Algebraic Group Actions

Exercise 2.2.4.Show that the degree of contact or right determinacy is not upper semicontinuous (see Theorem 2.6 forμandτ).

Hint. Show that x^2p−y^2p is 4p-determined, while (ty−x^p)²−y^2p is not 4p- determined for sufficiently largep.

Exercise 2.2.5.Prove Theorem 2.28.

Hint. Show first that ifm^bj(f) =m^bj(g) and f−g∈m^bj(f) thenf∼^r gas in the proof of Theorem 2.26. Then show that the assumptions in (c) allow to reduce to this situation.

Exercise 2.2.6.Prove Theorem 2.29.

Exercise 2.2.7. (1) Show that any germ f ∈m^d⊂C{x, y} with a non- degenerate principald-form is right (2d−2)-determined.

(2)^∗ Show that any germ f ∈m^d⊂C{x1, . . . , xn}, n≥2, with a non- degenerate principald-form is right (nd−2n+ 2)-determined.

Hint.Use the fact that the Jacobian ideal of a non-degenerated-form innvariables containsm^nd−2n+1.

where jet(ϕ, k)(xi) = jet(ϕ(xi), k) is the truncation of the power series of the component functions ofϕ.

As we shall show below, R^(k)and K^(k)are algebraic groups acting alge- braically on the jet space J^(k), which is a finite dimensional complex vector space. The action is given by

ϕ·f = jet(ϕ(f), k), (u, ϕ)·f = jet(u·ϕ(f), k),

forϕ∈ R^(k), (u, ϕ)∈ K^(k). Hence, we can apply the theory of algebraic groups to the action of R^(k)and K^(k). If k is bigger or equal to the determinacy of g (see Definition 2.21), then g∼^r f (respectively g∼^c f) iff g∈ R^(k)f (re- spectively iff g∈ K^(k)f). Hence, the orbits of these algebraic groups are in one-to-one correspondence with the corresponding equivalence classes.

Before we make use of this point of view, we recall some basic facts about algebraic group actions. For a detailed study we refer to [Bor, Spr, Kra].

Definition 2.31. (1) An (affine) algebraic group G (over an algebraically closed field K) is a reduced (affine) algebraic variety over K, which is also a group such that the group operations are morphisms of varieties. That is, there exists an element e∈G(the unit element) and morphisms of varieties overK

G×G−→G , (g, h) →g·h (the multiplication), G−→G , g →g⁻¹ (the inverse)

satisfying the usual group axioms.

(2) Amorphism of algebraic groups is a group homomorphism, which is also a morphism of algebraic varieties over K.

Example 2.31.1.(1) GL(n, K) and SL(n, K) are affine algebraic groups.

(2) For any field K, the additive group (K,+) and the multiplicative group (K^∗,·) ofK are affine algebraic groups.

(3) The groups R^(k) and K^(k) are algebraic groups for any k≥1. This can be seen as follows: an element ϕof R^(k) is uniquely determined by

ϕ⁽ⁱ⁾:=ϕ(xi) = n j=1

a⁽ⁱ⁾_j xj+ k

|α|=2

a⁽ⁱ⁾_α x^α, i= 1, . . . , n,

such that det a⁽ⁱ⁾_j

= 0. Hence,R^(k)is an open subset of a finite dimensional K-vectorspace (with coordinates the coefficients a⁽ⁱ⁾_j and a⁽ⁱ⁾α ). It is affine, since it is the complement of the hypersurface defined by the determinant.

The elements of the contact groupK^(k)are given by pairs (u, ϕ),ϕ∈ R^(k), u=u0+k

|α|=1uαx^α with u0= 0, hence K^(k) is also open in some finite dimensional vectorspace and an affine variety.

The group operations are morphisms of affine varieties, since the com- ponent functions are rational functions. Indeed the coefficients of ϕ·ψ are polynomials in the coefficients ofϕ, ψ, while the coefficients ofϕ⁻¹ are deter- mined by solving linear equations and involve det

a⁽ⁱ⁾_j

(respectively det a⁽ⁱ⁾_j andu₀) in the denominator.

Proposition 2.32.Every algebraic groupGis a smooth variety.

Proof. Since G is a reduced variety, it contains smooth points by Corollary 1.111. For anyg∈Gthe translationh →ghis an automorphism ofGand in this way G acts transitively on G. Hence, a smooth point can be moved to

any other point ofGby some automorphism ofG.

Definition 2.33. (1) An (algebraic) action of G on an algebraic varietyX is given by a morphism of varieties

G×X −→X , (g, x) →g·x, satisfyingex=xand (gh)x=g(hx) for allg, h∈G, x∈X.

(2) Theorbit ofx∈X under the action ofGonX is the subset Gx:=G·x:={g·x∈X|g∈G} ⊂X ,

that is, the image ofG× {x} inX under the orbit mapG×X →X. (3) Gactstransitively onX ifGx=X for some (and then for any)x∈X.

(4) Thestabilizerofx∈Xis the subgroupGx:={g∈G|gx=x}ofG, that is, the preimage ofxunder the induced mapG× {x} →X.

In this sense,R^(k)andK^(k)act algebraically onJ^(k). Note that the somehow unexpected multiplication on K^(k) as a semidirect product (and not just as direct product) was introduced in order to guarantee (gh)x=g(hx) (check this!).

For the classification of singularities we need the following important prop- erties of orbits.

Theorem 2.34.Let G be an affine algebraic group acting on an algebraic varietyX, andx∈X an arbitrary point. Then

(1)Gx is open in its (Zariski-) closureGx.

(2)Gx is a smooth subvariety ofX.

(3)Gx\Gx is a union of orbits of smaller dimension.

(4)Gx is a closed subvariety ofG.

(5) IfGis connected, thendim(Gx) = dim(G)−dim(Gx).

Proof. (1) By Theorem 2.35, below, Gx contains an open dense subset of Gx; in particular, it contains interior points ofGx. For anyg∈G, g·Gx is closed and containsGx. Hence,Gx⊂g·Gx. Replacing gwith g⁻¹and then

multiplying with g we also obtain g·Gx⊂Gx. It follows that Gx=g·Gx, that is,Gxis stable under the action ofG.

Now, consider the induced action ofGonGx. SinceGacts transitively on Gx and Gx contains an interior point of its closure, every point ofGx is an interior point ofGx, that is,Gxis open in its closure.

(2) Gx with its reduced structure contains a smooth point and, hence, it is smooth everywhere by homogeneity (see the proof of Proposition 2.32).

(3) Gx\Gxis closed, of dimension strictly smaller than dimGxandG-stable, hence a union of orbits.

(4) follows, sinceG_x is the fibre, that is, the preimage of a closed point, of a morphism, and since morphisms are continuous maps.

(5) Consider the mapf:G× {x} →Gxinduced byG× {x} →X. Thenf is dominant andGx=f⁻¹(x). SinceGis connected,G× {x} andGxare both irreducible. Since for y=gx∈Gx we have Gx=Gy and Gy =gGxg⁻¹, the statement is independent of the choice of y∈Gx. Hence, the result follows

from (2) of the following theorem.

We recall that a morphismf:X →Y of algebraic varieties is calleddominant if for any open dense set U ⊂Y, f⁻¹(U) is dense inX. When we study the (non-empty) fibres f⁻¹(y) of any morphism f:X →Y we may replace Y by f(X), that is, we may assume that f(X) is dense inY. If X and Y are irreducible, then f is dominant ifff(X) =Y.

The following theorem concerning the dimension of the fibres of a mor- phism of algebraic varieties has many applications.

Theorem 2.35.Let f:X →Y be a dominant morphism of irreducible vari- eties,W ⊂Y an irreducible, closed subvariety andZan irreducible component of f⁻¹(W). Putr= dimX−dimY.

(1) If Z dominatesW thendimZ≥dimW+r. In particular, fory∈f(X), any irreducible component off⁻¹(y)has dimension≥r.

(2) There is an open dense subset U ⊂Y (depending only on f) such that U ⊂f(X)and dimZ = dimW+ror Z∩f⁻¹(U) =∅. In particular, for y∈U, any irreducible component off⁻¹(y)has dimension equal tor.

(3) If X and Y are affine, then the open set U in (2) may be chosen such thatf:f⁻¹(U)→U factors as follows

f⁻¹(U) ^π

f

U ×A^r

pr1

U

withπ finite andpr₁ the projection onto the first factor.

Proof. See [Mum1, Ch. I,§8] and [Spr, Thm. 4.1.6].

Observe that the theorem implies that for dominant morphisms f:X →Y there is an open dense subsetU ofY such thatU ⊂f(X)⊂Y.

Recall that a morphism f:X →Y of algebraic varieties with algebraic structure sheaves OX and OY is finite if there exists a covering of Y by open, affine varietiesU_i such that for eachi,f⁻¹(U_i) is affine and such that OX

f⁻¹(U_i)

is a finitely generatedOY(U_i)-module.

Iff:X →Y is finite, then the following holds:

(1) f is a closed map,

(2) for eachy∈Y the fibref⁻¹(y) is a finite set,

(3) for every open affine setU ⊂Y, f⁻¹(U) is affine and OX

f⁻¹(U) is a finitely generatedOY(U)-module.

(4) IfX andY are affine, thenf is surjective if and only if the induced map of coordinate ringsOY(Y)→ OX(X) is injective.

(5) Moreover, if f: X→Y is a dominant morphism of irreducible varieties and y∈f(X) such that f⁻¹(y) is a finite set, then there exists an open, affine neighbourhood U of y in Y such that f⁻¹(U) is affine and f:f⁻¹(U)→U is finite.

For proofs see [Mum1, Ch. I,§7], [Spr, Ch. 4.2] and [Har, Ch. II, Exe. 3.4–3.7].

Now, let f:X →Y be a morphism of complex algebraic varieties and let f^an: X^an→Y^an be the induced morphism of complex spaces. It follows that f finite implies thatf^anis finite. The converse, however, is not true (see [Har, Ch. II, Exe. 3.5(c)]).

Let f: X→Y be a morphism of algebraic varieties,x∈X a point and y=f(x). Then the induced map of local rings f^#:OY,y → OX,x induces a K-linear map mY,y/m²_Y,y →mX,x/m²_X,x of the cotangent spaces and, hence, its dual is aK-linear map of tangent spaces

T_xf:T_xX −→T_yY

whereTxX = HomK(mX,x/m²_X,x, K) is the Zariski tangent space of X at x.

Observe that the cotangent and, hence, the tangent spaces coincide, inde- pendently of whether we considerX as an algebraic variety or as a complex space. Hence, iff^an:X^an→Y^anis the induced map of complex spaces, then the induced map (f^an)^#:OY^an,y→ OX^an,x induces the same map as f^# on the cotangent spaces and, hence, on the Zariski tangent spaces.

Proposition 2.36.Let f: X→Y be a dominant morphism of reduced, irre- ducible complex algebraic varieties. Then there is an open dense subsetV ⊂X such that for eachx∈V the mapTxf:TxX−→Tf(x)Y is surjective.

Proof. By Theorem 2.35 there is an open dense subset U ⊂Y such that the restrictionf:f⁻¹(U)→U is surjective.

By deleting the proper closed set A= Sing(f⁻¹(U))∪f⁻¹(Sing(U)) and consideringf:f⁻¹(U)\A→U\Sing(U), we obtain a mapf between com- plex manifolds. The tangent map off is just given by the (transpose of the)

Jacobian matrix off with respect to local analytic coordinates, which is sur- jective on the complement of the vanishing locus of all maximal minors.

Another corollary of Theorem 2.35 is the theorem of Chevalley. For this recall that a subsetY of a topological spaceX is calledconstructibleif it is a finite union of locally closed subsets ofX. We leave it as an exercise to show that a constructible setY contains an open dense subset ofY. Moreover, the system of constructible subsets is closed under the Boolean operations of taking finite unions, intersections and differences.

If X is an algebraic variety (with Zariski topology) and Y ⊂X is con- structible, then Y =s

i=1L_i with L_i locally closed, and we can define the dimension of Y as the maximum of dimL_i,i= 1, . . . , s. The following theo- rem is a particular property of algebraic varieties. In general, it does not hold for complex analytic varieties.

Theorem 2.37 (Chevalley). Let f:X→Y be any morphism of algebraic varieties. Then the image of any constructible set is constructible. In partic- ular, f(X) contains an open dense subset off(X).

Proof. It is clear that the general case follows if we show that f(X) is con- structible. Since X is a finite union of irreducible varieties, we may assume that X is irreducible. Moreover, replacingY byf(X) we may assume that Y is irreducible and that f is dominant.

We prove the theorem now by induction on dimY, the case dimY = 0 being trivial. Let the open set U ⊂Y be as in Theorem 2.35, then Y \U is closed of strictly smaller dimension. By induction hypothesis,f(f⁻¹(Y \U)) is constructible inY \U and hence inY. Thenf(X) =U∪f(f⁻¹(Y \U)) is

constructible.

We return to the action of R^(k)andK^(k)onJ^(k)=C{x1, . . . , xn}/m^k+1, the affine space ofk-jets. Note thatR^(k)andK^(k)are both connected as they are complements of hypersurfaces in some C^N.

Proposition 2.38.Let Gbe either R^(k), or K^(k), and forf ∈J^(k) letGf be the orbit off under the action ofGonJ^(k). We denote byTf(Gf)the tangent space toGf atf, considered as a linear subspace of J^(k). Then, for k≥1,

Tf(R^(k)f) =

m·j(f) +m^k+1 /m^k+1, Tf(K^(k)f) =

m·j(f) +f+m^k+1 /m^k+1.

Proof. Note that the orbit map and translation byg∈Ginduce a commuta- tive diagram

T_eG

∼=

Tf(Gf)

∼=

TgG T_gf(Gf).

Since the orbit mapG× {f} →Gf satisfies the assumptions of Proposition 2.36,TgG→Tgf(Gf) and, hence, TeG→Tf(Gf) are surjective. Hence, the tangent space to the orbit atf is the image of the tangent map ate∈Gof the mapR^(k)→J^(k),Φ →f◦Φ, respectivelyK^(k)→J^(k), (u, Φ) →u·(f◦Φ).

Let us treat only the contact group (the statement for the right group follows with u≡1): consider a curve t →(u_t, Φ_t)∈ K^(k) such that u₀= 1, Φ₀= id, that is,

Φ(x, t) =x+ε(x, t) : (Cⁿ×C,(0,0))−→(Cⁿ,0) u(x, t) = 1 +δ(x, t) : (Cⁿ×C,(0,0))−→ C,

withε(x, t) =ε¹(x)t+ε²(x)t²+. . .,εⁱ= (εⁱ₁, . . . , εⁱ_n) such thatεⁱ_j∈m, and δ(x, t) =δ1(x)t+δ2(x)t²+. . ., δi∈C{x}. The image of the tangent map are all vectors of the form

∂

∂t

(1 +δ(x, t))·f(x+ε(x, t)

t=0 modm^k+1

=δ1(x)·f(x) + n j=1

∂f

∂xj

(x)·ε¹_j(x) modm^k+1,

which proves the claim.

Of course, instead of using analytic curves, we could have used the inter- pretation of the Zariski tangent space T_xX as morphisms T_ε→X, where Tε= Spec

C[ε]/ε² .

In view of Proposition 2.38 we callm·j(f), respectivelym·j(f) +fthe tangent space atf to the orbit off under the right actionR ×C{x} →C{x}, respectively the contact actionK ×C{x} →C{x}.

Corollary 2.39.Forf ∈C{x1, . . . , xn}, f(0) = 0, the following are equiva- lent.

(a)f has an isolated critical point.

(b)f is right finitely-determined.

(c)f is contact finitely-determined.

Proof. (a)⇒(b). By Corollary 2.24, f is μ(f) + 1-determined. On the other hand,μ(f)<∞due to Lemma 2.3. Since the implication (b)⇒(c) is trivial, we are left with (c)⇒(a). Letf be contactk-determined andg∈m^k+1. Then f_t=f+tg∈ K^(k+1)f modm^k+2and, hence,

g= ∂ft

∂t

t=0∈m·j(f) +fmodm^k+2,

by Proposition 2.38. By Nakayama’s lemmam^k+1⊂m·j(f) +f, the latter being contained inj(f) +f. Hence,τ(f)<∞andf has an isolated critical

point by Lemma 2.3.

Lemma 2.40.Let f ∈m²⊂C{x1, . . . , xn} be an isolated singularity. Let k satisfy m^k+1⊂m·j(f), respectivelym^k+1⊂m·j(f) +f, and call

r-codim(f) := codimension ofR^(k)f inJ^(k), respectively c-codim(f) := codimension ofK^(k)f inJ^(k)

the codimension of the orbitoff inJ^(k)under the action ofR^(k), respectively K^(k). Then

r-codim(f) =μ(f) +n , c-codim(f) =τ(f) +n.

Proof. In view of Proposition 2.38 and the definition of μ(f) andτ(f), one has to show that

dim_C j(f)

mj(f)

= dim_C(j(f) +f)

(mj(f) +f) =n . (2.3.1) Both linear spaces in question are generated by the partials _∂x^∂f

1, . . . ,_∂x^∂f

n, and it is sufficient to prove that none of these derivatives belongs to the ideal mj(f) +f. Arguing to the contrary, assume that _∂x^∂f

1 ∈mj(f) +f. This implies

∂f

∂x1

= n i=2

α_i(x)∂f

∂xi

+β(x)f for some α₂, . . . , α_n∈m,β∈C{x}.

The system of differential equations dx_i

dx1

=−α_i(x₁, . . . , x_n), x_i(0) =y_i, i= 2, . . . , n , has an analytic solution

xi=ϕi(x1, y2, . . . , yn)∈C{y2, . . . , yn}{x1}, i= 2, . . . , n ,

convergent in a neighbourhood of zero. In particular, we can define an iso- morphismC{x₁, x₂, . . . , x_n} ∼=C{x₁, y₂, . . . , y_n}which sendsf(x) to

f(x₁, y₂, . . . , y_n) =f(x₁, ϕ₂(x₁, y₂, . . . , y_n), . . . , ϕ_n(x₁, y₂, . . . , y_n)) such that

∂f

∂x1

=

∂f

∂x1 − n i=2

α_i(x)∂f

∂xi

xi=ϕi(x1,y2,...,yn) i=2,...,n

=β(x₁, y₂, . . . , y_n)·f .

This equality can only hold iffdoes not depend onx₁. But thenfand _∂x^∂f

i, i= 1, . . . , n, all vanish along the line{(t,0, . . . ,0)|t∈C}, contradicting the assumption that f (and, hence, f) has an isolated singularity at the origin.

Remark 2.40.1.In Section 3.4, we will study another important classification of (plane curve) singularities: the classification with respect to (embedded) topological equivalence. Unlike the classifications studied above, the topolog- ical classification has no description in terms of an algebraic group action.

Exercises

Exercise 2.3.1.(1) Let f =y²+x^2p2∈C{x, y}. Show that the R^(k)- and K^(k)-orbits off contain an element of degree 2p, wherek≥2p².

(2)^∗Letf =x^2p₁ ⁿ+x²₂+. . .+x²_n,n≥3. Show that theR^(k)- andK^(k)-orbits off contain an element of degree 2p, wherek≥2pⁿ.

Hint.See [Wes].

(3)^∗Show that the R^(k)-orbit (resp. K^(k)-orbit) of a germ f ∈C{x, y} with Milnor numberμ(f) =μ <∞contains an element of degree less than 4√

μ (resp. less than 3√

μ).

Hint.See [Shu].

(4)^∗∗(Unsolved problem).Is it true that theR^(k)- andK^(k)-orbits of a germ f ∈C{x1, . . . , xn},n≥3, with Milnor numberμ(f) =μ <∞contain an element of degree less thanαn√ⁿμ, whereαn>0 depends only onn?

(5)^∗∗(Unsolved problem). Given an integer p≥10 such that √p∈Z, does there exist a series of semiquasihomogeneous fm∈C{x, y}, m≥1, of type (p,1; 2mp) whose K^(k)-orbits contain elements of degree less than m√p(1 +o(m))?

Exercise 2.3.2.Introduce the right-left group RL= Aut(C,0)×Aut(Cⁿ,0) with the product

(ψ, ϕ)·(ψ, ϕ) = (ψ ◦ψ, ϕ◦ϕ),

acting onm⊂C{x} by (ψ, ϕ)(f) =ψ(f(ϕ)), and define theright-left equiva- lence

f ∼^rlg :⇐⇒ g=Φ(f) for someΦ∈ RL. (1) Show the implications

f∼^rg =⇒ f ∼^rlg =⇒ f∼^c g ,

and that the right-left equivalence neither coincides with the right, nor with the contact equivalence.

(2) Determine Tf(RL^(k)f) forf ∈m⊂C{x} andksufficiently large.

Exercise 2.3.3.Show that the right classification of the germs f ∈C{x}= C{x1, . . . , xn} of order d with a non-degenerate d-form depends on N−n parameters (moduli), where

N = # (i1, . . . , in)∈Zⁿ i1+. . .+in≥d, max{i1, . . . , in} ≤d−2 . Exercise 2.3.4.LetC(w,d)(x) be the space of semiquasihomogeneous germs f ∈C{x} with a non-degenerate quasihomogeneous part of type (w, d), and letR(w,d)⊂ R be the subgroup leaving C(w,d)(x) invariant. Determine Tf(R(w,d)f) and compute the number of moduli in the right classification of the above germs.

No documento Introduction to (páginas 146-155)