conferencia internacional en memo- ria de pilar pison...

Conferencia Internacional en Memo-ria de Pilar Pison Casares – Sevilla 2008

Cremona Maps

as related to

Linear Syzygies and Ideals of Linear Type

(In collaboration with F. Russo)

Aron Simis

Universidade Federal de PernambucoRecife, Brazil

Conferencia en Memoria de Pilar Pison Casares – p. 1/50

Sample references

• I. Dolgachev, Polar Cremona transformations, Mich.

Math. J. 48 (2000), 191–202.


Sample references


Math. J. 48 (2000), 191–202.

• C. Ciliberto, F. Russo and A. Simis, Homaloidalhypersurfaces and hypersurfaces with vanishing Hessian,arXiv:math.AG/0701596 21 Jan 2007.


Sample references


Math. J. 48 (2000), 191–202.


• L. Ein and N. Sheperd Barron, Some specialCremona transformations, Amer. J. Math. 111 (1989),

783-800.


Sample references


Math. J. 48 (2000), 191–202.



783-800.

• D. Eisenbud and B. Ulrich , Row Ideals and Fibers ofMorphisms, arXiv:math.AG/0705.3931 27 May 2007.


Sample references


Math. J. 48 (2000), 191–202.



783-800.

• D. Eisenbud and B. Ulrich , Row Ideals and Fibers ofMorphisms, arXiv:math.AG/0705.3931 27 May 2007.

• E. Guardo and B. Harbourne, Resolutions of idealsof six fat points inP

2, J. Algebra, to appear.Conferencia en Memoria de Pilar Pison Casares – p. 2/50

Sample references

• B. Harbourne, On Nagata’s conjecture, J. Algebra

236 (2001), 692-702.


Sample references


236 (2001), 692-702.

• K. Hulek, S. Katz and F.-O. Schreyer, Cremonatransformations and syzygies, Math. Z. 209 (3)(1992),

419–443.


Sample references


236 (2001), 692-702.


419–443.

• F. Russo and A. Simis, On birational maps andJacobian matrices, Compositio Math. 126 (2001),

335–358.


Sample references


236 (2001), 692-702.


419–443.


335–358.

• A. Simis, Cremona transformations and some relatedalgebras, J. Algebra 280 (2004), 162–179.


Sample references


236 (2001), 692-702.


419–443.


335–358.

• A. Simis, Cremona transformations and some relatedalgebras, J. Algebra 280 (2004), 162–179.

• A. Simis, B. Ulrich and W. Vasconcelos, Jacobiandual fibrations, Amer. J. Math. 115 (1993), 47–75.


FIRST PART

ALGEBRAICPRELIMINARIES


Ideals of linear type

An ideal I ⊂ R in a ring R is said to be of linear typeif the

natural R-algebra homomorphism SR(I) ։ RR(I) is

injective.





injective.

Here SR(I) and RR(I) denote the symmetric and the Rees

algebra of I , respectively.





injective.



Behold:





injective.



Behold:

• If I contains a regular element then I being of linear type

means that SR(I) is a torsionfree R-algebra.





injective.



Behold:



• If I is of linear type then µ(IP ) ≤ htP for every prime

ideal P ⊃ I ; equivalently (if I contains a regular element):





injective.



Behold:



• If I is of linear type then µ(IP ) ≤ htP for every prime

ideal P ⊃ I ; equivalently (if I contains a regular element):

• ht It(ϕ) ≥ m+ 1 − t, 1 ≤ t ≤ m− 1, where

Rp ϕ−→ Rm −→ I −→ 0 is a free presentation.


Rees algebras and their presentation

Throughout I 6= 0 is a homogeneous ideal in the standard

polynomial ring R = k[x] = k[x0, . . . , xn] over a field k.

We assume that I is generated by forms of one same degree

d ≥ 2; accordingly, one has a free graded presentation.







R(−(d+1))ℓ⊕∑

j≥2R(−(d+j))ϕ

−→ R(−d)m → I → 0







R(−(d+1))ℓ⊕∑

j≥2R(−(d+j))ϕ

−→ R(−d)m → I → 0

The image of R(−(d+ 1))ℓ by ϕ, denoted ϕ1, will be called

the linear part of ϕ.







R(−(d+1))ℓ⊕∑

j≥2R(−(d+j))ϕ

−→ R(−d)m → I → 0



We say that ϕ1 has maximal rank(or that ϕ has maximallinear rank) if rank(ϕ1) = rank(ϕ).







R(−(d+1))ℓ⊕∑

j≥2R(−(d+j))ϕ

−→ R(−d)m → I → 0



We say that ϕ1 has maximal rank(or that ϕ has maximallinear rank) if rank(ϕ1) = rank(ϕ).The maximal rank condition is trivially attained if ϕ1 = ϕ in

which case I is said to have linear presentationor is

linearly presentedConferencia en Memoria de Pilar Pison Casares – p. 6/50


Let J = ker (k[x, y] ։ R(I)) denote apresentation ideal of the Rees algebra ofthe ideal I ⊂ R = k[x].




J bihomogeneous in the standard bigradingof k[x, y] as we are assuming that I isgenerated in fixed degree d.





We fix throughout such a set of minimalgenerators – i.e., a k-vector basis of Id.





We fix throughout such a set of minimalgenerators – i.e., a k-vector basis of Id.

Then J =⊕

(l,k) J(l,k), where J(l,k) is thek-vector spanned by the biforms of bidegree(l, k).



Set

• J1,1 for the homogeneous subideal ofJ generated bythe biforms of bidegree(1, 1) – call it theideal ofbilinear relations;



Set


• J1,∗ for the homogeneous subideal ofJ generated bythe biforms of bidegree(1, r) for all r ≥ 1 – call it theideal of semi-linear relations.



Set



One has surjectiveR-algebra homomorphismsk[x, y]/J1,1 ։ k[x, y]/J1,∗ ։ R(I).



Set



One has surjectiveR-algebra homomorphismsk[x, y]/J1,1 ։ k[x, y]/J1,∗ ։ R(I).

Note that quite commonlyJ1,∗ = 0 and even whenJ1,∗ 6= 0, it frequently turns out thatJ1,1 = 0.


The semilinear codimension number

Define the semilinear codimension numberof J (orperhaps of the ideal I ⊂ R = k[x] itself) as theminimum value s ≥ 1 for which htJ1,∗ = htJ1,≤s,where the latter is the subideal generated by allsemi-linear relations of bidegree (1, r) with r ≤ s.




If htJ1,∗ = 0 we say for convenience that thesemilinear codimension number is zero.




If htJ1,∗ = 0 we say for convenience that thesemilinear codimension number is zero.

This invariant expresses the measure of how far outthrough the successive powers of the ideal I one hasto go in order to pick a sufficient number of linearsyzygies of these powers to attain the codimensionof J1,∗.



There is a related question that has been raised byWolmer Vasconcelos in connection with eliminationtheory:




Question Let r stand for the maximum degree of aminimal generator of the subideal ofJ generated by thebiforms of bidegree(0, s), with s ≥ 1. When isr at leastthe semilinear codimension number ofI?




Question Let r stand for the maximum degree of aminimal generator of the subideal ofJ generated by thebiforms of bidegree(0, s), with s ≥ 1. When isr at leastthe semilinear codimension number ofI?

As it turns this question has an affirmative answer in thecase whereI is a3-generated ideal in2-dimensionalpolynomial ring; this has recently been proved in lowdegrees byJ. Hong, W. Vasconcelos and thisspeaker and in full generality byD. Cox, J. W.Hoffman andH. Wang and, independently, byL.Buse. Conferencia en Memoria de Pilar Pison Casares – p. 10/50

SECOND PART

IDEAL THEORETICMETHODS

forRATIONAL MAPS


Geometric setup

Fix an arbitrary field k. A rational map F : Pn

99K Pm may

be described by m+ 1 forms f = {f0, . . . , fm} of a fixed

degree ≥ 1.


Geometric setup


99K Pm may


degree ≥ 1.

Canceling any non-trivial common factor of the forms f will

not change F ; the ideal I ⊂ k[x] generated by the resulting

forms is a uniquely defined ideal of codimension ≥ 2associated to F - called the base idealof the map F .


Geometric setup


99K Pm may


degree ≥ 1.




The imageof F is the closure of the image of a non-empty

subset of Pn where F is defined.


Geometric setup


99K Pm may


degree ≥ 1.




The imageof F is the closure of the image of a non-empty

subset of Pn where F is defined.

In algebraic terms, the image of F is the closed subvariety

Y ⊂ Pm whose homogeneous coordinate ring is the

k-subalgebra k[f ] ⊂ k[x], up to degree renormalization.


Geometric setup

Recall that F is birational (onto its image Y ) if there is an

inverse rational map G : Y ⊂ Pm

99K Pn.


Geometric setup



99K Pn.

If G is given by g = {g0, . . . , gn} ⊂ k[y] = k[y0, . . . , ym](forms of the same degree) then the above condition is

equivalent to having the conditions:


Geometric setup



99K Pn.



(g0(f) : . . . : gn(f)

)= (x0 : . . . : xn)

and


Geometric setup



99K Pn.



(g0(f) : . . . : gn(f)

)= (x0 : . . . : xn)

and

(f0(g) : . . . : fm(g)

)≡ (y0 : . . . : ym) (mod I(Y ))


Geometric setup



99K Pn.



(g0(f) : . . . : gn(f)

)= (x0 : . . . : xn)

and

(f0(g) : . . . : fm(g)

)≡ (y0 : . . . : ym) (mod I(Y ))

as tuples of homogeneous coordinates in Pmk(x) (resp. P

nK(Y ))

where K(Y ) is the field of fractions of k[y]/I(Y ).Conferencia en Memoria de Pilar Pison Casares – p. 13/50

Characteristic free criterion

These conditions can be translated into one single ring

isomorphism, namely:





The identity map of k[x, y] induces a bigraded k-isomorphism

Rk[x]

((f))≃ Rk[y]

((g, I(Y ))

I(Y )

)

of Rees algebras.






Rk[x]

((f))≃ Rk[y]

((g, I(Y ))

I(Y )

)

of Rees algebras.

This isomorphism in turn yields a module theoretic criterions

as follows.






Rk[x]

((f))≃ Rk[y]

((g, I(Y ))

I(Y )

)

of Rees algebras.

This isomorphism in turn yields a module theoretic criterions

as follows.

Recap the previous notation: J ⊂ k[x, y] is the presentation

ideal of Rk[x](f) and J1,∗ is the subideal generated by all

biforms of bidegree (1, r), with r ≥ 1Conferencia en Memoria de Pilar Pison Casares – p. 14/50


Consider a minimal set of biforms generating J1,∗ and let ψdenote the Jacobian matrix of these biforms with respect to

the x-variables.




the x-variables.

Clearly, ψ is a matrix with n+ 1 columns and entries in k[y]and it is graded since its rows are homogeneous vectors in

k[y].




the x-variables.


k[y].

Theorem. Let S = k[y]/I(Y ) ≃ k[f ], the homogeneous

coordinate ring of the image Y ⊂ Pm of the rational map

F : Pn

99K Pm defined by f . Assume that dimS = n+ 1.

The following condition are equivalent:




the x-variables.


k[y].



F : Pn



(i) F is birational onto Y




the x-variables.


k[y].



F : Pn



(i) F is birational onto Y(ii) rankS(ψ) = n and Im((ψ)t) = Im(ψ)∗, where t denotes

transpose and ∗ denotes S-dual.Conferencia en Memoria de Pilar Pison Casares – p. 15/50


Supplement. Moreover, when condition (ii) holds, the

coordinates of any homogeneous syzygy of ψ over S define the

inverse map to F .





inverse map to F .

Here is a more tangible special case:





inverse map to F .

Here is a more tangible special case:

Corollary. Assumptions as above. Suppose thatrankS(ψ) = n, that the ideal of n-minors of ψcontains a regular sequence of length ≥ 2 over Sand that the cokernel of the transpose ψt over S istorsionfree. Then F is birational onto Y .


The role of the linear syzygies

We now deal with the linear syzygies of the baseideal more closely in order to establish theirgeometric meaning.




Lemma (Assume k is algebraically closed)Let F : P

n99K P

m be a rational map with imageY ⊂ P

m.





n99K P


m.

Set L = J1,1 + I(Y ) ⊂ k[x, y], where J1,1 is the

previously introduced ideal of bilinear relations.





n99K P


m.

Set L = J1,1 + I(Y ) ⊂ k[x, y], where J1,1 is the

previously introduced ideal of bilinear relations.

Then the fiber of BiProj(k[x, y]/L) → Y over any

closed point p ∈ Y is a “vertical” linear subspaceL× {p} ⊂ P

n × {p}.Conferencia en Memoria de Pilar Pison Casares – p. 17/50


We note that, quite generally, BiProj(R(I)) is aclosed subvariety of BiProj(k[x, y]/L), where

I ⊂ k[x] denotes the base ideal of F .





Proposition (k algebraically closed)Let F : P

n99K P


m.






n99K P


m.

Suppose that BiProj(R(I)) is an irreduciblecomponent of BiProj(k[x, y]/L).






n99K P


m.

Suppose that BiProj(R(I)) is an irreduciblecomponent of BiProj(k[x, y]/L).

Then the fiber of BiProj(R(I)) → Y over a generalclosed point p ∈ Y is again a “vertical” linearsubspace L× {p} ⊂ P

n × {p}.Conferencia en Memoria de Pilar Pison Casares – p. 18/50


The interesting consequences of this method forbirationality are, unfortunately, not characteristicfree.




Corollary (k algebraically closed, char(k) = 0)Let F : Pn

99K Pm be a rational map with imageY ⊂ P

m.






m.

With the previous notation suppose that:






m.


• BiProj(R(I)) is an irreducible component ofBiProj(k[x, y]/L).






m.



• dimY = n






m.



• dimY = n

Then F is birational onto Y .Conferencia en Memoria de Pilar Pison Casares – p. 19/50


Checking whether a prime is minimal is not sohorrend. However, to stay strictly effective, thefollowing version may be used:




Corollary (k algebraically closed, char(k) = 0)Let F : P

n99K P


m.





n99K P


m.

In an earlier terminology suppose that:





n99K P


m.


• The presentation matrix ϕ of the base ideal hasmaximal linear rank.





n99K P


m.



• dimY = n





n99K P


m.



• dimY = n

Then F is birational onto Y .



Remark The last corollary follows from the precedent by

showing that whenever rank(ϕ1) = rank(ϕ) then:





• There is a natural isomorphism of Rees algebrasR(E) ≃ R(I), where E = coker(ϕ1);






• There is an isomorphism S(E) ≃ k[x, y]/J1,1,

hence ker (k[x, y]/J1,1 → R(I)) is the k[x]-torsion of

R(E);








R(E);

• The k[x]-torsion of R(E) is a minimal prime ofS(E).








R(E);

• The k[x]-torsion of R(E) is a minimal prime ofS(E).

• It follows that ker (k[x, y]/J1,1 → R(I)) is a

minimal prime of k[x, y]/L as well.Conferencia en Memoria de Pilar Pison Casares – p. 21/50

THIRD PART

Cremona transformations


The general algebraic criterion

Definition. A Cremona transformation of Pn is a

birational map of Pn onto itself. The designation honors

Luigi Cremona, winner of the Steiner prize - not the

homonymous city in Italy.







We may read out the earlier criterion in this case in the

following way, where hd(M) denotes the homological

dimension of a module M :







We may read out the earlier criterion in this case in the

following way, where hd(M) denotes the homological

dimension of a module M :

F : Pn

99K Pn is a Cremona transformation if and only if f

is algebraically independent over k, the matrix ψ has rank nand hd(coker(ψ)t)P ) ≤ ht (P ) − 1 for every nonzero prime

ideal P ⊂ k[x].


Syzygetic criteria

Let F : Pn

99K Pn be a rational map defined by forms

f ⊂ k[x]. Denote by θ(f) the Jacobian matrix of f . As an

immediate application of the previous results one has

conditions in order that F be a Cremona map solely in terms

of the two main matrices attached to f :


Syzygetic criteria

Let F : Pn






Proposition (k algebraically closed, char(k) = 0) If

rank(θ(f)) = n + 1 and rank(ϕ1) = n then F is a

Cremona map.


Syzygetic criteria

Let F : Pn






Proposition (k algebraically closed, char(k) = 0) If

rank(θ(f)) = n + 1 and rank(ϕ1) = n then F is a

Cremona map.

Observe that the role of the two matrices is asymmetrical: the

condition on θ(f) is necessary, while the one on ϕ is

(partially) sufficient.


Syzygetic criteria

Under a condition stronger than algebraic independence, the

condition on the syzygies of the base ideal is just right in order

that F : Pn

99K Pn be a Cremona map.


Syzygetic criteria



that F : Pn


Proposition (k algebraically closed, char(k) = 0)

Suppose that the ideal (f) ⊂ k[x] is of linear type. Then

F : Pn

99K Pn is a Cremona map if and only if the syzygies

of f have maximal linear rank.


Syzygetic criteria



that F : Pn


Proposition (k algebraically closed, char(k) = 0)

Suppose that the ideal (f) ⊂ k[x] is of linear type. Then

F : Pn

99K Pn is a Cremona map if and only if the syzygies

of f have maximal linear rank.

Supplement Moreover, if Im(ϕ1) ⊂ k[x]n+1 is a free

submodule of rank n then the base ideal of the inverse map

F−1 is a codimension two perfect ideal generated by forms of

degree n.


Examples, I

1. – For an integer m ≥ 1 let X (respectively, X̃ ) stand for

the m×m generic (respectively, generic symmetric) matrix.


Examples, I



Let f (respectively, f̃ ) denote the submaximal minors of X

(respectively, of X̃ ).


Examples, I





The rational map F : Pm2−1

99K Pm2−1 (respectively,

F̃ : P(m+1

2 )−199K P(m+1

2 )−1) defined by f (respectively, by

f̃ ) is a Cremona map.


Examples, I





The rational map F : Pm2−1

99K Pm2−1 (respectively,

F̃ : P(m+1

2 )−199K P(m+1

2 )−1) defined by f (respectively, by

f̃ ) is a Cremona map.

We note that in both cases the partial derivatives of the

respective determinant coincide with the (signed) cofactors of

the matrix; these are, therefore, examples of polar Cremona

maps, to be seen in the next part.Conferencia en Memoria de Pilar Pison Casares – p. 26/50

Examples, II

2. – Consider the generic 3 × 3 Hankel matrix


Examples, II


H =

x0 x1 x2

x1 x2 x3

x2 x3 x4


Examples, II


H =

x0 x1 x2

x1 x2 x3

x2 x3 x4

Let f stand for the partial derivatives of detH and let

F : P499K P

4 denote the rational map defined by f .


Examples, II


H =

x0 x1 x2

x1 x2 x3

x2 x3 x4

Let f stand for the partial derivatives of detH and let

F : P499K P

4 denote the rational map defined by f .

Note that the cofactors define now a rational map

G : P499K P

5 which is birational onto its image, the

Grassmannian quadric G(1, 3) ⊂ P5. This is either classical

or a consequence of the minors generating a linearly presented

ideal.


Examples, II

On the other hand, F is the composition of G and the linear

projection P : P599K P

4 from the point

(0 : 0 : 1/3 : −1 : 0 : 0) ∈ G(1, 3).


Examples, II



4 from the point

(0 : 0 : 1/3 : −1 : 0 : 0) ∈ G(1, 3).

Moreover, the restriction of P to the quadric is generically a

double covering.


Examples, II



4 from the point

(0 : 0 : 1/3 : −1 : 0 : 0) ∈ G(1, 3).


double covering.

It follows that F is not a Cremona map.


Examples, II



4 from the point

(0 : 0 : 1/3 : −1 : 0 : 0) ∈ G(1, 3).


double covering.

It follows that F is not a Cremona map.

Still in this case the partial derivatives f generate an ideal of

linear type. Thus, by the previous criterion, it must be the

case that the syzygies of this ideal are not of maximal linear

rank. And indeed, this can be laboriously proved by

specializing from the generic symmetric case or else by a

computer calculation.


FOURTH PART

Polar Cremonatransformations


Generalities

Definitions. A homogeneous polynomial f ∈ k[x]determines a polar mapPf : P

n99K P

n defined by itspartial derivatives.


Generalities


n99K P


One says that f is homaloidalif Pf is a Cremonatransformation of P

n, in which case one refers to Pf

as being a polar Cremona map.


Generalities


n99K P




as being a polar Cremona map.For n = 2 the classification was given by I.Dolgachevand the possible degrees are ≤ 3 = n+ 1


Generalities


n99K P




as being a polar Cremona map.For n = 2 the classification was given by I.Dolgachevand the possible degrees are ≤ 3 = n+ 1

For n ≥ 3 any degree is attainable by irreduciblehomaloidal polynomials (work of C. Ciliberto, F.Russoand this speaker)


The class of sub–Hankel matrices

We start with a class of homaloidal polynomials that admit an

explicit description and are, moreover, absolutely irreducible.





The generic sub–Hankel matrix of orderr on the variables

x0, . . . , xn is the matrix





The generic sub–Hankel matrix of orderr on the variables

x0, . . . , xn is the matrix

x0 x1 x2 ... xn−2 xn−1

x1 x2 x3 ... xn−1 xn

x2 x3 x4 ... xn 0

. . . ... . .

. . . ... . .

. . . ... . .

xn−2 xn−1 xn ... 0 0

xn−1 xn 0 ... 0 0



Theorem. Let n ≥ 2. Let f stand for thedeterminant of the sub–Hankel matrix and setJ = (∂f/∂x0, . . . , ∂f/∂xn). Then:




• For every value of i in the range 1 ≤ i ≤ n− 1, thepartial derivatives ∂f/∂x0, . . . , ∂f/∂xi divided bytheir g.c.d. define a Cremona transformation of P

i;





i;

• In addition, both base loci of this Cremonatransformation and of its inverse map are defined bycodimension two perfect ideals of linear type,generated in degree i;





i;

• In addition, both base loci of this Cremonatransformation and of its inverse map are defined bycodimension two perfect ideals of linear type,generated in degree i;

• The Hessian of f has the form h(f) = c x(r+1)(r−2)r ,

for some c ∈ k, c 6= 0Conferencia en Memoria de Pilar Pison Casares – p. 32/50


Moreover:



Moreover:

• The syzygies of J have maximal linear rank;



Moreover:


• As a consequence of previous criteria, if the fieldhas characteristic zero then f is a homaloidalpolynomial.



Moreover:


• As a consequence of previous criteria, if the fieldhas characteristic zero then f is a homaloidalpolynomial.

Comment. The proof of the first of these resultsrelies on a mix of fairly technical non-trivialinductive formulas involving partial derivatives. Itwould be curious to have a geometric approachshowing that the corresponding polar map isCremona.



• The degree of a sub–Hankel determinant is the dimension of

the ambient space, while many of the known classical

examples of homaloidal polynomials have a small degree as

confronted with the number of variables;







• The singular locus of a sub–Hankel hypersurface is a

multiple structure over the codimension 2 linear subspace

xr−1 = xr = 0 off the codimension 3 subspace

xr−2 = xr−1 = xr = 0, while the generators of any Ji define

a generalized de Jonquieres transformation;







• The singular locus of a sub–Hankel hypersurface is a

multiple structure over the codimension 2 linear subspace

xr−1 = xr = 0 off the codimension 3 subspace

xr−2 = xr−1 = xr = 0, while the generators of any Ji define

a generalized de Jonquieres transformation;

• The Hessian of a sub–Hankel determinant is essentially a

power of a linear form. Ciliberto has asked whether it is

possible to characterize the class of homaloidal polynomials

whose Hessian is of this form.


Homology of a sub–Hankel matrix

In this slide we deal with further algebraic properties of

sub–Hankel matrices. The main result of this part is as follows.





Theorem Let J ⊂ R = k[x] denote the gradient ideal of

the sub–Hankel determinant f as above. Then







• The minimal graded free resolution of J over R is the

mapping cone of a map between the minimal graded free

resolution of Jn−1 and the graded free resolution of the shifted

complete intersection R

(xn,xn−1

n−1)(−(n− 1));







• The minimal graded free resolution of J over R is the

mapping cone of a map between the minimal graded free

resolution of Jn−1 and the graded free resolution of the shifted

complete intersection R

(xn,xn−1

n−1)(−(n− 1));

• J is a (codimension two non-perfect) ideal of linear type.


Shifted sub–Hankel matrices

As it turns, the shifted versions of a sub–Hankel matrix

produce irreducible homaloidal determinants as well. Namely,

for a fixed integer r, 1 ≤ r ≤ n− 1, look at the following

(r + 1) × (r + 1) matrix



As it turns, the shifted versions of a sub–Hankel matrix

produce irreducible homaloidal determinants as well. Namely,

for a fixed integer r, 1 ≤ r ≤ n− 1, look at the following

(r + 1) × (r + 1) matrix

x0 x1 x2 . . . xr−1 xr

xn−r xn−r+1 xn−r+2 . . . xn−1 xn

xn−r+1 xn−r+2 xn−r+3 . . . xn 0...

......

......

...

xn−2 xn−1 xn . . . 0 0

xn−1 xn 0 . . . 0 0



As before, we denote by J the gradient ideal of its

determinant. The first case (r = 2, n = 4) does not present

any new phenomena as the syzygies of J have maximal linear

rank and the Hessian of g is of the form cx54, with a nonzero

c ∈ k;







c ∈ k;

Next relevant case (r = 3, n = 5) has some definite

anomalies:







c ∈ k;


anomalies:

• The linear rank of the syzygies of g is one less than the

maximal value;







c ∈ k;


anomalies:


maximal value;

• g is nevertheless homaloidal and the inverse to its polar map

Pg has a perfect codimension two base ideal;







c ∈ k;


anomalies:


maximal value;

• g is nevertheless homaloidal and the inverse to its polar map

Pg has a perfect codimension two base ideal;

• Neither the base ideal of Pg nor its inverse base ideal is of

linear type. Conferencia en Memoria de Pilar Pison Casares – p. 37/50

Sporadic examples, I. The circulant

There is yet another way in which the “schiebung” principle

can be applied and that is as “the thickened circulant”. We

will comprise only the case of the circulant itself:



There is yet another way in which the “schiebung” principle

can be applied and that is as “the thickened circulant”. We

will comprise only the case of the circulant itself:

x0 x1 . . . xn−1 xn

x1 x2 . . . xn x0

x2 x3 . . . x0 x1...

xn−1 xn . . . xn−3 xn−2

xn x0 . . . xn−2 xn−1



Here is the expected structural result on the homaloidal

nature of the circulant:





• The partial derivatives of the circulant determinant are the

(signed) cofactors of (any) row (or column) up to a constant

multiplier;







multiplier;

• The partial derivatives of the circulant determinant generate

a codimension 2 linearly presented perfect ideal;







multiplier;



• The circulant determinant is a squarefree homaloidal

polynomial.







multiplier;



• The circulant determinant is a squarefree homaloidal

polynomial.

Remark The first of these says that the circulant

determinant is a free divisor in the theory of K. Saito – it is

actually a Koszul free divisor in the terminology of the

Sevilla school. Conferencia en Memoria de Pilar Pison Casares – p. 39/50


Alas, the circulant determinant factors completelyover C:




Assume that k contains a primitive (n+ 1)th root ofunity ζ and let σ denote the permutation of theroots {1, ζ, ζ2, . . . , ζn} such that σ(1) = 1, σ(ζn) = ζand σ(ζ i) = ζ i+1, for 1 ≤ i ≤ n− 1. Then thecirculant determinant factors as




Assume that k contains a primitive (n+ 1)th root ofunity ζ and let σ denote the permutation of theroots {1, ζ, ζ2, . . . , ζn} such that σ(1) = 1, σ(ζn) = ζand σ(ζ i) = ζ i+1, for 1 ≤ i ≤ n− 1. Then thecirculant determinant factors as

(x0 + · · · + xn)n−1∏

j=0

(n∑

i=0

σj(ζ i)xi

)



Since the factors are all distinct, it is just a Steinerdeterminant up to a collineation.




However, over R, for example, the residual factorofx0 + · · · + xn is irreducible and can be seen to betotally Hessian, hence is homaloidal. Our presentknowledge of the theory, as vindicated in this talk,is not enough to decide whether the inverse polarmap is defined over R.




However, over R, for example, the residual factorofx0 + · · · + xn is irreducible and can be seen to betotally Hessian, hence is homaloidal. Our presentknowledge of the theory, as vindicated in this talk,is not enough to decide whether the inverse polarmap is defined over R.Thus, one may wonder if there is a genuine theoryof “rational” homaloidal polynomials; if this is case,one would expect the classification results to bemuch harder.


Sporadic examples, II. Dual of ratio-nal normal curve

The following instance is a lot speculative; it hasbeen verified up to n ≤ 5.




The rational normal curve in Pn is known to be

non-defective. Let fn stand for its dualhypersurface.






• The partial derivatives of fn generate acodimension two perfect ideal of linear type;







• fn is a Koszul free divisor for all n ≥ 3;







• fn is a Koszul free divisor for all n ≥ 3;

• fn is homaloidal if and only if n = 3.


Sporadic examples, III. Projecting the elementary

Gordan–Noether hypersurface

The elementary Gordan–Noether polynomialfd ofdegree d+ 1 ≥ 3 is defined as





(xd0 xd−1

0 x1 . . . xd1) · (xd+2 − xd+1 . . . (−1)d+2x2)

t





(xd0 xd−1

0 x1 . . . xd1) · (xd+2 − xd+1 . . . (−1)d+2x2)

t

=∑

j=0

(−1)jxd+2−jxd−1−j0 xj

1





(xd0 xd−1

0 x1 . . . xd1) · (xd+2 − xd+1 . . . (−1)d+2x2)

t

=∑

j=0

(−1)jxd+2−jxd−1−j0 xj

1

• The partial derivatives of fd have a nice recurrencein which ∂fd/∂x0, ∂fd/∂x1 are again elementaryGordan–Noether polynomials, whilst the remainingones are the signed monomials of degree d in x0, x1




This makes the Hessian matrix of fd prone to easyinspection; in particular it has rank 4, hence thedimension of the polar image of fd is only 3





Instead, we look for suitable projections to P3 of the

hypersurface V (fd) defined by fd and pose:







Puzzle For a convenient projection Pd+2

99K P3 the

image of V (fd) is a homaloidal surface.







Puzzle For a convenient projection Pd+2

99K P3 the

image of V (fd) is a homaloidal surface.

For low values of d this is directly checkable; ingeneral, it would suffice by a result of Ciliberto et.al. to prove that V (fd) is projectively equivalent tothe dual of the normal scroll of type (1, d).


FIFTH PART(apocryphal)

Peripheral questions


The depth of the base ideal

Once one starts digging into the more algebraic side of a

Cremona map, one stumbles in the usual questions. We will

restrain ourselves from going to far ashore, but here is a

puzzle:






puzzle:

Question Let F : Pn

99K Pn be a Cremona map

defined by forms f ⊂ R = k[x] of degree ≥ 2. When

is the ideal (f) an (x)-saturated ideal?






puzzle:

Question Let F : Pn

99K Pn be a Cremona map

defined by forms f ⊂ R = k[x] of degree ≥ 2. When

is the ideal (f) an (x)-saturated ideal?

Remark One reason to ask is that by and large questions on

linear syzygies and fibers go astray to questions on the

cohomological side. It naturally puzzles us to know whether

there are any restrictions of the second nature at all for a

bunch of forms to define a birational map.



Even for n = 2 the above question has a negative answer in

general, as the following result tells us.





Quintics through six points Let pj ∈ P2(1 ≤ j ≤ 6) be

six points in general position in the sense that no three of

them lie on the same line and no conic goes through all six

points. Let Pj ⊂ R = k[x0, x1, x2] denote the prime ideal of

pj and set J :=⋂6

j=1 P2j (“fat points”). Then:









pj and set J :=⋂6


J is a perfect ideal with free graded resolution

0 −→ R(−7)3 −→ R(−6) ⊕R(−5)3 −→ R −→ R/J ;









pj and set J :=⋂6


J is a perfect ideal with free graded resolution

0 −→ R(−7)3 −→ R(−6) ⊕R(−5)3 −→ R −→ R/J ;

Any independent 5-forms f0, f1, f2 ∈ J define a plane

Cremona map such that depthR/I = 0, where

I = (f0, f1, f2).Conferencia en Memoria de Pilar Pison Casares – p. 47/50


No proofs here, but some highlights may be in place:




(i) Because of the general position assumption, J is generated

in degrees 5 or higher – this relates to the classical Nagata

lower bound δ(m, r) for the degree of a curve to pass with

multiplicity at least m through r plane points in general

position;








position;

(ii) J is generated by 3 quintics f0, f1, f2 and one sextic –

this has been first shown by S. Fitchett(2001);








position;



(iii) I = J locally off the irrelevant maximal ideal (x), in

particular deg(R/I) = deg(R/J) = 18;








position;



(iii) I = J locally off the irrelevant maximal ideal (x), in

particular deg(R/I) = deg(R/J) = 18;

(iv) R/I is not Cohen–Macaulay – otherwise a calculation

with Hilbert–Burch matrices in this range yields

deg(R/I) = 19, 21. Conferencia en Memoria de Pilar Pison Casares – p. 48/50


Some comments to finish:




• Cremona maps in which all the virtual multiplicities

coincide are known as symmetric






• According to M. Alberich-Carramiñana(LNM, Springer,

2002)there are only four characteristic types of symmetric

Cremona transformations of degree ≥ 2 and exactly one in

degree 5 – however, this knowledge does not seem to yield the

depth value of R/I










depth value of R/I

• We conjecture that (perhaps, wonder if) there is no such

behaviour in degrees ≤ 4










depth value of R/I

• We conjecture that (perhaps, wonder if) there is no such

behaviour in degrees ≤ 4

• By Dolgachev there is no such behavior for polar Cremona

maps in degree 2; I know of no “interesting” non-saturated

behavior in higher degrees.Conferencia en Memoria de Pilar Pison Casares – p. 49/50

THE END

THANK YOU!


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