of R R ( I ) .Since this result is for arbitrary maps, one cannot expect that the obtained upperbound be attained without further assumptions on the base ideal. Yet, one can also obtaina lower bound in certain cases (Theorem 4.1), in terms of other invariants such as theCastelnuovo–Mumford regularity and the so-called Jacobian dual rank of [DHS].The overall strategy of the main result is to reduce it to finding an upper bound forthe multiplicity of a one-dimensional standard domain over the field of fractions of thehomogeneous coordinate ring of the closed image of Φ . Since the latter ring is identifiedwith the special fiber (or fiber cone) of the base ideal, there ensues a subtle algebraic pathto be followed in the argument.An important case of the theorem relates to the knowledge of the generating degrees ofthe syzygies of I . Upper bounds in terms of these degrees have been established before inparticular cases of I , such as when it has dimension one or when it is Cohen–Macaulay ofcodimension two (see, e. g., [SUV], [BCD], [CidSi]) – exact formulas of varying complexityare also known in terms of certain local multiplicities. A built-in reason that the upperbounds in Theorem 3.3 typically fail to be attained is that they are a product structure,while exact formulas usually include differences of products.The contents of the sections are as follows.The first section brings in the required algebraic preliminaries and their mutual rela-tionship. The basic ingredients are a standard graded algebra R over an arbitrary fieldand an ideal I ⊂ R generated by forms of a fixed degree.The second section is a collection of preliminaries regarding the degree of structures incertain situations and some prime avoidance like results regarding generators of homoge-neous ideals of R . The contents are pretty technical to be explained here, so the reader isreferred to the details of the section.The third section contains the main bulk of the paper. The main result is Theorem 3.3,giving an upper bound for the degree of a generically finite rational map Φ : X Y asa product of the degree of X and certain standard degrees coming from the Rees algebra R R ( I ) of the base ideal I ⊂ R of Φ , where R denotes the homogeneous coordinate ring of2 . Getting such degrees is pretty effective in the sense that they depend on the generatingstructure of the defining ideal J of R R ( I ) or of some subideal of J over which J is a minimalprime. One of the consequences of the main theorem is an upper bound for deg(Φ) in termsof the syzygy degrees of the base ideal I or of certain graded submodules of maximal rankmapping to the syzygy module of I . The section ends with some comparison with thedetails of the Jacobian dual matrix method as developed in [DHS]. The ideas are thenillustrated through a couple of examples.The fourth section deals with lower bounds. These are expressed as sums involving theso-called Jacobian dual rank of Φ and the regularity of the one-dimensional graded domainmentioned above. It is shown, in particular, that the defect of the Jacobian dual rank of Φ relative to its maximal possible value is a lower bound for deg(Φ) .The last section recovers some of the consequences of Theorem 3.3 in terms of theso-called row ideals introduced in [EU]. Let R denote a Noetherian ring and I ⊂ R an ideal.The Rees algebra of I is defined as the R -subalgebra R R ( I ) := R [ Iu ] = M n ≥ I n u n ⊂ R [ u ] . Assume that ( R, m ) is a standard graded algebra over a field k with maximal irrelevantideal m = ([ R ] ) . Then, just as for a local ring, one defines the fiber cone (or special fiber )of I to be F R ( I ) := R R ( I ) / m R R ( I ) , and the analytic spread of I , denoted by ℓ ( I ) , to be the (Krull) dimension of F R ( I ) .Let I ⊂ R be a homogeneous ideal generated by forms { f , . . . , f m } ⊂ R of the samedegree d > – in particular, I = ( I d ) . Consider the bigraded k -algebra S := R ⊗ k k [ y , . . . , y m ] = R [ y , . . . , y m ] , where k [ y , . . . , y m ] is a polynomial ring over k and the bigrading is given by bideg([ R ] ) =(1 , and bideg( y i ) = (0 , . Setting bideg( u ) = ( − d, , then R R ( I ) = R [ Iu ] inherits abigraded structure over k . One has a bihomogeneous R -homomorphism S −→ R R ( I ) ⊂ R [ u ] , y i f i u. Thus, the bigraded structure of R R ( I ) is given by R R ( I ) = M c,n ∈ Z [ R R ( I )] c,n and [ R R ( I )] c,n = [ I n ] c + nd u n . R -grading of the Rees algebra, namely, [ R R ( I )] c = L ∞ n =0 [ R R ( I )] c,n , and of particular interest is [ R R ( I )] = ∞ M n =0 [ I n ] nd u n = k [ I d u ] ≃ k [ I d ] = ∞ M n =0 [ I n ] nd ⊂ R. Clearly, R R ( I ) = [ R R ( I )] ⊕ (cid:0)L c > [ R R ( I )] c (cid:1) = [ R R ( I )] ⊕ m R R ( I ) . Therefore, one gets k [ I d ] ≃ [ R R ( I )] ≃ R R ( I ) / m R R ( I ) (1.1)as graded k -algebras. We will use this identification whenever needed without further ado.Finally, consider the canonical surjective mapSym R ( I ) ։ R R ( I ) , (1.2)where the source denotes the symmetric algebra of I . Fix algebra presentations R R ( I ) = S/J and Sym R ( I ) = S/L , where S = R [ y , · · · , y m ] is as above and L ⊂ J are bihomoge-neous ideals of S – so named presentation ideals . It is well-known, or easy to see, that L isgenerated by the biforms g y + · · · + g m y m of bidegree ( ∗ , such that g f + · · · + g m f m = 0 – in other words, L = I ( ψ ) where ψ is the defining matrix of a free R -presentation of I : R s ψ −→ R m +1 −→ I → . (1.3)It follows that L is identified with the bihomogenous ideal ( J ( ∗ , ) = M r > J ( r, ! . Form now on, assume that R is a domain.Moreover, if I has a regular element a then J = L : ( a ) ∞ . Supposing, for simplicity,that R is a domain then J is an associated prime of L . But, since Sym R ℘ ( I ℘ ) = R R ℘ ( I ℘ ) forevery ℘ ∈ Spec R \ V ( I ) , a power of I kills the kernel of (1.2), hence any prime containing L contains either I or J . It follows that J is actually a minimal prime of L . Remark 1.1.
There are many graded R -submodules N of Im ( ψ ) in (1.3) such that J isobtained by saturating the corresponding bihomogeneous subideal of L – e.g., the submod-ule generated by the Koszul syzygies or, for that matter, any N having maximal rank m .For any such N with corresponding subideal I ⊂ L , one might assume as well that J is aminimal prime thereof. At the other end, by a similar token, any squeezed bihomogeneoussubideal L ⊂ I ⊂ J will serve the same purpose.4his observation clarifies the nature of the hypothesis in the main theorem of this work. The following result will play an important role in the sequel. Similar consideration hasappeared in [SUV, (ii), p. 251].
Lemma 2.1.
Let A = L ∞ i =0 A i denote a standard graded Noetherian domain of dimension over a field F = A . Let K ( A ) be the subfield of homogeneous fractions of degree zero inthe field of fractions of A . Then K ( A ) is a finite extension of k of degree [ K ( A ) : F ] = e ( A ) . Proof.
Set c := e ( A ) . Assume that dim F ( A i ) = c for i > d and let { a , · · · , a c } be a F -vector basis for A d . Let H ∈ A be a linear form such that Q ( A ) = K ( A )( H ) i.e. H is transcendental over K ( A ) . Since dim( A ) = 1 , trdeg F Q ( A ) = 1 hence trdeg F K ( A ) = 0 .We show that K ( A ) is generated by { a /H d , · · · , a c /H d } as a F -vector space.We first show that fractions of the form β/H ld belongs to F h a /H d , · · · , a c /H d i where β ∈ A ld for some l . Since dim F ( A ld ) = c , there exist γ , · · · , γ c ∈ F such that β = P γ i H ( l − d a i . Clearly one has, β/H ld = P γ i a i /H d . Now we show that H ld /β belongs to F h a /H d , · · · , a c /H d i . To this order, notice that K ( A ) is algebraic over F , so that there exist θ i ∈ F such that βH ld ( X θ i ( βH ld ) i ) = 1 . Since the terms in the parenthesis belong to F h a /H d , · · · , a c /H d i , we conclude that H ld /β as well belongs.Now let β/γ be a homogenous fraction. With out loss of generality we may assume thatboth β and γ have degree ld . We can then write β/γ = ( H ld /γ )( β/H ld ) which belongs to F h a /H d , · · · , a c /H d i .The next lemma looks a bit familiar in some of its parts, but we could find no precisestatement in the literature. Lemma 2.2.
Let R = L i > R i be a standard N -graded *local Noetherian ring with *max-imal ideal m , over a local base ring ( R , m ) having infinite residue field. Let a be ahomogeneous ideal of R generated by r homogeneous elements of degrees d > · · · > d r > and p be a minimal prime of a of height m . Then: (1) There exists a subset { α , · · · , α r } ⊆ a such that deg( α i ) = d i and ht ( α , · · · , α i ) p = i for i = 1 , · · · , m . In particular, p is a minimal prime of ( α , · · · , α m ) . If, moreover, R is a domain and m > , then a contains { α , · · · , α r } with deg( α i ) d i such that ht( α , · · · , α i − , α r ) p = i for i m . (3) If further R is a factorial domain, R is a field, m > and d r = 1 . Then a contains { α , · · · , α r } with deg( α i ) d i such that ht( α , · · · , α i − , α r − , α r ) p = i for i m .Proof. As usual, for a graded R -module M and an integer j , M j stands for the j th gradedcomponent of M , a module over the ground local ring R . Let { f , · · · , f r } be a homoge-neous generating set of a with d t = deg f t for all t .(1) We will induct on m . The case m = 0 is trivial. Assume that m > and let P , · · · , P n ⊂ R denote the minimal primes of R contained in p . Claim. a d \ S ni =1 P id = ∅ .Since p is minimal over a and m > , one has a P i and P i = m for every i n . Byprime avoidance, let i j ∈ { , · · · , r } such that f i j / ∈ P i . In particular, we have P id ij = R d ij ,(recall that, by assumption, d i j > ). Set c i := d − d i j . Since R is a standard positivelygraded ring, for any graded prime ideal P and any µ > , P µ = R µ if and only if P = R .Hence, R c i \ P ic i = ∅ and for any r i ∈ R c i \ P ic i , r i f i j ∈ a d \ P id . Thus, a d = P id T a d for every i n . By Nakayama’s lemma, a d = P id T a d + m a d .Since R / m is aninfinite field, then a d = S ni =1 ( P id T a d + m a d ) . In particular, a d \ S ni =1 P id = ∅ , asrequired in the claim.Now, let α ∈ a d \ S ni =1 P id . The avoidance-like result in the last step above canactually be accomplished by choosing α of the form f + P i > l i f i , for suitable ground co-efficients l i ∈ R . Therefore, the ideal a / ( α ) is generated by elements of degrees d , · · · , d r .Moreover, since α P i for every i , p / ( α ) is a minimal prime of a / ( α ) of height m − .The result now follows by induction.(2) As f r = 0 and R is a domain, f r is a non zero divisor and one can apply the resultof (1) to the ring R/ ( f r ) and the image of a in this ring, thus proving the assertion. Notethat, since there is no assumption on the minimality of the generating set { f , · · · , f r } , itis possible that some of the f i ’s vanish modulo ( f r ) . Thus in the argument of part (1), thechosen element α may have degree at most d and so on.(3) As d r = 1 the element f r is irreducible. Since R is a factorial domain, the ideal ( f r ) is prime. Hence, applying (2) to the ring R/ ( f r ) and the image of a in this ring yieldsthe assertion.The previous lemma admits the following natural generalization.. Lemma 2.3.
Let R = L i > R i be a standard N -graded *local Noetherian domain, with*maximal ideal m over a local base ring ( R , m ) having infinite residue field. Let a be a omogeneous ideal of R generated by r homogeneous elements of degrees d > · · · > d r > and P be the set of minimal primes of R/ a . Then there exists a subset { α , · · · , α r } ⊂ a with deg( α i ) d i such that ht( α , · · · , α i − , α r ) p = i for all p ∈ P and i = 1 , · · · , ht( p ) .Moreover, if R is not a domain then the result still holds by changing α r to α i and deg( α i ) = d i .Proof. Let N denote the number of generators of a d as an R -module. In the courseof the proof of Lemma 2.2, one notes that for the ideal p , there is an open subset U p of A NR / m such that any element α ∈ a d generated by using the coordinates of any pointin U p satisfies the inductive basics. In the present context, one can find an element α that simultaneously works for all p ∈ P by considering the intersection T p ∈ P U p of finitelymany nonempty open sets.An alternative is to adapt the proof of Lemma 2.2 by changing at the outset to theminimal primes of R contained in set S p ∈ P p (of course, the containment condition isvacuous if R is a domain).Next, consider a closed subscheme Z of P nk . Let Supp( Z ) = Z ∪ · · · ∪ Z s be a de-composition of Supp( Z ) into irreducible closed subschemes. In other words, the Z i ’s arethe schemes defined by the minimal primes corresponding to the defining ideal of Z . Wedefine δ ( Z ) := P si =1 deg( Z i ) where deg( Z i ) is the standard definition of the degree of anirreducible variety (the multiplicity with respect to the maximal ideal ( x ) ). Notice that δ ( Z ) = deg( Z ) if and only if Z is equidimensional and reduced. Lemma 2.4.
With the above notations. Let Z be any closed subscheme of P nk and let H , · · · , H t be hypersurfaces ( not necessarily reduced or irreducible ) in P nk . Then δ ( Z ∩ H ∩ · · · ∩ H t ) δ ( Z ) Y i δ ( H i ) . Proof.
By induction it is enough to show the assertion for t = 1 . Thus, set H = H andlet Z = Z ∪ · · · ∪ Z s be a decomposition of Z into irreducible closed subschemes. Then δ ( Z ∩ H ) P δ ( Z i ∩ H ) . Similarly, let H = H ∪ · · · ∪ H s be a decomposition of H into irreducible closed subschemes. Then δ ( Z i ∩ H ) P δ ( Z i ∩ H j ) . Given indices i, j ,either δ ( Z i ∩ H j ) = δ ( Z i ) = deg( Z i ) if H j ⊃ Z i , or else, the standard Bézout theorem [Ha,Theorem I.7.7] gives δ ( Z i ∩ H j ) = X l deg( T l ) X l i ( Z i , H j , T l ) deg( T l ) = deg( Z i ) deg( H j ) , where the T l ’s are the irreducible component of Z i ∩ H j .7herefore, δ ( Z i ∩ H ) P j deg( Z i ) deg( H j ) = deg( Z i ) δ ( H ) , and hence δ ( Z ∩ H ) P i deg( Z i ) δ ( H ) = δ ( Z ) δ ( H ) , as was to be shown. Lemma 2.5.
Let k be a field and V ⊂ P nk be a reduced irreducible closed subscheme and T ⊆ V be a closed subscheme defined by equations of degrees d > · · · > d r > . Let T i ’sbe the irreducible components of Supp( T ) . Then, for any p r , X { i | codim V ( T i ) p } deg( T i ) d · · · d p − d r deg( V ) . Proof.
By the flat extension k ⊂ k ( u ) , where u is a variable over k , one can assume that k is infinite.Let R be the homogeneous coordinate ring of V ⊂ P nk and let a ⊂ R be the ideal ofdefinition of T in V . Let p i be the minimal prime ideal of R/ a defining the component T i .Applying Lemma 2.3, one can choose a subset { α , · · · , α p − , α r } ⊂ a such that all p i ’s areminimal over ( α , · · · , α ht( p i ) − , α r ) . Since ( α , · · · , α ht( p i ) − , α r ) ⊆ ( α , · · · , α p − , α r ) ⊆ p i , p i must be minimal over ( α , · · · , α p − , α r ) . That is to say, every T i is an isolatedcomponent of V ∩ H · · · ∩ H p where H i ⊆ P nk is the reduced hypersurface defined by α i .The result now follows from Lemma 2.4. Remark 2.6.
If furthermore R is factorial and p := codim T i > , then for any i , deg( T i ) d · · · d p − d r − d r deg( V ) . Indeed, a minimal generator of minimal degree f of the defining ideal of T i is of degree atmost d r and f must be irreducible. One can therefore replace R by R/ ( f ) and apply thelemma. Let k denote a field of arbitrary characteristic and let X ⊆ P nk = Proj( k [ x , · · · , x n ]) be areduced and irreducible closed projective variety. One is given a rational map Φ : X P mk = Proj( k [ y , · · · , y m ]) , with (closed) image Y .We will often write k [ x ] = k [ x , · · · , x n ] and k [ y ] = k [ y , · · · , y m ] . Given a closedprojective (respectively, biprojective) variety X ⊂ P nk (respectively, X ⊂ P nk × k P mk ), we8enote by K ( X ) (respectively, K ( X ) ) its field of functions.The degree of Φ : X Y is the degree of the field extension K ( X ) | K ( Y ) . Φ is generically finite if [ K ( X ) : K ( Y )] < ∞ .The following result is well-known; we isolate it for the reader’s convenience. Lemma 3.1.
Let
Φ : X P mk be a rational map with image Y ⊂ P mk . The following areequivalent: (i) Φ is generically finite. (ii) dim( X ) = dim( Y ) . (iii) There exists a Zariski dense open subset U ⊂ Y such that Φ − ( U ) → U is a finitemorphism. So much for the geometric side.The map Φ is represented by a set of homogeneous elements { f , · · · , f m } ⊂ R of acertain degree d – called coordinates of Φ – where R stands for the homogeneous coordinatering of the given embedding X ⊆ P nk . Such a representation is unique up to proportionalityof the coordinates within R . The ideal I := ( f , · · · , f m ) ⊂ R is often called a base ideal of Φ .The degree deg( X ) of the projective variety X in its embedding is the graded Hilbertmultiplicity e ( R ) of the standard graded k -algebra R .An essential role will be played in this work by the Rees algebra R R ( I ) = R [ Iu ] of I and by its special fiber R R ( I ) / m R R ( I ) , where m = ( R ) . Recall from the preliminaries(Section 1) that R R ( I ) / m R R ( I ) ≃ k [ I d ] as graded k -algebras. Normalizing degrees, thecorresponding projective subvariety of P mk coincides with the closed image Y of Φ . Remark 3.2.
It is important to note heretofore that we will fix a base ideal of Φ , but anytwo base ideals will have graded isomorphic respective Rees algebras and special algebras.This will give certain slight instability in the main result, but this is a small price to payas far as applications are concerned.As a final piece of notation, recall from Section 1 that R R ( I ) is a bigraded k -algebrawith the induced bidegree of k [ x ] ⊗ k k [ y ] = k [ x , y ] . Thus, one can write R R ( I ) ≃ S/J ,where S := R ⊗ k k [ y ] = R [ y ] , for some bihomogeneous presentation ideal J . Again, J isnot uniquely defined by I , much less by Φ , but its main numerical invariants are uniquelydetermined by I . 9 .2 Main theorem Theorem 3.3.
Let
Φ : X Y ⊂ P mk be a generically finite rational map with source anon-degenerate reduced and irreducible variety X ⊆ P nk . Fix a base ideal I ⊂ R of Φ and let J ⊂ S as above be a bihomogeneous presentation ideal of the Rees algebra R R ( I ) of I . Let I ⊂ J be a bihomogeneous subideal, with a set of generators consisting of bihomogeneouselements of x -degrees d > · · · > d r > . If J is a minimal prime of I , then deg(Φ) d · · · d t − · d r · e ( R ) , where t = dim X .Proof. First, note that the statement at least makes sense since t r ; indeed, since afinitely generated domain over a field is catenary, one has t = dim X = dim Y m = ht( J ) r, by Krull’s prime ideal theorem.The idea of the proof consists in reducing the question to a one dimensional standardgraded domain over a field and using Lemma 2.1. We will first collect some overall factsabout the present dealings, then introduce the role of the given subideal I .From the above prolegomena and the general preliminaries in Section 1, one can write R R ( I ) = [ R R ( I )] ⊕ m R R ( I ) , a direct decomposition of graded modules over [ R R ( I )] ≃ k [ I d ] . By a previous identification, the latter is the special fiber F ( I ) of I and also thehomogeneous coordinate ring of the closed image Y of Φ . In particular, the field of fractionsof [ R R ( I )] is identified with the field Q ( Y ) of fractions of this ring.For a lighter notation, set B := k [ y ] ⊂ S := R [ y ] and take respective presentations [ R R ( I )] ≃ B/ q and R R ( I ) ≃ S/J . In this notation, one has R R ( I ) ⊗ [ R R ( I )] Q ( Y ) ≃ R R ( I ) ⊗ B/ q B q /qB q ≃ ( R R ( I ) ⊗ B B/ q ) ⊗ B/ q B q /qB q ≃ R R ( I ) ⊗ B (cid:0) B/ q ⊗ B/ q B q /qB q (cid:1) ≃ R R ( I ) ⊗ B B q /qB q ≃ R R ( I ) ⊗ B Q ( Y ) ≃ ( S/J ) ⊗ B Q ( Y ) . Claim 1. R R ( I ) ⊗ [ R R ( I )] Q ( Y ) is a one-dimensional domain.We first argue that this ring is a domain because it is a ring of fractions of R R ( I ) with respect to a multiplicatively closed set. Namely, using the notation ( S/J ) ⊗ B Q ( Y ) ,consider the multiplicatively closed set T := B \ q ⊂ S and its image ¯ T in S/J . Then thenatural map
S/J → ( S/J ) ⊗ B ⊂ ( S/J ) ⊗ B Q ( Y ) induces a map of fractions ¯ T − ( S/J ) → ( S/J ) ⊗ B Q ( Y )) since an element ¯ b ∈ ¯ T maps to ¯ b ⊗ B ⊗ B ¯ b , where ¯ b is invertible10egarded as an element of Q ( Y ) . On the other hand, by the universal property of tensorproducts, there is map going the other way in a similar fashion. It is clear that these mapsare inverses of each other.To see the dimension, since R is a domain, one has by [SV, Lemma 1.1.2]: dim R + 1 = dim R R ( I ) = dim[ R R ( I )] + ht m R R ( I ) = dim F ( I ) + ht m R R ( I ) . Since Φ is generically finite, dim F ( I ) = dim R (Lemma 3.1). Then ht m R R ( I ) = 1 .But R R ( I ) ⊗ [ R R ( I )] Q ( Y ) = (cid:0) [ R R ( I )] ⊗ [ R R ( I )] Q ( Y ) (cid:1) M (cid:0) m R R ( I ) ⊗ [ R R ( I )] Q ( Y ) (cid:1) = Q ( Y ) M m R R ( I ) ⊗ [ R R ( I )] Q ( Y )= Q ( Y )[ m R R ( I ) ⊗ [ R R ( I )] Q ( Y )] Since m R R ( I ) ⊗ [ R R ( I )] Q ( Y ) is the maximal irrelevant ideal of R R ( I ) ⊗ [ R R ( I )] Q ( Y ) , itsuffices to see that it has height one. But this follows from the fact that it is fractions ofthe prime ideal m R R ( I ) and the latter has height one.So much for the claim.Let Γ be (the closure of) the graph of Φ , namely, Γ = BiProj( R R ( I )) . Let Q (Γ) byabuse denote the field of fractions of R R ( I ) , while K (Γ) will denote the function field of Γ asa biprojective variety, i.e., the subfield of Q (Γ) consisting of the fractions of bihomogeneouselements of the same bidegree.Note the usual diagram of maps: Γ π (cid:31) (cid:31) ❄❄❄❄❄❄❄ X (cid:127) β ⑦⑦⑦⑦ Φ / ❴❴❴❴❴❴❴ Y (3.1)The map β is a rational in the context of abstract non-embedded varieties, but one canretrieve its degree as [ K (Γ) : K ( X )] . Likewise, the degree of π is [ K (Γ) : K ( Y )] – notethat both degrees are well-defined integers since dim Γ = dim R R ( I ) − R − X = dim Y . Since β is birational, its degree is , hence deg(Φ) = deg( π ) .Thus, we can move over to the rational map π . Claim 2. deg( π ) = [ K (( S/J ) ⊗ B Q ( Y )) : Q ( Y )] , where K (( S/J ) ⊗ B Q ( Y )) denotesthe subfield of Q (Γ) of fractions of homogeneous elements with same x -degree.To see this, recall that Q ( Y ) | K ( Y ) and K ( R R ( I ) ⊗ B Q ( Y )) | K (Γ) are both purelytranscendetal extensions of degree one, hence Q ( Y ) = K ( Y )( H ) and K ( R R ( I ) ⊗ B Q ( Y )) = (Γ)( H ) for a sufficiently general k -linear form = H ∈ B/ q ⊂ Q ( Y ) . Clearly, then deg( π ) = [ K (Γ) : K ( Y )] = [ K (( S/J ) ⊗ B Q ( Y )) : Q ( Y )] . The above two claims imply, via Lemma 2.1, that deg( π ) = e (( S/J ) ⊗ B Q ( Y )) .Therefore, we have attained our preliminary goal of reducing the problem to estimatingthe multiplicity of a one-dimensional standard graded domain over a field.It is now time to bring over the ideal I .One has natural inclusions of biprojective schemes over Spec ( k )Γ = BiProj k ( S/J ) ⊂ Z := BiProj k ( S/ I ) ⊂ X := BiProj k ( S ) ⊂ P nk × k P mk , inducing similar inclusions at the level of the associated projective schemes over Spec ( Q ( Y )) : e Γ = Proj Q ( Y ) (( S/J ) ⊗ B Q ( Y )) ⊂ e Z = Proj Q ( Y ) (( S/ I ) ⊗ B Q ( Y )) ⊂ e X = Proj Q ( Y ) ( S ⊗ B Q ( Y ))) ⊂ P nQ ( Y ) . On the other hand, S ⊗ B Q ( Y ) = ( R ⊗ k k [ y ]) ⊗ k [ y ] Q ( Y ) = R ⊗ k Q ( Y ) , hence e X =Proj Q ( Y ) ( R ⊗ k Q ( Y ))) . Since the Hilbert function is stable under base field extension, onehas deg( X ) = deg( e X ) .Finally, note that e Γ is a minimal component of e Z . Indeed, since J is a minimal primeof I , the ideal J ⊗ B B q ⊂ S ⊗ B B q is a minimal prime of ( I , q ) ⊗ B B q . Then the imageof J ⊗ B B q is still a minimal prime of the image of ( I , q ) ⊗ B B q over the residue ring S ⊗ B Q ( Y ) = ( S ⊗ B B q ) / ( q S ⊗ B B q ) . Claim 3. ht S ⊗ B Q ( Y ) (( J ⊗ B B q ) / ( q S ⊗ B B q )) t .Recall that t = dim X = dim R − . Since R and B = k [ y ] are in disjoint variables,one has dim R ⊗ k Q ( Y ) = dim R . On the other hand S ⊗ B Q ( Y ) = R ⊗ k Q ( Y ) . Since dim(
S/J ) ⊗ B Q ( Y ) = 1 , as seen above, then ht S ⊗ B Q ( Y ) ( J ⊗ B B q / q S ⊗ B B q ) dim S ⊗ B Q ( Y ) − dim( S/J ) ⊗ B Q ( Y ) = dim R − , as claimed.Now apply Lemma 2.5 with V = e X , T = e Z and p = ht S ⊗ B Q ( Y ) ( J ⊗ B B q / q S ⊗ B B q ) .Since e Z is defined in (standard) degrees d > · · · > d r > , it follows that deg(Φ) = deg( π ) = deg( e Γ) δ ( e Z ) d · · · d p − · d r · deg( X ) d · · · d t − · d r · deg( X ) , as was to be shown. 12 emark 3.4. (a) The ideal I is really optimal for the purpose of the stated bound whenit is generated by a subset of a minimal set of bihomogeneous generators of J .(b) If further R is factorial and t > , it follows from Remark2.6 that deg(Φ) d · · · d t − · d r − d r · e ( R ) . (c) With the same notation as above, a similar argument to reduce the degree of Φ to a local algebraic multiplicity is shown in [KPU, Theorem 5.3], namely, that deg(Φ) = e ( R R ( I ) ( x R R ( I )) ) . Thus, Theorem 3.3 provides an upper bound for the multiplicity of thelocal ring R R ( I ) ( x R R ( I )) . A notable case of the previous theorem is when I is closely related to the presentation idealof the symmetric algebra of I , in which case the bound is expressed in terms of syzygies. Corollary 3.5. (Syzygy rank criterion)
Let Let
Φ : X Y ⊂ P mk be a generically finiterational map with source a non-degenerate reduced and irreducible variety X = Proj( R ) ⊆ P nk . Fix a base ideal I ⊂ R of Φ and its minimal graded presentation F ψ −→ F = m +1 M R ( − d ) → I → . Let G be a finitely generated graded R -submodule of Im( ψ ) ⊂ F of rank m , generated indegrees d i > · · · > d i s . If grade R ( I ) > , then deg(Φ) d i · · · d i t − · d i s · e ( R ) , where t = dim X and s = µ ( G ) .Proof. Let N be the ( m + 1) × s ( s r ) matrix whose columns are the generators of G .First note that r > s > m > dim Y = dim X = t . Since N has rank m , we can choose an ( m + 1) × m submatrix M of N of rank m . With a previous notation, let J ⊂ S = R [ y ] denote a presentation ideal of the Rees algebra of I . Then one has inclusions of ideals L M := I ( y · M ) ⊂ L N := I ( y · N ) ⊂ L := I ( y · ψ ) ⊂ J. Claim. J is a minimal prime over L N .It suffices to prove that J is a minimal prime over L M . To see this, let ∆ , · · · , ∆ m bethe maximal minors of M . Since grade R ( I ) > , [HS, Lemma 3.9] that hI = (∆ , · · · , ∆ m ) ,for some non-zero element h ∈ R . Localizing at the powers of h , I h = (∆ , · · · , ∆ m ) h and grade R h (∆ , · · · , ∆ m ) h > as well, yielding the free presentation (resolution) → m M R h M −→ m +1 M R h → I h → . ( L M ) h is the defining ideal of the symmetric algebra of I h . But, R hI ( R ) = R I ( R ) as R is a domain. Therefore, J h is a presentation ideal of the Rees algebra of I h on R h ,and hence J h is a minimal prime over ( L M ) h . Since h J , J is a minimal prime over L M thence over L N too.Now the result follows from Theorem 3.3.Curiously, Corollary 3.5 provides a numerical obstruction for integrality in the contextof a Noether normalization. Corollary 3.6.
Let R be a standard graded Noetherian ring of dimension t + 1 over afield k and let { f , · · · f t } ⊂ R be a set of algebraically independent elements of the samedegree d generating an ideal of grade at least . Let N be a submatrix of maximal rank ofthe syzygy matrix of ( f , · · · f t ) . If k [ f , · · · f t ] ⊂ R is integral then t √ d · · · d t > d , where d , · · · , d t are the column degrees of N .Proof. Set X =Proj ( R ) and let Φ : X P t denote the rational map represented by f , · · · f t . If A := k [ f , · · · f t ] ⊂ R is an integral extension then, in particular, Φ is agenerically finite map.Now, Corollary 3.5 implies the inequality d · · · d t e ( R ) > deg(Φ) . On the other hand, deg(Φ) = e ( R ( d ) ) = e ( R ) d t by an application of [KPU, Observation 2.8 ] to the inclusion A ⊂ R ( d ) , where R ( d ) is the d -th Veronese subring of R . Therefore, t √ d · · · d t > d , asdesired.The next result profits from the existence of common factors among the generators ofa base ideal. We give but the simplest case. Corollary 3.7.
In the setting of
Corollary 3.5 , assume in addition that f and f have acommon factor of degree δ . Then deg(Φ) ( d − δ ) d m − · e ( R ) . Proof.
In the setting of Corollary 3.5, take G to be the submodule generated by the columnsof the reduced Koszul relations N := f /h f · · · f m − f /h · · · − f · · · ... ... · · · − f , where h is the given common factor.It is known that a regular sequence of forms of degree > cannot be the coordinatesof a birational map. The degree of the rational map it defines comes out as an equality in14he previous corollary. This value is known (see, e.g., [KPU, Observation 3.2]). We giveanother argument in the spirit of this work. Proposition 3.8.
Let
Φ : X = Proj( R ) P mk ( m > be a generically finite rationalmap with source a non-degenerate reduced and irreducible variety X ⊆ P nk . If Φ is moreoverregular and dominant and R is Cohen–Macaulay, then deg(Φ) = e ( R ) d m , where d is thegenerating degree of a base ideal.Proof. Let R be the coordinate ring of X and let I = ( f , · · · , f m ) ∈ R be a base idealgenerated in degree d . Since Φ is regular, codim I = dim( R ) , and since it is dominantand finite, dim( R ) = m + 1 . Thus, { f , · · · , f m } is a complete intersection, since R isCohen-Macaulay. In particular, grade I = m + 1 > , hence the base ideal is uniquelydefined ([Si, Proposition 1.1]), and so is the integer d . Also R R ( I ) = S R ( I ) = R [ y , · · · , y m ]( f i y j − f j y i ) . Using the notation in the proof of Theorem 3.3, tensoring with Q ( Y ) over B , allows toinvert y i ’s. Hence modulo the ideal ( f j y − f y j | j = 1 , . . . , m ) , every generator f i y j − f j y i with i, j = 0 is zero. This shows that R R ( I ) ⊗ B Q ( Y ) is a quotient of R [ y , · · · , y m ] ⊗ B Q ( Y ) by a regular sequence of length m , generated in x -degree d . Thus e ( R R ( I ) ⊗ B Q ( Y )) = e ( R ) d m and the result follows from the same argument as in the proof of Theorem 3.3. Remark 3.9.
The above is the case of a zero-dimensional base ideal. Equalities withbase ideal of dimension have been proved by various authors. For example, the followingformula is given in [BCD, Theorem 3.3] for a generically finite rational map Φ : X Y with base ideal of dimension , generated in degree d : d t e ( R ) − X p ∈ Proj ( R/I ) [ κ ( p ) : k ] ℓ ( R p /I p ) = deg(Φ) e ( S ) where R and S are the coordinate rings of X and Y and t = dim( X ) . A similar resultcan be found in [BJ, Theorem 2.5] and [SUV, Theorem 6.6]. In the case of polar mapsthis formula has been obtained in [DP] and [FM]. In addition, the conjecture of Dimcafor base ideal of dimension one has been proved by J. Huh ([Huh, Theorem 4]), namely, tothe effect that no hypersurface of degree > with only isolated singularities is homaloidal.In dim > , there is a beautiful general formula in [Xie, Theorem 4.1] generalizing theformulas of [SUV] by introducing the j -multiplicity.15 .4 Relation to the Jacobian dual rank In order to state a few more consequences, we now relate the present approach to the
Jacobian dual method as introduced in detail in [DHS].Recall that B = k [ y ] ⊂ S = R [ y ] , J ⊂ S was a presentation ideal of the Rees algebra R R ( I ) and q ⊂ B was a presentation ideal of the special fiber of I . By the identificationdiscussed in Section 1, one can view q ⊂ J sitting in bidegrees (0 , r ) , for r > , and thequotient field of the homogeneous coordinate ring of the closed image of Φ (after degreenormalization) is identified with the quotient field B q / q B q of the special fiber.Consider the ideal ( J/ q S ) ⊗ B B q ⊂ ( S/ q S ) ⊗ B B q ≃ S ⊗ B Q ( Y ) , which is naturallygraded over Q ( Y ) . The Q ( Y ) -vector space ( J/ q S ) ⊗ B B q ) in degree one will play a role inthe next result. Essentially, its elements are the x -linear bihomogeneous forms among thedefining equations of the Rees algebra of I . Its dimension s is an invariant closely relatedto the Jacobian dual rank. Remark 3.10.
The invariant s := dim Q ( Y ) (( J/ q S ) ⊗ B B q ) is at least the linear rank ofthe presentation matrix of I (see Corollary 3.5). As a consequence, Corollary 3.11 belowwill provide another proof of [DHS, Theorem 3.2]. If X ⊆ P n is non-degenerated, thenin fact s coincides with the Jacobian dual rank. It is shown in [DHS, Thorem 2.15] that s n + dim( Y ) − dim( X ) = n with equality if and only if Φ is birational. Corollary 3.11also provides an alternative argument for the implication ( b ) ⇒ ( a ) in [DHS, Theorem2.18] in the case where X = P n . Corollary 3.11.
With the above notation and the hypotheses of
Theorem 3.3 one has deg(Φ) d · · · d t − s · e ( R ) , if s t − e ( R ) , if s > t, where t = dim X .Proof. This is actually, a consequence of the proof of the main theorem. Let L ⊂ J ⊗ B B q be the subideal generated by the residues of the x -linear bihomogeneous forms from J .Since J is a minimal prime of I by assumption, J ⊗ B B q is a minimal prime of I ⊗ B B q ,hence of ( L, I ⊗ B B q ) as well. Now apply, Lemma 2.2, item (3), repeatedly s times in theproof of Theorem 3.3. Remark 3.12.
One may notice that Lemma 2.2, item (3), and Theorem 3.3 give animproved upper bound, deg(Φ) d · · · d t − s − d r · e ( R ) , if s t − , I d r L , where L is as above.We next give some simple examples regarding the upper bound established in Corol-lary 3.11. Example 3.13. (Complete intersection base ideal) Here the upper bound is attained, ascomes out of the proof of Proposition 3.8.
Example 3.14. (Monomial base ideal)(a) Consider the map
Φ : P k P k defined by the monomials { x , yz, z } . For mapsdefined by monomials, [SiVi, Proposition 2.1] gives an easy criterion of birationality interms of the gcd of the maximal minors of the corresponding log-matrix, namely, that thelatter has to be ± d , where d is the common degree of the monomials. In this case it isnearly trivial to see that Φ is not birational because the matrix is square of determinant .Alternatively, a computation with Macaulay2 will give that the base ideal I = ( x , yz, z ) has only one linear syzygy and is of linear type. Therefore, by [DHS, Proposition 3.2], Φ is not birational. It also follows that s = 1 , taking for granted that s coincides with theJacobian dual rank of I .As for the degree of Φ , one can easily sees that a general point such as (1 : ab : b ) has exactly two pre-images, so that deg(Φ) = 2 . Alternatively, one can directly computethe field extension degree [ k (( x, y, z ) ) : k ( x , yz, z )] . Namely, one has k (( x, y, z ) ) = k ( x , y/x, z/x ) and ( z/x ) = z /x ∈ k ( x , yz, z ) , while y/x = ( yz/z )( z/x ) . There-fore, [ k (( x, y, z ) ) : k ( x , yz, z )] = 2 . Finally, the full minimal graded free resolution of ( x , yz, z ) is → R ( − → R ( − ⊕ R ( − → R ( − → . Therefore, according to Corollary 3.11, deg(Φ) attains the upper bound d e ( R ) = 2 · .We note that the improved bound in Remark 3.12 is not attained here. Actually, theobstruction I d r ⊆ ( L ) takes place, where L is the presentation ideal of the symmetricalgebra(b) Let Φ : P k P k be defined by { x y, xz , y z } This belongs to the class of mono-mial rational maps whose base ideal has radical ( xy, xz, yz ) . It can be shown that theonly birational one (i.e., Cremona) is the classical involution defined by { xy, xz, yz } . Itis an easy exercise to see that the syzygies of { x y, xz , y z } are generated by the threereduced Koszul relations, all of standard degree . On the other hand, I = ( x y, xz , y z ) is an almost complete intersection that is a complete intersection locally at its mini-mal primes. Therefore, it is an ideal of linear type ([HSV, Proposition 5.5 and propo-sition 9.1]). It follows that s = 0 , hence the bounds give deg(Φ) . . . Fi-nally, by Corollary 4.3, deg(Φ) > . To decide between or , requires a direct cal-culation. Namely, y/x satisfies the equation T − ( y z ) / ( xz · x y ) of degree over17 ( x y, xz , y z ) . This equation is irreducible as y/z / ∈ k ( x y, xz , y z ) – to see this, oth-erwise, x z = ( x /y ) y z ∈ k ( x y, xz , y z ) , hence also z/x = xz /x z ∈ k ( x y, xz , y z ) .Since, by a similar token, x = x y ( x/y ) ∈ k ( x y, xz , y z ) , we would have k (( x, y, z ) ) = k ( x , y/x, z/x ) = k ( x y, xz , y z ) , which would say that Φ is birational. To conclude,since is not a multiple of , one must have deg(Φ) = 3 . Thus, the upper bound ofCorollary 3.11 is missed by .It may be interesting to stress the case where the source is two-dimensional. Corollary 3.15.
With same assumptions as in
Corollary 3.11 , let moreover dim X = 2 .Let d and d r respectively denote the highest and the lowest degrees among the degrees ofa minimal set of generating syzygies of a base ideal. Then deg(Φ) = = 1 , if s > d e ( R ) , if s = 1; d d r e ( R ) if s = 0 . The previous corollary generalizes the inequality assertion of the following result:
Proposition 3.16. ([BCD, Proposition 5.2])
Let
Φ : P P be a dominant rationalmap with a dimension base ideal I that is moreover saturated, with minimal graded freeresolution → R ( − d − µ ) ⊕ R ( − d − µ ) → R ( − d ) → . Then, deg( F ) µ µ withequality if and only if I is locally a complete intersection at its minimal primes. See [CidSi, Theorem 6.3] for more encompassing work assuming the base ideal is perfectof codimension two satisfying a certain condition on the local number of generators.
There are known lower bounds for the degree of a polar rational map for isolated singulari-ties (see, e.g., [Huh, Theorem 1]). In the case of arbitrary rational maps, any exact formulasuch as the ones quoted in Remark 3.9 implies trivially both upper and lower bounds. Adifferent story is to recover these trivial bounds in terms of other interesting and moreconcrete invariants. In this part we wish to communicate a lower bound in the spirit ofthis work, bringing up some computable invariants. For this effect we will draw upon aknown affirmative case of the Eisenbud-Goto conjecture ([EG]).Recall the notation employed in Theorem 3.3 and in its proof. Take a presentation ( S/J ) ⊗ B Q ( Y ) ≃ Q ( Y )[ x , · · · , x n ] / J .
18n the notation of Subsection 3.4, let s = dim Q ( Y ) (( J/ q S ) ⊗ B B q ) denote the dimensionof the x -linear forms. As stated in Remark 3.10, s coincides with the Jacobian dual rank jdrank(Φ) of Φ . We have seen that ( S/J ) ⊗ B Q ( Y ) is a one dimensional domain, hence isin particular Cohen–Macaulay. Theorem 4.1.
In the notation and hypotheses of
Theorem 3.3 one has deg(Φ) > n + 1 − jdrank(Φ) + reg( J ) − where reg( J ) is the Castelnuovo–Mumford regularity of J .In particular, one always has deg(Φ) > n + 1 − jdrank(Φ) .Proof. According to the proof of Theorem 3.3, to compute deg(Φ) one has to compute themultiplicity of the one dimensional domain ( S/J ) ⊗ B Q ( Y ) .We notice that the Eisenbud-Goto conjecture holds for Cohen-Macaulay rings [Eis,Corollary 4.15]. Therefore, e (( S/J ) ⊗ B Q ( Y )) > reg(( S/J ) ⊗ B Q ( Y )) + edim(( S/J ) ⊗ B Q ( Y )) − dim(( S/J ) ⊗ B Q ( Y )) . Since reg((
S/J ) ⊗ B B q ) = reg( J ) − and s = jdrank(Φ) , it suffices to show that edim(( S/J ) ⊗ B Q ( Y )) = n + 1 − s . To this end, note that ( S/J ) ⊗ B Q ( Y ) ≃ S ⊗ B Q ( Y )(( J/ q S ) ⊗ B B q ) , while S ⊗ B Q ( Y ) ≃ R ⊗ k Q ( Y ) is non-degenerated since R is so. Thus, canceling s linearforms out of the denominator, the sought embedding dimension is n + 1 − s .As for the last claim, notice that if J consists merely of linear forms then jdrank(Φ) = n ,hence in this case Φ is a birational map by [DHS], so the statement is trivial. Otherwise, reg( J ) > and the assertion follows from the proved inequality. Remark 4.2.
The lower bound of Theorem 3.3 is attained in Example 3.14 (b). Indeed, jdrank(Φ) = 1 since the base ideal is of linear type with minimal syzygy degree > , while reg( J ) = 2 as J is linearly presented perfect of codimension . Corollary 4.3.
In the notation and hypotheses of
Theorem 3.3 , if J (1 , ∗ ) = 0 then deg(Φ) > n + 1 .Proof. If J (1 , ∗ ) = 0 then jdrank(Φ) = 0 , in the notation of the Theorem 4.1. Therefore deg(Φ) > n + 1 .Let us note that an immediate application of the above corollary is to the case wherethe base ideal I is generated by a regular sequence of d -forms, with d > . In general, the19esult of the corollary was only known in the case where Φ is birational onto its image (see[DHS]).The next result provides an upper bound for the regularity of J . Proposition 4.4. ([CP])
In the above notation and the hypotheses of
Theorem 3.3 , assumethat X = P n . Then reg( J ) d + · · · + d n − + d r − n + 1 Proof.
The proof is a consequence of Lemma 2.2 and [CP, Theorem A]. The main point isthe fact mentioned in the proof of Theorem 3.3 that e Γ is an irreducible isolated componentof e Z defining a zero dimensional scheme. Although syzygies are very important, they are not the only data covered by Theorem 3.3.Actually, if one is only interested in an upper bound given by the product of the highestdegrees of the syzygies of I , then there will be other ways to see this fact, although obtainingthe same upper bound has more cost.Here, we recall the notion of a row ideal introduced in [EU].Consider the polynomial ring R [ T ] = R [ T , . . . , T m ] and a specialization map R [ T ] → R with T i q i . Given elements { f , . . . , f m } ⊂ R , let I q ⊂ R denote the ideal of R generatedby the image of the Koszul relations { f i T j − f j T i | i < j m } .The following Lemma is a basic fact about row ideals. We provide a proof for the sakeof completeness. Lemma 5.1.
Let R be a Noetherian ring and let I = ( f , · · · , f m ) be an ideal of R withpresentation matrix ψ . With the above notation, if some element among the q i ’s is a unit,then I ( q · ψ ) = I q : R I, where q = ( q · · · q m ) .Proof. Let ψ = ( a ij ) be an ( m +1) × r matrix presenting I . Let J = ( j , · · · , j r ) := I ( q · ψ ) .Note that j t = P ri =0 q i a it .For any f k ∈ I , one has j t f k = m X i =0 q i a it f k ≡ m X i =0 q k a it f i mod I q But P mi =0 q k a it f i = q k P mi =0 a it f i vanishes since ( a t , · · · , a mt ) is a syzygy of f , · · · , f m .This shows the inclusion I ( q · ψ ) ⊆ I q : R I .20o see the reverse inclusion, assume without loss of generality that q is unit. Let s ∈ ( I q : R I ) . Writing out sf ∈ I q gives sf = b ( q f − q f ) + · · · + b m ( q m f − q f m ) , for certain { b , · · · , b m } ⊂ R . Hence, ( s − P mi =1 b i q i ) f + P mi =1 q b i f i = 0 , i.e., the vector ( s − P mi =1 b i q i , q b , · · · , q b m ) is a syzygy of I . Therefore, ( s − m X i =1 b i q i , q b , · · · , q b m ) t = t C + · · · + t r C r , for some t , · · · , t r ∈ R , where C , · · · , C r denote the columns of ψ . Multiplying out bythe vector ( q · · · q m ) yields (( q s − m X i =1 b i q i q ) + m X i =1 b i q i q ) = r X i =1 t i j i . Thence q s ∈ J . Since q is a unit, s ∈ J as desired.We now return to the setting of the main Theorem 3.3 with the notation of Corol-lary 3.5. For any point q ∈ Y , the fiber Φ − ( q ) is determined by the ideal a = ( I q : R I ∞ ) .In particular, deg(Φ) = e ( R/ a ) for a general choice of q , see for example [KPU, Proposition3.6].The next proposition has already been obtained as a corollary of Theorem 3.3 underweaker conditions. Here, we give an alternative proof using the idea of row ideals. Proposition 5.2.
Let
Φ : X Y be a generically finite dominant rational map ofprojective varieties and let r M R ( − d − d i ) ψ −→ m +1 M R ( − d ) → I → stand for the minimal graded presentation of the base ideal I = ( f , · · · , f m ) of Φ .If the base locus of X has dimension and the homogeneous coordinate ring of X isCohen–Macaulay, then deg(Φ) e ( R ) d · · · d t − d r , where t = dim X .Proof. Without loss of generality we may assume that the base field is infinite. Let R denote the homogeneous coordinate ring of X . Pick a general point q = ( q , · · · , q m ) ∈ Y and let I q be as above. Write I q = Q ∩ · · · ∩ Q l ∩ Q l +1 ∩ · · · ∩ Q s for the primarydecomposition of I q where √ Q i I for i l and √ Q i ⊃ I for i > l . Clearly, then ( I q : R I ) = ( I q : R I ∞ ) ∩ ( Q l +1 : R I ) ∩ · · · ∩ ( Q s : R I ) . ( I q : I ∞ ) = t . Also, codim ( Q i : I ) > codim ( I ) > dim X = t . Thus, codim ( I q : I ∞ ) = codim ( I q : I ) = t . Now let J := I ( q · ψ ) . Note that J is generated by elements ofdegrees d , · · · , d r . By Lemma 5.1, J = ( I q : I ) . Since R is Cohen-Macaulay, J contains aregular sequence, say α , of length t . According to Lemma 2.2, α can be chosen in degrees d , · · · , d t − , d r . The result now follows from the fact that e ( R/ ( I q : R I ∞ )) e ( R/ J ) e ( R/ α ) = e ( R ) d · · · d t − d r . Our examples show that without the standing assumptions in Proposition 5.2 thelast inequalities in the above proof do not hold. Although the inequality deg(Φ) e ( R ) d · · · d t − d r , still holds by Corollary 3.5. Acknowledgments.
The first and second authors thank the Franco-Brazilian Net-work in Mathematics for providing facilities to mutual interchange visits.
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Marc Chardin
Institut de Mathématiques de Jussieu, CNRS & Sorbonne Université23 Place Jussieu 75005 Paris, France e-mail : [email protected]
Seyed Hamid Hassanzadeh
Centro de Tecnologia - Bloco C, Sala ABCCidade Universitária da Universidade Federal do Rio de Janeiro,21941-909 Rio de Janeiro, RJ, Brazil e-mail : [email protected]
Aron Simis
Departamento de Matemática, CCENUniversidade Federal de Pernambuco50740-560 Recife, PE, Brazil e-maile-mail