Workshop Teoria Algébrica de Singularidades. Isabel S. Labouriau Maria Aparecida S.Ruas CMUP ICMC USP Porto, Portugal S.

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1 Workshop Teoria Algébrica de Singularidades Isabel S. Labouriau Maria Aparecida S.Ruas CMUP ICMC USP Porto, Portugal S. Carlos, SP Brasil UFF, Niterói, RJ 22 a 24 de Fevereiro de 2010

2 Set-up f : R n,0 R n,0 smooth map-germ finitely determined corank 1 under A-equivalence general form f (x,y) = (x,g(x,y))) x R n 1 Stabilisation of f a map-germ F : R n+1 R n+1 of the form F(t,x,y) = (t,x,g(x,y,t)) with G(x,y,0) = g(x,y) such that for t 0 the only singularities of F t (x,y) = F(t,x,y) are of type A j, j n. Aim Obtain the number of A n -singularities in a stabilisation of f.

3 Simple example f : R 2,0 R 2,0 f (x,y) = (x,g(x,y)) Stabilisation where g(x,y) = y 4 + xy F(t,x,y) = (t,x,y 4 + xy ty 2 )

4 Path formulation (x, y) (x, y 4 ) Versal unfolding (x,y) (x,y 4 + u 1 y u 2 y 2 ) Bifurcation set 27u1 2 8u3 2 = 0 u 1 = 0 u 2 > 0 cusps A 2 double-fold (2 A 1, same value) t >0 double-folds cusps t <0

5 Path formulation (x, y) (x, y 4 ) map-germ path (x,y 4 + xy) p(x) = (u 1,u 2 ) = (x,0) stabilisation F t = (x,y 4 + xy ty 2 ) P(x) = (u 1,u 2 ) = (x,t) t >0 double-folds cusps t <0

6 Path formulation (x, y) (x, y 4 ) map-germ path (x,y 4 + xy) p(x) = (u 1,u 2 ) = (x,0) stabilisation F t = (x,y 4 + xy ty 2 ) P(x) = (u 1,u 2 ) = (x,t) t >0 double-folds cusps t <0

7 Path formulation (x, y) (x, y 4 ) map-germ path (x,y 4 + xy) p(x) = (u 1,u 2 ) = (x,0) stabilisation F t = (x,y 4 + xy ty 2 ) P(x) = (u 1,u 2 ) = (x,t) t >0 double-folds cusps t <0 For t < 0 two cusps and one double-fold For t > 0 no cusps, no folds.

8 well known results: Given f (x,y) = (x,g(x,y))), consider the ideal g S(f ) = y (x,y)... n g y n (x,y) E x,y E x,y Lemma For a finitely determined map-germ f : R n,0 R n,0 of corank 1, S(f ) has finite codimension.

9 well known results: S(f ) = g y (x,y)... n g y n (x,y) E x,y E x,y Algebraic number of A n singularities σ(f ) = cod Ex,y (S(f )) Theorem For a finitely determined map-germ f : R n,0 R n,0 of corank 1, the number σ(f ) is an invariant for A-equivalence giving the number of A n -singularities appearing in a stable deformation of the complexification of f.

10 F(t,x,y) = (t,x,g(x,y,t)) a stabilisation of f. G y (x,y,t) = 0... n G (x,y,t) = 0 (1) yn Algebraic number of A n singularities σ(f ) = cod Ex,y (S(f )) counts simple solutions (x,y) C n of (1) for t 0 Effective number of A n singularities s + (F) s (F) counts simple solutions (x,y) R n of (1) for t > 0 t < 0. s + (F) and s (F) usually depend on the choice of F. s + (F) and s (F) σ(f ).

11 Simple example f : R 2,0 R 2,0 f (x,y) = (x,g(x,y)) where g(x,y) = y 4 + xy Algebraic number of A n singularities S(f ) = 4y 3 + x,y 2 E x,y E x,y σ(f ) = cod Ex,y (S(f )) = 2 Effective number of A n singularities Stabilisation F(t,x,y) = (t,x,y 4 + xy ty 2 ) From path formulation: s + (F) = 2 s (F) = 0

12 Given f (x,y) smooth, finitely determined, corank 1. Given F(t, x, y) a stabilisation, obtain a general method for computing s + (F). Maximality Show that there is a stabilisation F(t,x,y) for which s + (F) = σ(f ). or, failing this, Geometric number of A n singularities compute s max (f ) the maximum value of s + (F) for any stabilisation F of f. This is work in progress!

13 Given f (x,y) smooth, finitely determined, corank 1. Given F(t, x, y) a stabilisation, obtain a general method for computing s + (F). Maximality Show that there is a stabilisation F(t,x,y) for which s + (F) = σ(f ). or, failing this, Geometric number of A n singularities compute s max (f ) the maximum value of s + (F) for any stabilisation F of f. This is work in progress!

14 Given f (x,y) smooth, finitely determined, corank 1. Given F(t, x, y) a stabilisation, obtain a general method for computing s + (F). Maximality Show that there is a stabilisation F(t,x,y) for which s + (F) = σ(f ). or, failing this, Geometric number of A n singularities compute s max (f ) the maximum value of s + (F) for any stabilisation F of f. This is work in progress!

15 Given f (x,y) smooth, finitely determined, corank 1. Given F(t, x, y) a stabilisation, obtain a general method for computing s + (F). Maximality Show that there is a stabilisation F(t,x,y) for which s + (F) = σ(f ). or, failing this, Geometric number of A n singularities compute s max (f ) the maximum value of s + (F) for any stabilisation F of f. This is work in progress!

16 Given f (x,y) smooth, finitely determined, corank 1. Given F(t, x, y) a stabilisation, obtain a general method for computing s + (F). Maximality Show that there is a stabilisation F(t,x,y) for which s + (F) = σ(f ). or, failing this, Geometric number of A n singularities compute s max (f ) the maximum value of s + (F) for any stabilisation F of f. This is work in progress!

17 Previous results on maximality M-deformations of plane curve germs (S.M. Gusein-Zade, 1974; N. ACampo, 1975); Deformations of K-simple map-germs (J. Damon & A. Galligo, 1976); M-morsifications of real function-germs (V. I. Arnold, 1992; M.R. Entov, 1993); Good real perturbations (T. Cooper, D. Mond & R.W. Atique, 2002; W.L. Marar & D. Mond, 1996); M- deformations of A-simple map-germs of corank 1 ( & J.H. Rieger, 2005).

18 Counter example maximality M- deformations of A-simple map-germs of corank 2 ( & J.H. Rieger, 2005) f (x,y) = (x 2 y 2 + x 3,xy) Algebraic number of singularities Any stabilisation has 3 complex cusps 2 complex double-folds Effective number of singularities Any stabilisation has 3 real cusps 0 real double-folds All known examples have the maximum number of cusps!

19 Theorem If F(t,x,y) = (t,x,g(x,y,t)) is a stabilisation of f (x,y), a smooth, finitely determined, corank 1, map-germ f : R n,0 R n,0, then s + (F) + s (F) = 2degree(Φ 1 ); s + (F) s (F) = 2degree(Φ 2 ). where Φ 1 and Φ 2 : R n+1,0 R n+1,0 are ( ) G Φ 1 (x,y,t) = y,..., n G y n,tj (x,y)h Φ 2 (x,y,t) = ( ) G y,..., n G y n,j (x,y)h and H : R n+1,0 R n,0 J (x,y) H = det(d (x,y) H) H(x,y,t) = ( ) G y,..., n G y n.

20 Corollary If F is a stabilisation of a finitely determined map-germ f : R n,0 R n,0 of corank 1, then s + (F) s (F) (mod 2).

21 Similar methods count: Branches in bifurcation problems (T. Nishimura, T. Fukuda & K. Aoki, 1989); Branches in symmetric bifurcations (J. Damon, 1991); Pitchforks in symmetric bifurcations (S.D.S. Castro & L. 2000); Folds in stabilisations of bifurcation diagrams (L. & R. 2006).

22 Simple example f : R 2,0 R 2,0 f (x,y) = (x,g(x,y)) where g(x,y) = y 4 + xy Stabilisation: F(t,x,y) = (t,x,y 4 + xy ty 2 ) = (t,x,g) H(x,y,t) = ( G/ y, 2 G/ y 2) H(x,y,t) = ( 4y 3 + x 2yt,12y 2 2t ) J xy H = 24y Φ 1 = 4y 3 + x 2yt 12y 2 2t 24yt T Φ 2 = 4y 3 + x 2yt 12y 2 2t 24y T

23 Φ : R n+1 R n+1 Φ 2 = degree(φ 2 ) = 1 Φ 1 = degree(φ) = 4y 3 + x 2yt 12y 2 2t 24y 4y 3 + x 2yt 12y 2 2t 24yt ξ Φ 1 (η) T T η a regular value of Φ signjφ(ξ). J (x,y,t) Φ 2 = 48 J (x,y,t) Φ 1 = 24(24y 2 + 2t) regular value (0, 2,0) at (x,y,t) = (0, 2,0) degree(φ 1 ) = 1 s + (F) = 2 s (F) = 0 as in path formulation.

24 Theorem If F(t,x,y) = (t,x,g(x,y,t)) is a stabilisation of f (x,y), a smooth, finitely determined, corank 1, map-germ f : R n,0 R n,0, then s + (F) + s (F) = 2degree(Φ 1 ); s + (F) s (F) = 2degree(Φ 2 ). where Φ 1 and Φ 2 : R n+1,0 R n+1,0 are ( ) G Φ 1 (x,y,t) = y,..., n G y n,tj (x,y)h Φ 2 (x,y,t) = ( ) G y,..., n G y n,j (x,y)h and H : R n+1,0 R n,0 J (x,y) H = det(d (x,y) H) H(x,y,t) = ( ) G y,..., n G y n.

25 Idea of proof F(t,x,y) = (t,x,g(x,y,t)) a stabilisation of f. H : R n+1,0 R n be ( ) G H(x,y,t) = y (x,y,t),..., n G y n (x,y,t) A n -singularities in F are branches of. {(x,y,t) : H(x,y,t) = 0 (x,y,t) (0,0,0)}.

26 Counting branches (for proof) H : R n R,(0,0) R n,0 (z, t) H(z, t) Branches of germ of H 1 (0) {(0,0)} r + (H) = branches t > 0 r (H) = branches t < 0 Theorem (Nishimura, Fukuda & Aoki, 1989) H : R n+1,0 R n a smooth germ with ( ) E (z,t) dim R < H 1,...,H n,j z H Then r + (H) + r (H) = 2degree(H 1,...,H n,tj z H); r + (H) r (H) = 2degree(H 1,...,H n,j z H). where J z H(z,t) = det D z H(z,t)

27 Φ j : R n+1 R n+1 Φ j = (Φ j,1,...,φ j,n+1 ) ring E (x,y,t) Q(Φ j ) = Φ j,1,...,φ j,n+1 E(x,y,t) Theorem If F is a stabilisation of a finitely determined map-germ f : R n,0 R n,0 of corank 1, then s + (F) and s (F) are dim R Q(Φ 1 ) 2dim R I 1 ± (dim R Q(Φ 2 ) 2dim R I 2 )) where each I j Q(Φ j ) is an ideal that is maximal with respect to the property I 2 j = 0.

28 (for proof) Φ : R n+1 R n+1 Φ = (Φ 1,...,Φ n+1 ) ring E (x,y,t) Q(Φ) = Φ 1,...,Φ n+1 E(x,y,t) Theorem (Eisenbud & Levine 1977) Φ : R n+1 R n+1 a finite map-germ with local ring Q(Φ). If I Q(Φ) is an ideal that is maximal with respect to the property I 2 = 0 then degree(φ) = dim R (Q(Φ)) 2dim R (I).

29 Φ j : R n+1 R n+1 Φ j = (Φ j,1,...,φ j,n+1 ) ring E (x,y,t) Q(Φ j ) = Φ j,1,...,φ j,n+1 E(x,y,t) socle s j = J (x,y,t) Φ j λ j : Q(Φ j ) R linear l j (s j ) > 0. L j : Q(Φ j ) Q(Φ j ) R bilinear L j (p,q) = λ j (pq). Theorem If F is a stabilisation of a finitely determined map-germ f : R n,0 R n,0 of corank 1, then s + (F) = signature(l 1 ) + signature(l 2 ).

30 (for proof) Φ : R n+1 R n+1 Φ = (Φ 1,...,Φ n+1 ) ring E (x,y,t) Q(Φ) = Φ 1,...,Φ n+1 E(x,y,t) socle s = J (x,y,t) Φ λ : Q(Φ) R linear l(s) > 0. L : Q(Φ) Q(Φ) R bilinear L(p,q) = λ(pq). Theorem (Eisenbud & Levine 1977) Φ : R n+1 R n+1 a finite map-germ with local ring Q(Φ) then degree(φ) = signature(l).

31 We had, for F a stabilisation of f : s + (F),s (F) σ(f ) s + (F) = degree(φ 1 ) + degree(φ 2 ) s (F) = degree(φ 1 ) degree(φ 2 ) degree(φ j ) = dim R Q(Φ j ) 2dim R I j < dim R Q(Φ j ) Theorem If F is a stabilisation of a finitely determined map-germ f : R n,0 R n,0 of corank 1, then dim R Q(Φ 1 ) dim R Q(Φ 2 ) = σ(f )

32 for proof H = ( G y,..., n G y n ) Q(Φ 1 ) = E (x,y,t) H,tJ (x,y) H from intersection multiplicity: get dim R Q(Φ 1 ) = dim R E (x,y,t) H,J(x,y) H + dim R E (x,y,t) H,t }{{} }{{} Q(Φ 2 ) σ(f ) dim R Q(Φ 1 ) dim R Q(Φ 2 ) = σ(f ) because j G y j (x,y,t) n g (x,y) (mod t ) implies yn E (x,y,t) G y,..., n G y n,t E (x,y) g y,..., n g y n = E (x,y) S(f )

33 Where to, now? Multisingularities Results may be extended using formulation of W.L. Marar, J. Montaldi &, 1999 Special situations weighted homogeneous symmetric dimension = 2 Go back to maximality problem!

34 THE END

35 References N. A Campo, 1975, Le groupe de monodromie du déploiement des singularités isolées de courbes planes. I, Maht. Ann V.I. Arnol d, 1992, Springer numbers and Morsification spaces, J. Algebraic. Geom S.B.S.D. Castro, 2000, Counting persitent pitchforks, in Real and Complex CRC Press, T. Cooper, D. Mond and R.W. Atique, 2002, Vanishing topology of codimension 1 multi-germs over R and C, Compositio Math. 131 (2002) J. Damon, 1991, Equivariant bifurcations and Morsifications for finite groups in Singularity theory and its applications, Part II, SLNM

36 References (cont.) J. Damon, 1991, On the number of branches for real and complex weighted homogeneous curve singularities, Topology, J. Damon and A. Galligo, 1976, A topological invariant for stable map germs, Inventiones Math D. Eisenbud and H.I. Levine, 1977, An algebraic formula for the degree of a C map-germ, Ann.Math M.R. Entov, 1993, On real Morsifications of D µ singularities, Dokl. Math S.M. Gusein-Zade, 1974, Dynkin diagrams of singularities of functions of two variables, Funct. Anal. Appl. 8, I.S. Labouriau and, 2006, Invariants for bifurcations, Houston J. of Mathematics

37 References (cont.) W.L. Marar, J.A. Montaldi and, 1999, Multiplicities of zero-schemes in quasihomogeneous corank-1 singularities C n C n, in: Singularity theory, LMS Lec. Notes 263 W.L. Marar and D. Mond, 1996, Real topology of map-germs with good perturbations, Topology T. Nishimura, T. Fukuda and K. Aoki, 1989, An algebraic formula for the topological types of one parameter bifurcation diagrams, Arch. Rat. Mech. Anal J.H. Rieger and, 2005, M- deformation of A-simple Σ n p+1 -germs from R n to R p, n p, Math. Proc. Camb. Phil. Soc