On the geometry of line arrangements and polynomial vector ...jviusos.perso.univ-pau.fr/resources/limoges2015.pdf · A dynamical approach to line arrangements in the plane Conclusions

IntroductionA dynamical approach to line arrangements in the plane

Conclusions and perspectives

On the geometry of line arrangements and polynomial vectorfields

Juan Viu-Sos(A joined work with J. Cresson et B. Guerville-Balle)

Functional Equations in LIMoges 2015XLIM, University of Limoges, Faculty of Sciences and Techniques

22 mars 2015

Juan Viu-Sos (A joined work with J. Cresson et B. Guerville-Balle) On geometry of line arrangements and polynomial vector fields 1 / 25



Table of contents

1 IntroductionLogarithmic vector fieldsDynamical approach to geometry

2 A dynamical approach to line arrangements in the planeLine arrangementsModule of logarithmic vector fields in the planeFiniteness of derivations and combinatorial dataNon combinatoriallity of the minimal finite derivations

3 Conclusions and perspectivesTerao’s conjecture in the planePerspectives and continuation




Logarithmic vector fieldsDynamical approach to geometry

Introduction

Part I





Logarithmic vector fields and logarithmic differential forms

Introduced by K. Saito (∼ 80) in order to study divisors in complexmanifolds, generalizing P. Deligne. study of Gauss-Manin connection.

Dynamical interpretation: “Logarithmic vector fields are holomorphicvector fields tangent to the smooth locus of a divisor D of X”.

Gives topological information of the complement X \ D.

A sufficient regular object becomes an invariant set of the flow of thelogarithmic vector field.

Darboux polynomial of a polynomial differential system in C2.













































Idea: Dynamical approach to affine/projective geometry for K = R,C.Given a sufficiently regular geometric object O in An

K or PnK, one can study

the set denoted by D(O) of vector fields for which O is an invariant set.

K. Saito: analytic class H. Terao (∼ 82) for line arrangements: itsuffices to consider the algebraic class.

Classical problems of plane differential systems simplifying to algebraicvector fields fixing configurations of algebraic curves.

Dulac’s conjecture: “There is a finite number of (algebraic) limit cycles ofplane polynomial vector fields”.

Algebraic Hilbert’s 16th Problem: “Is there a bound C on the number ofalgebraic limit cycles of a polynomial vector field such that C ≤ dq forq > 0”.





































Line arrangementsModule of logarithmic vector fields in the planeFiniteness of derivations and combinatorial dataNon combinatoriallity of the minimal finite derivations

A dynamical approach to line arrangements

Part II

”Combinatorics of line arrangements and dynamics of polynomial vector fields.”arXiv:1412.0137, 14 pages, with B. Guerville-Balle. (Submitted)





Let K = R, C.

Definition

An affine (resp. projective) line arrangement A is a finite collectionL1, . . . , Ln of lines in A2

K (resp. P2K).

Defining polynomial: QA =∏

L∈A αL where αL is an affine (resp. linear)form verifying L = kerαL.

Combinatorial data: encoded in the intersection poset

L(A) = ∅ 6= Li ∩ Lj | Li , Lj ∈ A ∪ A

(partially ordered by reverse inclusion of subsets).

L1

L2

L3L4

P1 P2

P3

P4

A

A

L1 L2 L3 L4

P1 P2 P3 P4

L(A)





Let K = R, C.

Definition


K (resp. P2K).




L(A) = ∅ 6= Li ∩ Lj | Li , Lj ∈ A ∪ A


L1

L2

L3L4

P1 P2

P3

P4

A

A

L1 L2 L3 L4

P1 P2 P3 P4

L(A)





Let K = R, C.

Definition


K (resp. P2K).




L(A) = ∅ 6= Li ∩ Lj | Li , Lj ∈ A ∪ A


L1

L2

L3L4

P1 P2

P3

P4

A

A

L1 L2 L3 L4

P1 P2 P3 P4

L(A)





Let K = R, C.

Definition


K (resp. P2K).




L(A) = ∅ 6= Li ∩ Lj | Li , Lj ∈ A ∪ A


L1

L2

L3L4

P1 P2

P3

P4

A

A

L1 L2 L3 L4

P1 P2 P3 P4

L(A)





What is the influence of the combinatorics on the embedding of A ?

Topological invariants:Exterior: EA = A2

K \ A or P2K \ A.

Fundamental group: π1(EA) (interesting when K = C)

Milnor fiber: FA = Q−1A (1)

Algebraic geometrical objects:Cohomological algebras: H•(EA), H•(FA), . . .

Logarithmic differential forms: Ω•(logA)

Logarithmic vector fields: D(A) =(Ω1(logA)

)∗







K \ A or P2K \ A.






)∗







K \ A or P2K \ A.






)∗





Let S = K[x , y ], define the algebra of K–derivations of S as

DerK(S) = χ : S → S K− linear | χ(fg) = χ(f )g + f χ(g), ∀f , g ∈ S

A derivation can be viewed as a polynomial vector field in the plane

χ = P∂x + Q∂y , where P,Q ∈ S .

Definition

The S-module of logarithmic derivations of A

D(A) = χ ∈ DerK(S) | χQA ∈ IQA

where IQA is the ideal in S generated by QA.

Proposition

χ ∈ D(A)⇐⇒ ∃K ∈ S such that χQA = KQA⇐⇒ A ⊂ A2

K is an invariant set of χ









Definition




Proposition











Definition




Proposition







We can define an ascending filtration by degree of D(A) by vector spaces:

D(A) =⋃d∈N

FdD(A) with FdD(A) ⊂ Fd+1D(A)

whereFdD(A) = P∂x + Q∂y ∈ D(A) | degP, degQ ≤ d

Definition

We denote byDd(A) = FdD(A) \ Fd−1D(A)

the set of polynomial vector fields of degree d fixing A.





We can define an ascending filtration by degree of D(A) by vector spaces:

D(A) =⋃d∈N

FdD(A) with FdD(A) ⊂ Fd+1D(A)

whereFdD(A) = P∂x + Q∂y ∈ D(A) | degP, degQ ≤ d

Definition

We denote byDd(A) = FdD(A) \ Fd−1D(A)

the set of polynomial vector fields of degree d fixing A.





Efficiently characterization of line arrangements as invariant sets of polynomialvector fields.

When χ ∈ DerK(S) posses a finite family of invariant lines?

·································

Figure: Phase portraits of χc = x∂x + y∂y and χp = (x + 1)∂y .





Efficiently characterization of line arrangements as invariant sets of polynomialvector fields.

When χ ∈ DerK(S) posses a finite family of invariant lines?

·································

Figure: Phase portraits of χc = x∂x + y∂y and χp = (x + 1)∂y .





Definition

Let χ be a polynomial vector field in the plane. We said that a linearrangement A is maximal fixed by χ if any invariant line L by χ belongs to A.

Definition

We said that χ fixes only a finite set of lines if there exists a maximalarrangement fixed by χ. Conversely, we said that χ fixes an infinity of lines ifthere is no such maximal line arrangement.

Following this notion, we are interested to study the partition

Dd(A) = Dfd(A) ∪ D∞d (A)

and the number df (A) = mind ∈ N | Dfd(A) 6= ∅.





Definition


Definition









Definition


Definition









Characterization of D∞d (A)

Theorem

If χ ∈ D∞d (A), then χ belongs to one of these classes of vector fields

1 null.

2 radial: there exist a point (x0, y0) ∈ A2K such that

(y0 − y , x − x0) ⊥ χ(x , y), for any (x , y) ∈ A2K.

3 parallel: there exist a vector ~v ∈ A2K such that

~v ‖ χ(x , y), for any (x , y) ∈ A2K.






Theorem

If χ ∈ D∞d (A), then χ belongs to one of these classes of vector fields

1 null.

2 radial: there exist a point (x0, y0) ∈ A2K such that

(y0 − y , x − x0) ⊥ χ(x , y), for any (x , y) ∈ A2K.

3 parallel: there exist a vector ~v ∈ A2K such that

~v ‖ χ(x , y), for any (x , y) ∈ A2K.






·································

NO!





Influence of the combinatorics in D∞d (A)

Define the combinatorial data:

|A| the number of lines in A.

m(A) the maximal multiplicity of the singularities in A.

p(A) the maximal number of parallel lines in A.

Theorem

Let 0 6= χ ∈ Dd(A):

1 If d < m(A)− 1 then χ is a radial vector field.

2 If d < p(A) then χ is a parallel vector field.

Corollary (1)

Define ν∞(A) = maxm(A)− 1, p(A). Then

Dd(A) = D∞d (A), ∀d < ν∞(A)










Theorem

Let 0 6= χ ∈ Dd(A):



Corollary (1)


Dd(A) = D∞d (A), ∀d < ν∞(A)










Theorem

Let 0 6= χ ∈ Dd(A):



Corollary (1)


Dd(A) = D∞d (A), ∀d < ν∞(A)






Corollary (2)

We have df (A) ≥ ν∞(A).

Corollary (3)

Define η∞(A) = minm(A)− 1, p(A). Then

Dd(A) = ∅, ∀0 < d < η∞(A)






Corollary (2)


Corollary (3)


Dd(A) = ∅, ∀0 < d < η∞(A)






Corollary (2)


Corollary (3)


Dd(A) = ∅, ∀0 < d < η∞(A)





Influence of the combinatorics in Dfd(A)

Theorem

1 The minimal degree of a radial vector field in D(A) is |A| −m(A) + 1.

2 The minimal degree of a parallel vector field in D(A) is |A| − p(A).

Corollary (1)

Define νf (A) = min|A| −m(A) + 1, |A| − p(A). Then

Dd(A) = Dfd(A), ∀0 < d < νf (A)

Corollary (2)

Let ν(A) = minν∞(A), νf (A). Then

Dd(A) = ∅, ∀0 < d < ν(A)






Theorem



Corollary (1)


Dd(A) = Dfd(A), ∀0 < d < νf (A)

Corollary (2)


Dd(A) = ∅, ∀0 < d < ν(A)






Theorem



Corollary (1)


Dd(A) = Dfd(A), ∀0 < d < νf (A)

Corollary (2)


Dd(A) = ∅, ∀0 < d < ν(A)






Theorem



Corollary (1)


Dd(A) = Dfd(A), ∀0 < d < νf (A)

Corollary (2)


Dd(A) = ∅, ∀0 < d < ν(A)





Strong combinatorics: The poset L(A).

Weak combinatorics: The tuple (|A|,SA,PA) where SA = (sm)m∈Nand PA = (pm)m∈N with:

sm being the number of singularities in A of multiplicity m.pm being the number of families of exactly m lines in A which are parallel.

df (A) is determined by the weak/strong combinatorics of A ?

NO! Two explicit counter-examples.
































L1

L2

L3

L4

L5

L6

L7

L8

L1 L2 L3 L4

L5

L6

L7

L8

Figure: The Pappus and Non-Pappus arrangements P1 and P2.

Same weak combinatorics: 8 lines, 6 triple points, 7 double points and 3couples of parallel lines.

Using a suite of functions coded in Sage in order to study the filtration ofD(A) and using the previous results:

Proposition

1 dimF3(P1) = 0, dimF4(P1) > 0 and df (P1) = 4.

2 dimF4(P2) = 0 , dimF5(P2) > 0 and df (P2) = 5.





L1

L2

L3

L4

L5

L6

L7

L8

L1 L2 L3 L4

L5

L6

L7

L8




Proposition







L1

L2

L3

L4

L5

L6

L7

L8

L1 L2 L3 L4

L5

L6

L7

L8




Proposition







L1L2 L3

L4

L5L6

L7

L8

C

Figure: The Ziegler arrangement Z1.

The 4 singularities of maximalmultiplicity 3 in Z1 are containedin a conic C.

We construct a line arrangementZ2 by a perturbation displacingthe triple point L1 ∩ L3 ∩ L7

outside the conic, and such thatL(Z1) ' L(Z2).

Again, studying the filtration with Sage and with the previous results:

Proposition

1 dimF4(Z1) = 0, dimF5(Z1) > 0 and df (Z1) = 5.






L1L2 L3

L4

L5L6

L7

L8

C






Proposition







L1L2 L3

L4

L5L6

L7

L8

C






Proposition






Terao’s conjecture in the planePerspectives and continuation


Part III





In general, for hyperplane arrangements in AnK, the module of logarithmic

derivations is not completely determined by the combinatorics of thearrangement.

Terao’s Conjecture (’80s)

Consider A,A′ ⊂ AnK be two hyperplane arrangements containing the origin

such that L(A) ' L(A′):

1 In this setting: D(A) =⊕

d∈NDd(A) by homogeneous components.

2 We said that A is free if D(A) is a free S-module.

Conjecture (Weak Terao’s conjecture)

If A is free then A′ is also free.

Conjecture (Strong Terao’s conjecture)

If A is free then D(A), D(A′) are isomorphic graded modules.





































































What happens for affine line arrangements in the plane ?

Theorem (Seshadri’58)

Any projective module over K[x , y ] is free.

⇒ D(A) is always free for affine line arrangements!

Two line arrangements A,A′ with the same combinatorics haveisomorphic D(A) ' D(A′) respecting the filtration?

No! We showed that L(Z1) ' L(Z2) but F5(Z1) 6= 0 = F5(Z2).













































Perspectives and continuation

(also with J. Valles) Use this approach to study the Terao’s conjecture forprojective line arrangements P2

K and central plane arrangements in A3K via

projectivization and taking cones from affine line arrangements.Study the meaning of freeness condition in the affine restriction.

Investigate (D(A), [·, ·]) seen as a Lie sub-algebra of DerK(S).

Developing of a complete package for Sage in order to study the moduleof logarithmic forms of line arrangements.

A mixed approach to the Algebraic Hilbert’s 16th Problem.

















































TH A N

K

Y O U !


Documents

On the geometry of line arrangements and polynomial vector ...jviusos.perso.univ-pau.fr/resources/limoges2015.pdf · A dynamical approach to line arrangements in the plane Conclusions