Equations of some embeddings of a projective space into another one
aa r X i v : . [ m a t h . A C ] D ec EQUATIONS OF SOME EMBEDDINGS OF A PROJECTIVE SPACEINTO ANOTHER ONE
MARC CHARDIN AND NAVID NEMATI
Abstract.
In [8], Eisenbud, Huneke and Ulrich conjectured a result on the Castelnuovo-Mumford regularity of the embedding of a projective space P n − ֒ → P r − determined bygenerators of a linearly presented m -primary ideal. This result implies in particular that theimage is scheme defined by equations of degree at most n . In this text we prove that theideal of maximal minors of the Jacobian dual matrix associated to the input ideal definesthe image as a scheme; it is generated in degree n . Showing that this ideal has a linearresolution would imply that the conjecture in [8] holds. Furthermore, if this ideal of minorscoincides with the one of the image in degree n - what we hope to be true - the linearity ofthe resolution of this ideal of maximal minors is equivalent to the conjecture in [8]. Introduction and Preliminaries
Let φ : P n − P r − be a rational map defined by homogeneous forms f , . . . , f r and I be the ideal generated by these forms. The Rees algebra associated to the ideal I , R ( I ),gives the bihomogeneous coordinate ring of the closure of the graph of φ inside P n − × P r − .The special fiber ring F ( I ) is the coordinate ring of the closed image of φ . The rings R ( I ) and F ( I ) are blowup rings associated to I . It is a fundamental problem to find thedefining equations of the Rees ring from generators of I , for significant classes of ideals.This problem has been studied extensively by commutative algebraists, algebraic geometers,and, more recently, by applied mathematicians in geometric modeling (see [1, 4, 10, 11, 12]).Rees algebras provide an algebraic realization for the concept of blowing up a variety alonga subvariety.Let S = k [ x , . . . , x n ] be a polynomial ring over a field k and I be an ideal of S . The Reesalgebra R ( I ) of an ideal I is defined as R ( I ) = S [ It ] = S ⊕ It ⊕ I t ⊕ · · · . The equations of the Rees algebra are elements in the kernel of the epimorphismΦ : S [ T , . . . , T r ] → R ( I ) given by Φ( T i ) = f i t where I = ( f , . . . , f r ). These equationsdo depend on the choice of generators of I ; however, their number and degrees do not, if onechooses minimal generators of a graded ideal I . Mathematics Subject Classification.
Key words and phrases.
Birational map, blowup algebra, Castelnuovo-Mumford regularity, Jacobian dualmatrix.
An important object for the study of Rees algebras is the symmetric algebra, Sym( I ), ofan ideal I . It could be defined asSym( I ) = S [ T , . . . , T r ] / L , where L = ([ T , . . . , T r ] · M ) and M is a presentation matrix of I . The map Φ above factorsthrough the symmetric algebra. To determine equations of R ( I ) of degrees at least two in the T i ’s is equivalent to study the kernel of the map Sym( I ) → R ( I ). Traditionally, techniquesfor computing the defining ideal of R ( I ) often revolved around the notion of Jacobian dual .For a polynomial ring S and an ideal I with a presentation S N M −→ S r → I →
0, the
Jacobian dual of M is defined to be a matrix Θ( M ) (we use Θ if the context makes noconfusion possible) with linear entries in S [ T , . . . , T r ] such that[ T , . . . , T r ] · M = [ x , . . . , x n ] . Θ( M ) . In this article, we focus on the Jacobian dual matrix of linearly presented m -primary ideals.The main theorem of this paper is the following: Main Theorem.
Let S = k [ x , . . . , x n ] be a polynomial ring over a field k , I = ( f , . . . , f r ) bea linearly presented m -primary ideal of S and Θ be the Jacobian dual matrix of a presentationmatrix of I . Then the image of the map P n − → P r − defined by the forms f , . . . f r isscheme-theoretically defined by the ideal of maximal minors of Θ . It is worth mentioning that it is not hard to show that the image is defined as a set bythe ideal of maximal minors (see Lemma 1); the more delicate point is to show that theimage is scheme-theoretically defined by the ideal of maximal minors. As a consequence, theimage of Φ is scheme-theoretically defined by equation of degree n . We finish the section bypresenting two conjectures about the ideal of maximal minors of Θ.In the last section, we present the connection between our work and a conjecture byEisenbud, Huneke, and Ulrich in [8]. Conjecture. [8, Conjecture 1.1]
Let S = k [ x , . . . , x n ] and I be an ideal of S . If I is alinearly presented m -primary ideal generated in degree d , then I n − = m d ( n − . Equivalently, reg ( I n − ) = ( n − d. In particular, we show that this conjecture holds if the ideal of maximal minors of Θ hasa linear resolution (see Theorem 5 ).2.
Jacobian dual matrix
For the rest of the article we adopt the following notations.
Notation 1.
Let S = k [ x , . . . , x n ] be a polynomial ring over a field k and I = ( f , . . . , f r ) be a linearly presented m -primary ideal, φ : P n − P r − be the map defined by φ ( x ) =( f ( x ) , · · · , f r ( x )) and denote the graph of φ by Γ ⊆ P n − × P r − . Set π : Γ → P n − and QUATIONS OF SOME EMBEDDINGS OF A PROJECTIVE SPACE INTO ANOTHER ONE 3 π : Γ → P r − for the two natural projections and W := Im( π ) . We write B = k [ T , . . . , T r ] .The defining ideal J of the symmetric algebra Sym ( I ) is generated by the bilinear forms L i ( x , T ) = P a i ( x ) T i where P a i ( x ) f i = 0 is a linear syzygy of I . The Jacobian dual Θ isgiven by Θ x ... x n = L ... L N , where the L i ’s are minimal generators of J (equivalently, the corresponding syzygies areminimal generators of the module of syzygies). Thus Θ is a matrix whose entries are linearforms in the T i ’s and we set W ′ := proj( ¯ B ) where ¯ B = B/I n (Θ) and I n (Θ) is the ideal ofmaximal minors of Θ (the n × n minors). Finally, if p ∈ W ⊆ W ′ and p is the correspondinghomogeneous prime in proj( B ) , we denote by M p the image of a matrix M by the naturalmap obtained by applying the map B → B p /I r ( M ) p to its entries, where r is the rank of M . Definition 1.
Let M be a p × q matrix over B . For a prime ideal p ⊇ I q ( M ) , consider thenatural map B → ( B/I q ( M )) p and its natural extension Mat p × q ( B ) → Mat p × q (( B/I q ( M )) p ) .We write M p for the image of M under this last map. Lemma 1.
With the above notations, p I n (Θ) = I ( W ) .Proof. As I is m -primary, Γ ⊆ P n − × P r − is scheme defined by the symmetric algebra, andis in particular, set defined by J . Consider the projection π : P n − × P r − → P r − and p = ( t : · · · : t r ) ∈ P r − . Let Θ( p ) be the image of Θ by the specialization T i t i . Then, p ∈ W ⇔ ∃ x = ( x , . . . , x n ); ( x, p ) ∈ Γ ⇔ ∃ x ; L i ( x, t ) = 0 ∀ i ⇔ ∃ x ; Θ( p ) x ... x n = 0 ⇔ t ∈ I n (Θ) . (cid:3) Lemma 2.
Assume that ψ : P n − \ X → P r − is defined by I = ( f , . . . , f r ) where X = V ( I ) .Let Γ ⊆ P n − × P r − be the closure of the graph of ψ . If π : Γ → Im( ψ ) is the projectionmap, then ∀ p ∈ Im( ψ ) , setting dim ∅ := − , dim π − ( p ) ≤ dim X + 1 . Proof.
Let p = ( p : · · · : p r ) ∈ Im ψ and let V be an irreducible component of π − ( p ). If V is contained in X it has dimension at most the one of X . Else ψ | V : V \ X → P r − is aconstant map with image p . Hence for all 1 ≤ i, j ≤ r , ( f i p j − f j p i ) ∈ I ( V ). Choose i with p i = 0, then f j − p j f i /p i ∈ I ( V ) for all j , and therefore I ⊆ I ( V ) + ( f i ), and the height of I ( V ) + ( f i ) is the one of I ( V ) plus one. (cid:3) MARC CHARDIN AND NAVID NEMATI
Lemma 3.
Adopt Notation 1. For a point p ∈ W ⊆ W ′ , the rank of Θ p is n − .Proof. By Lemma 2, π − ( p ) is a zero dimensional scheme. The assertion follows from thefact that this fiber is a linear space defined by the system of equationsΘ p ( p ) x ... x n = 0 . (cid:3) Remark 1.
Since reg I t = dt + 0 for t ≫ , the regularity of fibers of the projection π : Γ → P r − at any point is zero [7] . The only finite set of point(s) with zero regular-ity is a single reduced point. More generally, the regularity of stalks of π at any point is zerotoo [2] . Definition 2.
Let N be a m × n matrix. For ≤ r ≤ n and ≤ s ≤ m , define N ( i ,...i r )( j ,...,j s ) be the matrix obtained by deleting i , . . . , i r -th columns and j , . . . j s -th rows of N . Let N bea ( n − × n matrix, denote ∆ i ( N ) to be the ( − i det N ( i ) , with N ( i ) the matrix obtainedby deleting i -th column of N . For simplicity we write ∆ i instead of ∆ i ( N ) if it is clear whatmatrix we consider. Lemma 4.
Let N be a ( n − × n matrix, then ( − i adj( N ( i ) ) N = ∆ i − ∆ . . . ... ∆ i − ∆ i − − ∆ i +1 ∆ i ... . . . − ∆ n ∆ i , where only the non zero terms are displayed.Proof. The ( j, ℓ )-entry of adj( N ( i ) ) N is P k b j,k a k,ℓ where b j,k and a k,ℓ are the entries ofadj( N ( i ) ) and N .Notice that b j,k = ( − j + k det N ( i,j )( k ) if j < i and b j,k = ( − j + k det N ( i,j +1)( k ) else.Hence then the ( j, ℓ )-entry of adj( N ( i ) ) N is X k ( − j + k det N ( i,j )( k ) a k,ℓ if j < i and X k ( − j + k det N ( i,j +1)( k ) a k,ℓ if j ≥ i. First let ℓ < i . If ℓ = j , then it is equal to det N ( i ) = ( − i ∆ i . If ℓ = j < i , replace the j -th column of N with its ℓ -th column and call it N ′ . By expanding ∆ i ( N ′ ) along the j -th QUATIONS OF SOME EMBEDDINGS OF A PROJECTIVE SPACE INTO ANOTHER ONE 5 column we get 0 = ∆ i ( N ′ ) = ( − i X k ( − j + k det N ( i,j )( k ) a k,ℓ , and similarly if j ≥ i , by relacing the ( j + 1)-th column of N with its ℓ -th column.Second, if ℓ > i , similar arguments as in the case ℓ < i show that if j = ℓ −
1, the( j, j + 1)-entry is equal to ∆ i , and the ( j, ℓ )-entry is equal to 0 if j = ℓ − j < ℓ = i , the ( j, i )-entry of adj( N ( i ) ) N is X k ( − j + k det N ( i,j )( k ) a k,i . By expanding ∆ j = ( − j det N ( j ) along the i -th column we get∆ j = ( − j X k ( − i − k a k,i det N ( i,j )( k ) = ( − i − X k ( − j + k a k,i det N ( i,j )( k ) . Finally, if j ≥ ℓ = i then b j,k = ( − j + k det N ( i,j +1)( k ) so ( j, i )-th entry adj( N ( i ) ) N is X k ( − j + k det N ( i,j +1)( k ) a k,i . By expanding ∆ j +1 = ( − j +1 det N ( j +1) on the i -th column we get∆ j +1 = ( − j +1 X k ( − i + k a k,i det N ( i,j +1)( k ) = ( − i − X k ( − j + k a k,i det N ( i,j +1)( k ) . (cid:3) Lemma 5.
For a × n row vector R define L as a product of R and column vector ( x , . . . , x n ) . Let N be a ( n − × n matrix with row vectors R , . . . R n − and considercorresponding linear forms L , . . . , L n − . Let M obtained by bordering N with a row vector R n and corresponding linear form L n . Then for ≤ i, j ≤ n (1) ∆ i L n + ( − n +1 det( M ) x i ∈ ( L , . . . , L n − )(2) ( x i ∆ j − x j ∆ i ) ∈ ( L , . . . , L n − ) . (3) If j = n , ∆ i L j ∈ ( x ∆ i − x i ∆ , . . . , x n ∆ i − x i ∆ n ) .Proof. (1) Consider det( M ) x ... x n = adj( M ) M x ... x n = adj( M ) L ... L n . The ( i, M ) x i . On the other, it is equal to X ≤ k ≤ n ( − k + i det M ( i )( k ) L k = X ≤ k ≤ n − ( − k + i det M ( i )( k ) L k + ( − n ∆ i L n . MARC CHARDIN AND NAVID NEMATI (2) By choosing L n = e j in (1) we get∆ i x j + ( − n +1 det( M ) x i = ∆ i x j + ( − n +1 ( − n + j ( − j ∆ j x i = ∆ i x j − x i ∆ j ∈ ( L , . . . , L n − ) . (3) By Lemma 4∆ i L ... L n − = ( − i N ( i ) adj( N ( i ) ) N x ... x n = N ( i ) x ∆ i − x i ∆ ... x i − ∆ i − x i ∆ i − x i +1 ∆ i − x i ∆ i +1 ... x n ∆ i − x i ∆ n . (cid:3) Corollary 1.
For any ( n − × n submatrix of Θ which corresponds to L i , . . . , L i n − define ∆ i for ≤ i ≤ n as in Definition 2. If ∆ i = 0 for some i , then (1) ∆ i J ⊆ ( L i , . . . , L i n − ) modulo I n (Θ) , (2) ( L i , . . . , L i n − ) B (∆ i ) = I (cid:20) x · · · x n ∆ · · · ∆ n (cid:21) B (∆ i ) .Proof. Choose for N the submatrix of Θ such that N x ... x n = L i ... L i n − . (1) follows from Lemma 5 (1), by choosing for M the matrix corresponding to add anyminimal generator L j of J .(2) follows from Lemma 5 (2) and (3). (cid:3) Theorem 1.
Let S = k [ x , . . . , x n ] be a polynomial ring over a field k , I = ( f , . . . , f r ) bea linearly presented m -primary ideal of S and Φ : P n − P r − be the map defined by theforms f , . . . f r . Let Θ be the Jacobian dual matrix of a presentation matrix of I , then I n (Θ) sat = I ( W ) . Proof.
By Lemma 1, p I n (Θ) = I ( W ). Let p ∈ spec( B ) containing I n (Θ). We need to showthat ( B/I n ( M )) p ∼ = ( B/I ( W )) p . For this to hold, it suffices to prove that ¯ B p = ( B/I n ( M )) p is a domain. By Lemma 3, there exist L i , . . . L i n − and i with ∆ i / ∈ p , with notations as inCorollary 1. Now¯ B p [ x , . . . , x n ] /J ⊗ B ¯ B p = ¯ B p [ x , . . . , x n ] / ( L i , . . . L i n − ) ⊗ B ¯ B p by Corollary 1 (1)= ¯ B p [ x i ] by Corollary 1 (2). QUATIONS OF SOME EMBEDDINGS OF A PROJECTIVE SPACE INTO ANOTHER ONE 7 As J scheme defines Γ, proj( ¯ B p [ x , . . . , x n ] /J ⊗ B ¯ B p ) is the stalk of the isomorphism π : Γ → W over V ( p ); in particular, it is reduced and irreducible. This shows thatproj( ¯ B p [ x i ]) ∼ = spec( ¯ B p ) is reduced and irreducible. Hence ¯ B p is a domain. (cid:3) We finish this section with two conjectures.
Conjecture 1.
Let S = k [ x , . . . , x n ] be a polynomial ring and I = ( f , . . . , f r ) be a linearlypresented m -primary ideal of S . Suppose that Φ : P n − P r − is a rational map defined byforms f , . . . f r . Let Θ be the Jacobian dual matrix of a presentation matrix of I , then I n (Θ) = I ( W ) ≥ n . Conjecture 2.
Let S = k [ x , . . . , x n ] be a polynomial ring and I = ( f , . . . , f r ) be a linearlypresented m -primary ideal of S . Let Θ be the Jacobian dual matrix of a presentation matrixof I then I n (Θ) has a linear resolution. Asymptotic behavior of regularity
In this section, we study the conjecture of Eisenbud, Huneke, and Ulrich [8, Conjecture1.1] on the asymptotic behavior of regularity of linearly presented m -primary ideals. We willshow the relation between this conjecture and the ideal of maximal minors of Jacobian dualmatrices. We start this section by stating two equivalent definition of Castelnuovo-Mumfordregularity. Before that, we need to mention the definition of graded Betti numbers. For afinitely generated graded S -module M , a minimal graded free resolution of M is an exactsequence 0 → F p → F p − → · · · → F → M → , where every F i is a graded free S -module of the form F i = ⊕ j ∈ Z S ( − j ) β i,j ( M ) with the minimalnumber of basis elements, and every map is graded. The value β i,j ( M ) is called the i th gradedBetti number of M of degree j . Theorem 2.
Let M be a finitely generated S -module, the regularity of M is given by: reg ( M ) = max i { end( H i m ( M )) + i } . Although the regularity is a measure of vanishing of local cohomologies, it is also a measureof vanishing of graded Betti numbers. Eisenbud and Goto in [6] proved that reg ( M ) = max i { j − i | β i,j ( M ) = 0 } . The most significant simple result on the regularity of powers of graded ideals is thefollowing one, due independently to Kodiyalam [9] and to Cutkosky, Herzog and Trung [5].
Theorem 3.
Let I be an ideal of S = k [ x , . . . , x n ] . There exists t and b such that reg ( I t ) = td + b, ∀ t ≥ t with d := min { µ | ∃ t ≥ , ( I ≤ µ ) I t − = I t } . MARC CHARDIN AND NAVID NEMATI
Notice that when I is equigenerated the number d in the above theorem is the degree ofthe generators of I . We call smallest such t as the stabilization index of I and denote it byStab( I ). In [8], authors showed that if I is a linearly presented m -primary ideal, then thepowers of I eventually have a linear resolution. Theorem 4 ([8]) . Let S = k [ x , . . . , x n ] and I be an ideal of S . If I is a linearly presented m -primary ideal generated in degree d , then reg ( I t ) = td for t ≫ . In addition, they conjectured an upper bound for the stabilization index of I . Conjecture 3. [8, Conjecture 1.1]
Let S = k [ x , . . . , x n ] and I be an ideal of S . If I isa linearly presented m -primary ideal generated in degree d , then I n − = m d ( n − . In otherword, reg ( I n − ) = ( n − d. Remark 2.
Eisenbud, Huneke and Ulrich in [8] proved this conjecture when n = 3 or I is amonomial ideal. Remark 3.
Since the only m -primary ideals with linear resolution are the powers of m , foran m -primary ideal I generated in degree d , Stab( I ) = min { t | h I t ( dt ) = h m t ( dt ) } , where h M ( d ) := dim k M d is the Hilbert funtion of M at degree d . The following proposition provide a connection between regularity of powers of an ideal I = ( f , . . . , f r ) and the image of φ : P n − P r − defined by f , . . . , f r . Proposition 1.
Let S = k [ x , . . . , x n ] and I be an ideal of S . Adopting the Notation 1, If I is a linearly presented m -primary ideal generated in degree d , then (1) W is smooth of dimension n − , and e ( W ) = d r − . (2) reg I ( W ) = max { Stab( I ) + 1 , reg I ( V ( d ) n ) } = max { Stab( I ) + 1 , n − ⌈ nd ⌉} where V ( d ) n is the Veronese embedding of degree d .Proof. Part (1) follows from [3, Proposition 1.7(b)]. For proving part (2) we use the definitionof regularity via local cohomologies. Let I ( W ) ⊂ B = k [ T ] and I ( V ( d ) n ) ⊂ B ′ = k [ T ′ ]with B ⊂ B ′ and I ( W ) = I ( V ( d ) n ) ∩ B . As the natural map B/I ( W ) → B ′ /I ( V ( d ) n ) isan isomorphism locally on the punctured spectrum, H i T ( B/I ( W )) ∼ = H i T ( B ′ /I ( V ( d ) n )) ∼ = H i T ′ ( B ′ /I ( V ( d ) n )) for i ≥ → B/I ( W ) → B ′ /I ( V ) → H T ( B/I ( W )) → . Since h B/I ( W ) ( a ) = h I a ( da ) and h B/I ( V ) ( a ) = h m da ( da ), by Remark 3 and above exact se-quence, H T ( B/I ( W )) = 0 if and only if i ≥ Stab( I ). In other words, end H T ( B/I ( W )) =Stab( I ) −
1. The assertion follows from the definition of regularity and the fact thatreg I ( W ) = reg B/I ( W ) + 1. (cid:3) QUATIONS OF SOME EMBEDDINGS OF A PROJECTIVE SPACE INTO ANOTHER ONE 9
Conjecture 3 was the initial motivation of this work, we end this paper by showing thathow our conjectures are related to Conjecture 3.
Theorem 5.
Let S = k [ x , . . . , x n ] and I be an ideal of S . Adopting the Notation 1, If I isa linearly presented m -primary ideal generated in degree d , then (1) Conjecture 3 follows from Conjecture 2. (2)
If Conjecture 1 holds, then Conjecture 2 and Conjecture 3 are equivalent.Proof. (1) Assume I n (Θ) has a linear resolution, i.e. reg ( I n (Θ)) = n . By Theorem 1, I n (Θ) sat = I ( W ). Hence reg I ( W ) ≤ reg ( I n (Θ)) ≤ n. As reg ( I ( V ( d ) n ) = n , Proposition 1 implies that Stab( I ) ≤ n − I ) ≤ n − I ( W )) ≤ n . As we assume I n (Θ) = I ( W ) ≥ n , reg ( I ( W )) ≤ n if and only if I n (Θ) has a linearresolution. (cid:3) References [1]
Bus´e, L., Chardin, M., and Simis, A.
Elimination and nonlinear equations of Rees algebras.
J.Algebra 324 , 6 (2010), 1314–1333. With an appendix in French by Joseph Oesterl´e.[2]
Chardin, M.
Powers of ideals and the cohomology of stalks and fibers of morphisms.
Algebra NumberTheory 7 , 1 (2013), 1–18.[3]
Chiantini, L., Orecchia, F., and Ramella, I.
Maximal rank and minimal generation of someparametric varieties.
Journal of Pure and Applied Algebra 186 , 1 (2004), 21 – 31.[4]
Cox, D. A.
The moving curve ideal and the Rees algebra.
Theoret. Comput. Sci. 392 , 1-3 (2008), 23–36.[5]
Cutkosky, S. D., Herzog, J., and Trung, N. V.
Asymptotic behaviour of the Castelnuovo-Mumford regularity.
Compositio Math. 118 , 3 (1999), 243–261.[6]
Eisenbud, D., and Goto, S.
Linear free resolutions and minimal multiplicity.
Journal of Algebra 88 ,1 (1984), 89 – 133.[7]
Eisenbud, D., and Harris, J.
Powers of ideals and fibers of morphisms.
Math. Res. Lett. 17 , 2 (2010),267–273.[8]
Eisenbud, D., Huneke, C., and Ulrich, B.
The regularity of Tor and graded Betti numbers.
Amer.J. Math. 128 , 3 (2006), 573–605.[9]
Kodiyalam, V.
Asymptotic behaviour of Castelnuovo-Mumford regularity.
Proc. Amer. Math. Soc.128 , 2 (2000), 407–411.[10]
Kustin, A., Polini, C., and Ulrich, B.
The bi-graded structure of symmetric algebras with appli-cations to Rees rings.
J. Algebra 469 (2017), 188–250.[11]
Kustin, A. R., Polini, C., and Ulrich, B.
Blowups and fibers of morphisms.
Nagoya Math. J. 224 ,1 (2016), 168–201.[12]
Kustin, A. R., Polini, C., and Ulrich, B.
The equations defining blowup algebras of height threeGorenstein ideals.
Algebra Number Theory 11 , 7 (2017), 1489–1525.
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