Resolution of Plane Curve Singularities

Exercises

Exercise 3.2.1.Letf, g∈C{x, y}. Show that the intersection multiplicity of f and gis at least the product of the respective multiplicities, that is,

i(f, g) ≥ mt(f)·mt(g).

Moreover, show that the multiplicity off can be expressed in terms of inter- section multiplicities

mt(f) = min i(f, g)g∈ x, y ⊂C{x, y} ,

and that the minimum is attained forg=αx+βya general linear form¹⁹. In particular, iff is irreducible, then

mt(f) = min ordx(t),ordy(t) , wheret →

x(t), y(t)

is a parametrization of the germ

V(f),0 .

Exercise 3.2.2.For any n≥3 and f1, . . . , fn ∈C{x}=C{x1, . . . , xn}, in- troduce theintersection multiplicity

i(f1, . . . , fn) = dim_CC{x}/f1, . . . , fn. Show that

(1)i(f1, . . . , fn)<∞ ⇐⇒ V(f1)∩. . .∩V(fn)⊂ {0}, (2)i(f1, . . . , fn)≥mt(f1)·. . .·mt(fn).

Exercise 3.2.3.Let f ∈C{x, y} split into snonsingular irreducible compo- nents f₁, . . . , f_s which pairwise intersect transversally, and let g∈C{x, y} satisfyi(fj, g)≥sfor allj= 1, . . . , s. Show thatf+tgsplits inC{x, y}into sirreducible components for almost allt∈C.

Blowing Up 0∈C².We identify the projective lineP¹ with the set of lines L⊂C² through the origin 0 and define B0C² to be the closed complex subspace

B0C²:= (z, L)∈C²×P¹z∈L

= (x, y;s:t)∈C²×P¹tx−sy= 0

⊂C²×P¹.

Definition 3.16.The projection π:B0C²→C², (p, L) →p, is called a σ- process with centre 0∈C², or theblowing up of 0∈C².

We writeE:=π⁻¹(0)⊂B₀C²and call it the exceptional divisor²⁰ofπ.

Frequently, E={0} ×P¹ (which we identify withP¹) is also called the first infinitely near neighbourhood of the point0∈C².

Note that each point of E corresponds to a unique line through the origin in C². Each fibreπ⁻¹(z),z=0, consists of exactly one point (z, L) whereL⊂C² is the unique line through 0and z= (x, y), that is, π⁻¹(z) = (x, y;x:y)

. In particular, the preimages of any two lines L=L ⊂C² through0 do not have any point in common. In other words, blowing up0∈C²“separates lines through the origin”.

C² π

E

B0C²

Fig. 3.13. The blowing up of0∈C²

Remark 3.16.1.In the same manner, we define the blowing upπ:Bz₀U→U of z0 in an open neighbourhoodU of z0∈C². Let ϕ= (ϕ1, ϕ2) :U →C² be

20 A(Cartier) divisor D in a complex space is a subspace which is locally defined by one equationf= 0. We denote bymDthe divisor given byf^m= 0. IfD is another divisor, given byg= 0, thenD+Ddenotes the divisor given byf g= 0.

biholomorphic onto some open neighbourhood of0∈C²withϕ(z0) =0, and lets:t be homogeneous coordinates on P¹. Then we can describe Bz0U in coordinates:

Bz₀U :=B^ϕ_z0U := (z;s:t)∈U×P¹ϕ1(z)t−ϕ2(z)s= 0} ⊂U×P¹. As usual, we writex, yinstead ofϕ₁, ϕ₂and call themlocal coordinates ofU with centrez0

Note that ifψ:U →C²provides other local coordinates onU with centre z0 then we get a canonical isomorphism

B^ϕ_z

0U −→^∼= B^ψ_z

0U , (z;s:t) −→

ψ⁻¹◦ϕ(z);s:t .

In particular, the notationBz₀U is justified. We coverU×P¹by two charts, induced by the canonical chartsV0:={s= 0},V1:={t= 0}forP¹:

Chart 1. U×V0⊂U×P¹(with coordinates x, y;v=t/s).

In this chartBz₀U is the zero-set ofxv−y. In particular, it is smooth, and we can introduce coordinates u=x, v onBz₀U∩(U×V0). With respect to these coordinates, the morphismπcan be described as

π: B_z0U∩(U×V₀)−→U , (u, v) −→(u, uv).

The exceptional divisor in this chart is E∩(U ×V0) = (u, v) | u= 0

= {0} ×Cwith coordinatev.

Chart 2. U×V1⊂U×P¹(with coordinates x, y;u=s/t).

B_z0U∩(U×V₁) is the zero-set of x−yu. Hence, it is smooth, and we can introduce local coordinates ¯u,¯v=ysuch that

π: Bz0U∩(U×V1)−→U , (¯u,v)¯ −→(¯u¯v,v)¯ , with exceptional divisorE∩(U×V1) = (u, v)|v= 0

=C× {0}.

Note that the coordinate of E (v in Chart 1, resp. u in Chart 2) is not only local but affine. That is, ifU =U1×U2⊂C², then (U ×V1)∩Bz₀U =

(u, v)∈U1×Cvu∈U2

is an open neighbourhood of {0} ×C, and (U ×U2)∩Bz0U = (u, v)∈C×U2uv∈U1

is an open neighbourhood of C× {0}.

Sometimes, we want to make a point p= (β :α)∈P¹=π⁻¹(0) in the exceptional divisor the centre of the coordinate system (u, v), resp. (u, v).

SinceP¹=V₀∪ {(0 : 1)}, we have p= (1 :α),α∈C, orp= (0 : 1). In Chart 1, a pointp= (1 :α) has coordinates (0, α); in Chart 2, the pointp= (0 : 1) has coordinates (0,0).

If (u, v) = (u, v−α) are new coordinates in Chart 1 thenp= (1 :α) has coordinates (u, v) = (0,0) and in these coordinates we have

π: B_z0U∩(U×V₀)−→U , (u, v) −→

u, u(v +α) .

Lemma 3.17.Let U be an open neighbourhood of z∈C². Then BzU is a 2-dimensional complex manifold, and the restrictionπ:BzU\E→U\ {z} is an analytic isomorphism.

Proof. The transition function between U×V0 and U×V1 is given by (x, y, t) →(x, y,¹_t), hence, analytic, which implies that BzU is a complex manifold. In local coordinates x, y;s:t, the inverse morphism is given by

π⁻¹:U\ {z} −→BzU\π⁻¹(z), (x, y) −→(x, y; x:y). Blowing Up a Point on a Smooth Surface.Blowing up is a purely local process. Hence, we can generalize the blowing up of0∈C²to define the blow- ing up of a point in an arbitrary smooth complex surface (i.e., 2-dimensional complex manifold) M.

Letz∈M be a point. Then there exists an open neighbourhood U ⊂M of z being isomorphic to an open neighbourhood of 0∈C². Choosing local coordinates with centre0, we can apply the above construction and define the blowing up ofz∈U,π:BzU →U ⊂M.

Since the graph of the restriction π:BzU\π⁻¹(z)−→^∼= U \ {z} is obvi- ously closed inB_zU×(U\ {z}), the glueing lemma [GuR, Prop. V.5] allows to define the blowing up ofz∈M,

B_zM := B_zU∪π

M\ {z}

−→M ,

by glueingB_zU andM\ {z}. Again, for different choice of local coordinates the result will be canonically isomorphic.

Remark 3.17.1.The following statements follow easily from the definition and are left as exercises.

(1) Let π:BzM →M be the blowing up of z∈M. Then the exceptional divisorE:=π⁻¹(z)⊂Bz satisfies

• E∼=P¹;

• its complementBzM \E is dense inBzM;

• the restrictionBzM \E→M\ {z} is an analytic isomorphism.

(2) The blowing upπ:B_zM →M of a pointz∈M iswell-defined up to iso- morphism (overM), that is, ifπ :B_zM →Mis another blowing up ofz∈M (obtained from local coordinatesx, y onU ⊂M) then there exists a unique isomorphism ϕ:B_zM →B_zM making the following diagram commute

BzM _∼^ϕ

= π

B_zM

π

M.

Moreover,ϕinduces a linear projectivityπ⁻¹(z) =E−→^∼= E =π ⁻¹(z).

We callB_zM →M amonoidal transformation blowing upz∈M, or sim- ply the blowing up of the pointz∈M.

(3) In particular, we can define the blowing up of the germ (M, z) at z as the germ ofπ:BzM →M along E=π⁻¹(z) (that is, an equivalence class of morphisms²¹ defined on a neighbourhood ofE⊂B_zM). We write

π: BzM−→(M, z), or π: (BzM, E)−→(M, z).

More generally, if (M, V) is the germ of M along the subvariety V ⊂M containingz, then we define the blowing up of (M, V) at z as the germ of π:B_zM →M alongπ⁻¹(V).

(4) Analytic isomorphisms lift to the blown-up surfaces. More precisely, let ϕ:M →M be an analytic isomorphism of smooth complex surfaces. Then there exists a unique isomorphismϕ:BzM →B_ϕ(z)M making the follow- ing diagram commute

BzM ^ϕ_∼

= Bϕ(z)M

M _ϕ^∼= M .

(5) Let z=w∈M. Then the surfaces B_z,wM (obtained by blowing up z∈M first and then blowing up the point π⁻¹(w)∈B_zM) and B_w,zM (obtained by blowing up in opposite order) are isomorphic overM.

Blowing up Curves and Germs.In the following we study the effect of the blowing upπ:BzM →M on a curveC⊂M. We define thetotal transform ofCto be the pull-back

C := π⁻¹(C)⊂B_zM . As we shall see below, as a divisor we have

C=C+mE , m= mt(C, z),

whereE is the exceptional divisor andCis thestrict transform ofC, C := π⁻¹(C)\E⊂BzM

provided with the induced, reduced structure. Here denotes the closure²² in BzM. E being an irreducible component of C, it follows that the strict

21 Thecategory of germs of complex spaces along a subspaceconsists of pairs (X, E) of complex spaces withEa subspace ofX. Morphisms (X, E)→(Y, F) are equiv- alence classes of morphismsX→Y mappingEtoF, where two such morphisms are called equivalent if they coincide on some common neighbourhood ofE.

22 It follows from the equations ofC in Chart 1, respectively in Chart 2, that the topological closure is a complex curve, hence an analytic subvariety ofBzM.

transformC consists precisely of the remaining irreducible components of the total transform C.

Since the blowing up map is an isomorphism outside the exceptional di- visor E=π⁻¹(z), it suffices to study the induced total (respectively strict) transform of the germ (C, z). Note that the total transform of the curve germ (C, z) is not a curve germ but thegerm ofCalong E, while the strict trans- form of (C, z) is a multi-germ of plane curve singularities.

Remark 3.17.2.Let (C,0)⊂(C²,0) be a plane curve singularity with local equation f ∈C{x, y}. Then we can describe the total (respectively strict) transform of (C,0) w.r.t. the local coordinates introduced in Remark 3.16.1:

let

f =fm+fm+1+. . . , fj homogeneous of degreej,

fm= 0, that is,m= mtf. Then the total transform of (C,0) is the germ of the total transform of a representativeC alongEwith (local) equation:

• inChart 1: f(u, v) =f(u, uv) =u^m

fm(1, v) +ufm+1(1, v) +. . .

& '( )

=:f(u, v)

,

• inChart 2: f(¯u,¯v) =f(¯u¯v,¯v) = ¯v^m

fm(¯u,1) + ¯vfm+1(¯u,1) +. . .

& '( )

=:f(¯u,¯v)

.

Then u, respectively ¯v, are the local equation of the exceptional divisor, f(u, v), respectively f(u, v), are the local equations of C, respectively C in Chart 1, while f(¯u,v), respectively¯ f(¯u,¯v), are the local equations ofC, re- spectivelyC in Chart 2. It follows that, as a divisor,C=mE+C. In partic- ular, the total transform is non-reduced whenever m >1.

The intersection of the strict transformC with E consists of at most m points given by the local equations

u= 0 =f_m(1, v), respectively ¯v= 0 =f_m(¯u,1). (3.3.1) Recall that the points of E⊂B0C² correspond to lines in C² through the origin. The points of intersection of E with the strict transform of a plane curve correspond precisely to those lines being tangent to the curve at the origin, as we shall see in the following.

Definition 3.18.Let f ∈C{x, y}, m:= mt(f), and let f_m∈C[x, y] denote the tangent cone, that is, the homogeneous part of lowest degree. Then f_m decomposes into (possibly multiple) linear factors,

fm = 8s i=1

(αix−βiy)^mi,

with (βi:αi)∈P¹ pairwise distinct,m=m1+. . .+ms. We call the factors (αix−βiy), i= 1, . . . , s, the tangents of f, the mi are called multiplicities of the tangent. We also refer to (β_i :α_i)∈P¹, i= 1, . . . , s, as the tangent directions of the plane curve germ (C,0) =V(f)⊂(C²,0) (with respect to the chosen local coordinates).

The tangents off are in 1–1 correspondence with the points of intersection of the strict transformCof (C,0) with the exceptional divisorE (cf. (3.3.1)).

Moreover, the multiplicity of the tangent coincides with the intersection mul- tiplicity ofC andE at the respective point.

Let C be a representative of the curve germ at 0 defined by f = 0. We leave it as an exercise to show that the tangents off correspond uniquely to the limits of secant lines 0swiths∈C,s→0.

Fig. 3.14.Tangents are limits of secant lines.

Lemma 3.19.Each irreducible factor off ∈C{x, y} has a unique tangent.

Proof. After a linear coordinate change,f isy-general of orderb(cf. Exercise 1.1.6), hence, by the WPT we can assume that f is, indeed, a Weierstraß polynomial

f = y^b+a1(x)y^b−1+. . .+ab, ai(0) = 0. Letf =fm+fm+1+fm+2+. . . and consider the strict transform

f(u, v) := f(u, uv)

u^m ≡ fm(1, v) modu ·C{u, v}.

It follows thatf(u, v) ∈C{u}[v] is monic, andf(0, v) =f_m(1, v) is a complex polynomial inv of degreem. In particular, it decomposes into linear factors,

f(0, v) = (v−c₁)^m1·. . .·(v−c_n)^mn,

n i=1

m_i=m .

Hensel’s lemma 1.17 implies the existence of polynomialsfi∈C{u}[v] of de- greemisuch thatf=f1·. . .·fnandfi(0, v) = (v−ci)^mi,i= 1, . . . n. Hence, we can write

f(x, y) = x^mf(x, y/x) = x^m1f1(x, y/x)

& '( )

∈m⊂C{x, y}

·. . .·x^mnfn(x, y/x)

& '( )

∈m⊂C{x, y} .

In particular, each tangent corresponds to a unique (not necessarily irre-

ducible) factorfi.

Example 3.19.1.(1) f =x∈C{x, y}. The (local) equations of the total, re- spectively strict, transform are

in Chart 1: f=u, f= 1 , inChart 2: f= ¯u¯v, f= ¯u, u, respectively ¯v, being the local equation of the exceptional divisor.

It follows that the strict transform of a smooth germ is, again, smooth and intersects the exceptional divisor transversally.

(2) f =x^m−y^m∈C{x, y}. Then we obtain

inChart 1: f=u^m(1−v^m) , f= 1−v^m= +m k=0

(1−e^2πi/kv) , inChart 2: f= ¯v^m(¯u^m−1) , f= ¯u^m−1 =

+m k=0

(¯u−e^2πi/k) .

E

Fig. 3.15.Blowing up the curve germ defined byx⁴−y⁴.

The strict transform intersects the exceptional divisor in m different points (corresponding to themtangents off), the germ at each of these points being smooth (see Fig. 3.15 on page 188).

(3)f =x²−y³∈C{x, y}. Here, the strict transform is smooth (local equation f= ¯u²−v), but intersects the exceptional divisor (in the point corresponding¯ to the unique tangentx) with multiplicity 2, that is, not transversally:

E={v¯²= 0}

C={u¯²−¯v= 0

No documento Introduction to (páginas 192-200)