Arc spaces and some adjacency problems of plane curves.

Transcription

1 Arc saces and some adjacency roblems of lane curves. María Pe Pereira ICMAT, Madrid 3 de junio de 05 Joint work in rogress with Javier Fernández de Bobadilla and Patrick Poescu-Pamu

2 Arcsace of (C, 0). Arc (through the origin) of C : germ of arametrization through the origin: γ : (C, 0) (X, O) (C, O) t ( i a i ti,..., i an i ti ) 0 O Formal arcs are considered: the ower series may not converge. It is an infinite affine sace. It is irreducible.

3 Nash sets associated to divisors over O C. A divisor is a excetional comonent of a comosition of blow us in oints above O C. X 3 π Comosition of Blow-us C

4 Nash sets associated to divisors over O C. A divisor is a excetional comonent of a comosition of blow us in oints above O C. Im γ 3 X γ C γ π C Comosition of Blow-us Im γ

5 Nash sets associated to divisors over O C. A divisor is a excetional comonent of a comosition of blow us in oints above O C. Im γ 3 X Take the minimal model for i. N i = {γ : γ(0) i } The Nash set is its closure N i. γ C γ π C Comosition of Blow-us Im γ

6 Nash sets associated to divisors over O C. A divisor is a excetional comonent of a comosition of blow us in oints above O C. γ C γ Im γ 3 π C X Comosition of Blow-us Im γ Take the minimal model for i. N i = {γ : γ(0) i } The Nash set is its closure N i. N 0 is equal to the whole arc sace.

7 Nash sets associated to divisors over O C. A divisor is a excetional comonent of a comosition of blow us in oints above O C. γ C γ Im γ 3 π C X Comosition of Blow-us Im γ Take the minimal model for i. N i = {γ : γ(0) i } The Nash set is its closure N i. N 0 is equal to the whole arc sace. Nash sets are irreducible.

8 Nash sets associated to divisors over O C. A divisor is a excetional comonent of a comosition of blow us in oints above O C. γ C γ Im γ 3 π C X Comosition of Blow-us Im γ Take the minimal model for i. N i = {γ : γ(0) i } The Nash set is its closure N i. N 0 is equal to the whole arc sace. Nash sets are irreducible. They are cylindrical: they are determined in order k.

9 Nash sets associated to divisors over O C. A divisor is a excetional comonent of a comosition of blow us in oints above O C. γ C γ Im γ 3 π C X Comosition of Blow-us Im γ Take the minimal model for i. N i = {γ : γ(0) i } The Nash set is its closure N i. N 0 is equal to the whole arc sace. Nash sets are irreducible. They are cylindrical: they are determined in order k. They have finite codimension.

10 Nash sets associated to divisors over O C. A divisor is a excetional comonent of a comosition of blow us in oints above O C. γ C γ Im γ 3 π C X Comosition of Blow-us Im γ Take the minimal model for i. N i = {γ : γ(0) i } The Nash set is its closure N i. N 0 is equal to the whole arc sace. Nash sets are irreducible. They are cylindrical: they are determined in order k. They have finite codimension. They are all different: N i N j.

11 Nash sets. Take a comosition of blow us in oint above the origin. Take the minimal model for i. Im γ 3 X N i = {γ : γ(0) i } The Nash set is its closure N i. What is the closure N i? γ C γ π C Comosition of Blow-us Im γ

12 Nash sets. Take a comosition of blow us in oint above the origin. Take the minimal model for i. γ C γ Im γ 3 π C X Comosition of Blow-us Im γ N i = {γ : γ(0) i } The Nash set is its closure N i. What is the closure N i? If there is a family as for examle α s (t) = (t 5 + st 3 ; t 4 + st 4 ) with α s N F and α 0 N, then α 0 N F.

13 Nash sets. Take a comosition of blow us in oint above the origin. Take the minimal model for i. γ C γ Im γ 3 π C X Comosition of Blow-us Im γ N i = {γ : γ(0) i } The Nash set is its closure N i. What is the closure N i? If there is a family as for examle α s (t) = (t 5 + st 3 ; t 4 + st 4 ) with α s N F and α 0 N, then α 0 N F. GNRALISD NASH PROBLM: Determine when N N F.

14 Nash roblem. For a singular variety (X, Sing X ), the comonents of the sace of arcs centered at Sing X are of the form N for certain excetional comonents of a resolution of singularities. These comonents aear in any resolution.

15 Nash roblem. For a singular variety (X, Sing X ), the comonents of the sace of arcs centered at Sing X are of the form N for certain excetional comonents of a resolution of singularities. These comonents aear in any resolution. Surface singularities (Nash Conjecture, Theorem 0, J. Fernandez de Bobadilla, M. P. P.): The comonents of the arcsace are in bijection with excetional comonents of the minimal resolution. Higher dimensional case (artial result 04, T. de Fernex, R. Docamo). Comonents in terminal models give comonents of the sace of arcs. But there are more... still oen.

16 Nash roblem. For a singular variety (X, Sing X ), the comonents of the sace of arcs centered at Sing X are of the form N for certain excetional comonents of a resolution of singularities. These comonents aear in any resolution. Surface singularities (Nash Conjecture, Theorem 0, J. Fernandez de Bobadilla, M. P. P.): The comonents of the arcsace are in bijection with excetional comonents of the minimal resolution. Higher dimensional case (artial result 04, T. de Fernex, R. Docamo). Comonents in terminal models give comonents of the sace of arcs. But there are more... still oen. If some N isn t a comonent of the sace of arcs, then N N F for some F.

17 Generalised Nash Problem: describe the inclusions/adjacencies N N F.

18 Generalised Nash Problem: describe the inclusions/adjacencies N N F. Trivial inclusions: F < ( dominates F ) imlies N N F F X X

19 Generalised Nash Problem: describe the inclusions/adjacencies N N F. Trivial inclusions: F < ( dominates F ) imlies N N F F X X

20 How to check if N N F? Theorem (Fernández de Bobadilla, 009) Given two divisors above O C, the following are equivalent: N N F there exists a convergent family of convergent arcs α realising the inclusion with α 0 N and α s N F. 3 for any convergent arc γ N there exists a family of convergent arcs α realising the inclusion with α 0 = γ (and α s N F ).

21 N N F doesn t imly F <. (t 5 + s 3 t 3, ( + s 4 )t 4 ), two different tangents for s = 0 and s 0. 0 F s=0 s=0 4 3

22 N N F doesn t imly F <. (t 5 + s 3 t 3, ( + s 4 )t 4 ), two different tangents for s = 0 and s 0. 0 F s=0 s=0 4 3 Multile examles: F 3 3 (6,9,) 5 6

23 We want to describe N N F (artial order among divisors over O C ) A divisor is determined by the combinatorics + moduli... Minimal model of a divisor by blowing u a finite number of (free or satellite) oints

24 We want to describe N N F (artial order among divisors over O C ) A divisor is determined by the combinatorics + moduli... Minimal model of a divisor by blowing u a finite number of (free or satellite) oints

25 We want to describe N N F (artial order among divisors over O C ) A divisor is determined by the combinatorics + moduli... Minimal model of a divisor by blowing u a finite number of (free or satellite) oints. 0 3

26 We want to describe N N F (artial order among divisors over O C ) A divisor is determined by the combinatorics + moduli... Minimal model of a divisor by blowing u a finite number of (free or satellite) oints

27 We want to describe N N F (artial order among divisors over O C ) A divisor is determined by the combinatorics + moduli... Minimal model of a divisor by blowing u a finite number of (free or satellite) oints

28 We want to describe N N F (artial order among divisors over O C ) A divisor is determined by the combinatorics + moduli... Minimal model of a divisor by blowing u a finite number of (free or satellite) oints

29 We want to describe N N F (artial order among divisors over O C ) A divisor is determined by the combinatorics + moduli... Minimal model of a divisor by blowing u a finite number of (free or satellite) oints These drawings only kee combinatorics, not moduli for free oints.

30 We want to describe N N F (artial order among divisors over O C ) A divisor is determined by the combinatorics + moduli... Minimal model of a divisor by blowing u a finite number of (free or satellite) oints These drawings only kee combinatorics, not moduli for free oints. mbedded toology of lane curves is encoded in combinatorics of the minimal good embedded resolution.

31 We want to describe N N F (artial order among divisors over O C ) A divisor is determined by the combinatorics + moduli... Minimal model of a divisor by blowing u a finite number of (free or satellite) oints These drawings only kee combinatorics, not moduli for free oints. mbedded toology of lane curves is encoded in combinatorics of the minimal good embedded resolution. Observe: not all the divisors are the final one for the minimal embedded resolution of a branch, but only the blow us of satellite oints.

32 We want to describe N N F (artial order among divisors over O C ) A divisor is determined by the combinatorics + moduli... Minimal model of a divisor by blowing u a finite number of (free or satellite) oints These drawings only kee combinatorics, not moduli for free oints. mbedded toology of lane curves is encoded in combinatorics of the minimal good embedded resolution. Observe: not all the divisors are the final one for the minimal embedded resolution of a branch, but only the blow us of satellite oints. F smooth! F

33 We want to describe N N F (artial order among divisors over O C ) A divisor is determined by the combinatorics + moduli... Minimal model of a divisor by blowing u a finite number of (free or satellite) oints These drawings only kee combinatorics, not moduli for free oints. mbedded toology of lane curves is encoded in combinatorics of the minimal good embedded resolution. Observe: not all the divisors are the final one for the minimal embedded resolution of a branch, but only the blow us of satellite oints. F F F F

34 We want to describe N N F (artial order among divisors over O C ) Take into account the contact order between and F, and much more (moduli for free oints also count ariori!)... F F F 0 0 0

35 We want to describe N N F (artial order among divisors over O C ) Take into account the contact order between and F, and much more (moduli for free oints also count ariori!)... F F F Talk about combinatorics of the air (, F ). We write (, F ) (, F )...

36 We want to describe N N F (artial order among divisors over O C ) Take into account the contact order between and F, and much more (moduli for free oints also count ariori!)... F F F Talk about combinatorics of the air (, F ). We write (, F ) (, F )... Domination relation F < = Inclusion of nriques diagrams.

37 Valuative criterion in arc saces (A. Reguera, C. Plenat, S. Ishii...) Divisorial valuation ord = vanishing order along the divisor. Can be comuted intersecting with aroiate arcs γ in N : ord (f ) = I O (f, γ) = ord t (f γ(t)). Choosing a family of arcs with an aroriate α 0 N and α s N F we get ord F (h) ord t (h α s (t)) ord t (h α 0 (t)) = ord (h) forall h C[[x, y]].

38 Valuative criterium in arc saces (A. Reguera, C. Plènat, S. Ishii...) A. Reguera for rational surfaces in Manuscrita math C. Plénat in general in Annal Inst. Fourier 005: N N F imlies ord ord F S. Ishii in Maximal Divisorial Sets: if F is toric, also the converse is true: ord ord F imlies N N F. She found a counterexamle for the converse in general (ord F ord but N N F ). (:0) 5 4 (:) 3 F:= 5 := 3 0 3

39 F < N N F ordf ord

40 Other tools to rule out adjacencies? N N F imlies codim(n F ) < codim(n ). In [de Fernex, in, Ishii, Lazarsfeld, Mustata 00?]: codim(n ) = + disc(, C ). The discreancy of is the coefficient of in K X /C where π : X C is any model where aears. It is not a sufficient criterium (even with toric examles) Neither lus the valuative criterium (counterexamle of Ishii with the same discreancy). The roblem turns very difficult...

41 xamle of toological tyes. F ord F ord disc() + = codim(n ) = > codim(n F ) = disc(f ) + = 7 N N F

42 Main result about inclusions Recall that N N F deends on the relative osition of and F, but... we don t know a riori that it is a combinatorial roblem, it also deends on the moduli of the free oints!. Theorem Assume there exists a wedge α realising the adjacency N N F. If (, F ) (, F ) then there exists a wedge realising the adjacency N N F.

43 Main result about inclusions Recall that N N F deends on the relative osition of and F, but... we don t know a riori that it is a combinatorial roblem, it also deends on the moduli of the free oints!. Theorem Assume there exists a wedge α realising the adjacency N N F. If (, F ) (, F ) then there exists a wedge realising the adjacency N N F. Corollary Assume we have that N N F. Let i 0 be the contact order between and F. Then, we have that N N F i0 F i0 F where A i0 B means that A has the same combinatorics as B and their contact order is i 0.

44 Main result about inclusions Recall that N N F deends on the relative osition of and F, but... we don t know a riori that it is a combinatorial roblem, it also deends on the moduli of the free oints!. Theorem Assume there exists a wedge α realising the adjacency N N F. If (, F ) (, F ) then there exists a wedge realising the adjacency N N F. Corollary Assume we have that N N F. Let i 0 be the contact order between and F. Then, we have that N N F i0 F i0 F where A i0 B means that A has the same combinatorics as B and their contact order is i 0. We imrove the log-discreancy inequality in many cases.

45 Main result about inclusions Recall that N N F deends on the relative osition of and F, but... we don t know a riori that it is a combinatorial roblem, it also deends on the moduli of the free oints!. Theorem Assume there exists a wedge α realising the adjacency N N F. If (, F ) (, F ) then there exists a wedge realising the adjacency N N F. Corollary Assume we have that N N F. Let i 0 be the contact order between and F. Then, we have that N N F i0 F i0 F where A i0 B means that A has the same combinatorics as B and their contact order is i 0. We imrove the log-discreancy inequality in many cases. Other conjectures...

46 (, F ) (, F ) [N N F N N F ]

47 (, F ) (, F ) [N N F N N F ] To change the comlex structure we can use: Let g : X Y is a non-ramified covering of differentaible manifolds. A comlex structure on Y can be lifted to X so that g is a local biholomorhism.

48 (, F ) (, F ) [N N F N N F ] To change the comlex structure we can use: Let g : X Y is a non-ramified covering of differentaible manifolds. A comlex structure on Y can be lifted to X so that g is a local biholomorhism. (Grauert-Remmert) Let A be a normal analytic sace, let B A be a closed analytic subset such that A \ B is dense in A. Let f : U A \ B be a finite and étale analytic morhism. Then there exists a finite analytic extension f : V A from a normal analytic sace V. Moreover V is unique u to isomorhism.

49 (, F ) (, F ) [N N F N N F ] To change the comlex structure we can use: Let g : X Y is a non-ramified covering of differentaible manifolds. A comlex structure on Y can be lifted to X so that g is a local biholomorhism. (Grauert-Remmert) Let A be a normal analytic sace, let B A be a closed analytic subset such that A \ B is dense in A. Let f : U A \ B be a finite and étale analytic morhism. Then there exists a finite analytic extension f : V A from a normal analytic sace V. Moreover V is unique u to isomorhism. We can assume the wedge α : C C is algebraic, that is there exists olynomials F, F C[s, t, x, y] such that F (s, t, α (s, t)) = F (s, t, α (s, t)) = 0.

50 Coming back to the valuative criterium... Recall: N N F imlies there exists a family of arametrizations α(t, s) with α 0 (t) N and α s N F for all s Λ \ {0}. Deforming a little α, we can assume that The equation F (x, y, s) of α (O) = {0} Λ. Im[(t, s) (α(t, s), s) C Λ] gives a deformation of lane curves given by f s (x, y) := F (x, y, s) where f 0 (x, y) = 0 lifts transversally to and all f s (x, y) = 0 lift transversally to F for all s 0. These deformations have a secial roerty: for s 0 they can be resolved simultaneously by a sequence of blow us, they fix the free oints (for F ).

51 Valuative criterium. Let f s be a deformation fixing the free oints. If f 0 = 0 has strict transform transverse to some and f s = 0 have strict transforms transverse to a fixed F for all s 0, then ord F (h) ord (h) h C[[x, y]]. We have I O (h, f s) I O (h, f 0) but is not enough...

52 Valuative criterium. Let f s be a deformation fixing the free oints. If f 0 = 0 has strict transform transverse to some and f s = 0 have strict transforms transverse to a fixed F for all s 0, then ord F (h) ord (h) h C[[x, y]]. We have I O (h, f s) I O (h, f 0) but is not enough... Proof Take embedded resolution ( X, D = i D i) (C, O) of f s = 0 and f 0 = 0. Look at it in family X Λ C Λ. Let Y be the strict transform of F = 0 (F (x, y, s) := f s(x, y)). Observe Y s = {f s = 0} for s 0 Y 0 = {f 0 = 0} + k d k D k, with d k 0. We get Y 0 D i = Y s D i for any i. Putting M = (D i D j ), ( = D 0, F = D n), (, 0,..., 0) t + M(d,.., d n) t = (0,..., 0, ) t. M (, 0,..., 0, ) t = (d,..., d n) t 0. and the entries of M are exactly ord D (h D ) = I O (h D, h D ).

53 Valuative criterium. Let f s be a deformation fixing the free oints. If f 0 = 0 has strict transform transverse to some and f s = 0 have strict transforms transverse to a fixed F for all s 0, then ord F (h) ord (h) h C[[x, y]]. We have I O (h, f s) I O (h, f 0) but is not enough... Recirocally, if ord F (h) ord (h) h C[[x, y]], then, taking h = 0 and h F = 0 with strict transform transverse to and F in a model of + F, then h + s h F have strict transform transverse to F for s 0 small enough. (Also roved by M. Alberich y J. Roe).

54 Valuative criterium. Let f s be a deformation fixing the free oints. If f 0 = 0 has strict transform transverse to some and f s = 0 have strict transforms transverse to a fixed F for all s 0, then ord F (h) ord (h) h C[[x, y]]. We have I O (h, f s) I O (h, f 0) but is not enough... Recirocally, if ord F (h) ord (h) h C[[x, y]], then, taking h = 0 and h F = 0 with strict transform transverse to and F in a model of + F, then h + s h F have strict transform transverse to F for s 0 small enough. (Also roved by M. Alberich y J. Roe). Proof Check it works.

55 Valuative criterium. Let f s be a deformation fixing the free oints. If f 0 = 0 has strict transform transverse to some and f s = 0 have strict transforms transverse to a fixed F for all s 0, then ord F (h) ord (h) h C[[x, y]]. We have I O (h, f s) I O (h, f 0) but is not enough... Recirocally, if ord F (h) ord (h) h C[[x, y]], then, taking h = 0 and h F = 0 with strict transform transverse to and F in a model of + F, then h + s h F have strict transform transverse to F for s 0 small enough. (Also roved by M. Alberich y J. Roe). Summarizing: Proosition Let and F be two rime divisors. There exists a deformation f s of a curve f 0 = 0 that lifts transversal to that fixes the free oints for F (f s = 0 has strict transform transverse to F for s 0) if and only if ord F ord. So, the valuative criterion is a criterion for the existence of deformations of functions not of arametrizations!

56 Adjacency roblems. CLASSICAL ON: Given two toological tyes f = 0 and g = 0 in (C, 0), study when there exists a deformation f t = 0 where f 0 = 0 has the toological tye of f = 0 and f t = 0 the one of g = 0. OUR OBSRVATION: deformation fixing the free oints of the generic curves are characterized by the valuative criterium.

57 Valuative criterium. Proosition Let and F be rime divisors over O C. There exists a deformation f s of a curve f 0 = 0 that lifts transversal to that fixes the free oints for F (f s = 0 has strict transform transverse to F for s 0) if and only if ord F ord. Good things about the result and our roblem: It talks about concrete divisors, not only toological tyes. Takes into account the contact order of and F. They are very easy to check finite conditions (inequalities for h D with D in the minimal model of F ) => Algorithm! Also works for F a non-rime divisor: if F = i a if i then we the condition is ord F := i a iord Fi ord. Bad news: Not all the adjacencies are of this tye.

58 We recover many of the adjacencies from Arnol d s list. A A A 3 A 4 A 5 A 6 A 7 A 8... D 4 D 5 D 6 D 7 D Only 7 out of the 93 classical adjacencies between simle singularities of µ 8 are not realizable.

59 We recover many of the adjacencies from Arnol d s list. A A A 3 A 4 A 5 A 6 A 7 A 8... D 4 D 5 D 6 D 7 D For examle, ord A5 ord 6 but still there exists a deformation y 3 + x 4 + s y + sx y.

60 We recover many of the adjacencies from Arnol d s list. J 3,0 J 3, X 9 J 0 Z,0 Z, Z Z Z 3 Z 7 Z 8 Z 9 W W W W W,0 W, Some were not in Arnol d s list: Z = S,4,5 8, Z = S,4,6 J 0 = T,3,6, W 7 Z 3 = S,4,7 Some are not realizable: W 8 Z 7, Z J 0, X 9 7.

61 Relation to the study of δ constant stratum. Recall Teissier s Theorem: a deformation f t admits a arametrization in family if and only if it is δ-constant. (δ(c, 0) = dim C (O C, 0 /O C,0)). Describe all the N N F is equivalent to describe which of the deformations fixing the free oints are in the δ-constant stratum. Our roblem is slightly different to the classical study of the δ-constant stratum: may be easier?

62 Hay birthday and thank you!

63 Order and duality for toological tyes that are resolved in n blow-us.

64 Order and duality for toological tyes that are resolved in n blow-us. Take combinatorial information (/0) about n 3 edges (straight/curve) and n 3 vertices (broken between straight/smooth). Combinatorics induces a artial order: the more straight lines and broken vertices, the bigger. Vertices dges 0 * + >0, 0 > Vertices dges You get a duality that inverting the artial order just interchanging broken/curve and smooth/straight and reading backwards.

65 Order and duality for toological tyes that are resolved in n blow-us.

66 Order and duality for toological tyes that are resolved in n blow-us.

67 Order and duality for toological tyes that are resolved in n blow-us. It is just a combinatorial haening for the moment, will it aear in a deeer context? P. Poescu-Pamu, M. Pe Pereira, Fibonacci numbers and self-dual lattice structures for lane branches.bridging Algebra, Geometry, and Toology, Sringer Proceedings in Mathematics Statistics, 96