Intersection Multiplicity

Proposition 3.10 (Halphen’s formula¹⁶). Let f, g∈C{x, y} and assume f =

8m i=1

y−yi(x^1/N) , g =

m

8

j=1

y−y_j(x^1/N)

with y_i(t), y_j(t)∈ t ·C{t}, N some positive integer. Then the intersection multiplicity off andg is

i(f, g) = m i=1

m

j=1

ordt

yi(t)−y_j(t)

N . (3.2.1)

Proof. Since both sides are additive, it suffices to prove (3.2.1) in the case f being irreducible.

Note thatf isy-general of orderm. Hence, due to Proposition 3.4 there exists a convergent power seriesy(t)∈ t·C{t} and a unitu∈C{x, y} such that, inC{x^1/mN, y},

8m i=1

y−yi(x^1/N)

= f = u· 8m i=1

y−y

ξⁱx^1/m ,

ξ a primitive m-th root of unity. Since the power series ringC{x^1/mN, y} is factorial, we may assume

yi

x^1/N

=y

ξⁱx^1/m

, i= 1, . . . , m . In particular,s →

s^m, y(ξⁱs)

is a parametrization forV(f), and we obtain (t^N =x=s^m)

i(g, f) = ordsg

s^m, y(ξⁱs)

= m

N ·ordtg

t^N, y(ξⁱt^N/m)

= m

N ·ordtg

t^N, yi(t)

= m·

m

j=1

ordt

yi(t)−y_j(t)

N .

Since this holds for anyi= 1, . . . , m, we derive the equality (3.2.1).

As an immediate corollary, we obtain Corollary 3.11.Let f, g∈C{x, y}. Then (1)i(f, g) =i(g, f).

(2)i(f, g)<∞ ⇐⇒f andg have no common non-trivial factor.

16 This kind of formula was already used by Zeuthen [ZeP] to determine the mul- tiplicities of fixed points of one-dimensional algebraic correspondences. Hence, sometimes, it is also calledZeuthen’s formula.

Using the finite coherence theorem, we can give a completely different formula for the intersection multiplicity which does not involve a parametrization¹⁷: Proposition 3.12.Let f, g∈C{x, y}. Then

i(f, g) = dim_CC{x, y}/f, g, (3.2.2) in the sense, that if one of the two sides is finite then so is the other and they are equal. In particular,

i(f, g)<∞ ⇐⇒ V(f)∩V(g)⊂ {0},

with V(f), V(g)⊂(C²,0) denoting the plane curve germs defined byf, g.

For the proof of Proposition 3.12 we need the following

Lemma 3.13.Letf ∈C{x, y}be irreducible, let(C,0)⊂(C²,0)be the plane curve germ defined by f, and let ϕ: (C,0)→(C²,0) be a parametrization of (C,0). Then there exist open neighbourhoods of the origin, D⊂C and B ⊂C², and a holomorphic representativeϕ:D→Bfor the parametrization such that

ϕ(D),0

= (C,0)and (1)ϕ:D→B is finite;

(2)ϕ:D→ϕ(D) =:C is bijective;

(3)ϕ:D\ {0} →C\ {0} is biholomorphic.

Proof. Letϕ(t) =

x(t), y(t)

and assume thatx(t)= 0∈C{t}(otherwise the statement is obvious). After a reparametrization (Lemma 3.2), we may assume thatx(t) =t^b. Thenϕis quasifinite and, by the local finiteness theorem (The- orem 1.66), we can find DandB such that ϕ:D→B is finite.

By Corollary 1.68, the imageϕ(D)⊂B is a closed analytic subset which we endow with its reduced structure. Sincef◦ϕ= 0, the latter is contained in the plane curve (germ) C. Now, let (x0, y0)∈C,x0= 0, be sufficiently close to 0, and lett0∈Cbe a fixedb-th root ofx0. By Proposition 3.4,

f(x₀, y₀) =c₀· 8b j=1

y₀−y ξ^jt₀

,

with ξ a primitiveb-th root of unity andc0∈C\ {0}. Moreover, due to the identity theorem for univariate holomorphic functions, we may assume that y

ξ^jt0

=y ξⁱt0

fori=j. It follows that there exists a uniquej0 such that y₀=y

ξ^j0t₀

, hence (2).

Since the choice of theb-th root can be made holomorphically along a fixed branch near x₀∈C\ {0}, the mapt →

t^b, ξ^j0t₀

has a holomorphic inverse in a neighbourhood of (x₀, y₀)∈C\ {0}, which implies (3).

17 Alternatively to the use of the finite coherence theorem, we could use the reso- lution of plane curve singularities via blowing up (as introduced in Section 3.3) and the recursive formula (3.3.2) for the intersection multiplicities of the strict transforms, cf. Remark 3.29.1.

Proof of Proposition 3.12.First, letf be irreducible. Iff dividesg, both sides of (3.2.2) are infinite. Hence, assume that this is not the case, and choose a representativeϕ:D→B of a parametrization of f as in Lemma 3.13. Since ϕ:D→C is surjective and biholomorphic outside the origin, the induced mapOC→ϕ_∗OD is injective and we have an exact sequence

0−→ OC−→ϕ_∗OD−→ϕ_∗OD

OC−→0, where the quotient sheaf ϕ_∗OD

OC is supported at {0}. By the finite co- herence theorem (Theorem 1.67), ϕ_∗OD is coherent, hence the quotient ϕ_∗OD

OC is coherent, too (A.7, Fact 2). By Corollary 1.74, ϕ_∗OD

OC

0

is a finite dimensional complex vector space. Since OC,0∼=C{x, y}/f and (ϕ_∗OD)0∼=C{t}, we get a commutative diagram with exact rows

x, y x(t), y(t) 0 C{x, y}/f

·g

C{t}

·g

x(t),y(t) ϕ_∗OD

OC

0 π

0

0 C{x, y}/f C{t} ϕ_∗OD

OC

0 0.

Sincef does not divideg, multiplication byg is injective onC{x, y}/f, and the snake lemma gives an exact sequence

0→Ker(π)→C{x, y}"

f, g#

→C{t}"

g

x(t), y(t)#

→Coker(π)→0. Since dim_C

ϕ_∗OD

OC

0<∞, theC-vector spaces Ker(π) and Coker(π) have the same dimension. Hence,

dim_CC{x, y}"

f, g#

= dim_CC{t}"

g

x(t), y(t)#

. (3.2.3) Ifm= ordg

x(t), y(t)

theng

x(t), y(t)

=t^m·u(t) for a unitu∈C{t}. But this just means that the dimension on the right-hand side of (3.2.3) equals m=i(f, g).

Iff is reducible, and if

xi(ti), yi(ti)

,i= 1, . . . , r, are parametrizations of the irreducible factors off, then the same argument as before works, noting that thenD=9r

i=1Di and (ϕ_∗OD)0∼=%r

i=1C{ti}.

Finally, as the quotient sheafOB/f, g is coherent, Corollary 1.74 gives that the stalk at 0, C{x, y}"

f, g#

, is a finite dimensionalC-vector space iff the germ of the support ofOB/f, gat0is contained in{0}. As the support ofOB/f, gequalsV(f, g) =V(f)∩V(g), this implies the second statement

of the proposition.

Using the equality (3.2.2) and the principle of conservation of numbers (Sec- tion 1.6), we can give a beautiful geometric description of the intersection multiplicity of two (not necessarily reduced) plane curve germs (C,0),(D,0) as number of intersection points of two neighbouring curves (obtained by small deformations) in a small neighbourhoodU ⊂C² ofz:

Proposition 3.14.Let f, g∈C{x, y} have no common factor, and let F, G∈C{x, y, t} be unfoldings of f, respectivelyg.

Then, for all sufficiently small neighbourhoods U =U(0)⊂C², we can choose an open neighbourhood W =W(0)⊂C such that

• F andGconverge onU×W,

• the curvesC=V(f) =V(F0)andD=V(g) =V(G0)have the unique in- tersection point0inU,

• for allt∈W, we have

i(f, g) =iU(Ct, Dt) :=

z∈U

iz(Ct, Dt), (3.2.4) whereC_t=V(F_t)andD_t=V(G_t)in U.

In particular, if the curves Ctand Dt are reduced and intersect transversally in U theni(f, g)is just the number of points inCt∩Dt.

We call iU(Ct, Dt) the total intersection multiplicity of the plane curves Ct

andDtinU.

Example 3.14.1.We reconsider the intersection multiplicity of the ordinary cusp (f =x²−y³) with the smooth curve germs given by x, respectively αx+y. The unfolding Ft:=x²−y³−ty² turns the cusp into an ordinary node. Now, for t= 0 small, we can computei(f, x), respectivelyi(f, αx+y) as the number of (simple) intersection points of the curvesV(Ft) andV(x−t), respectively V(αx+y+t):

V(x)

V(f)

−→

V(x−t) V(Ft)

V(αx+y) V(f)

−→

V(αx+y+t) V(Ft)

Indeed, there are three, respectively two, simple intersection points appearing on the scene.

Proof of Proposition 3.14. Choose open neighbourhoodsU=U(0)⊂C²and W =W(0)⊂Csuch thatFandGconverge onU×W. Since0is an isolated point of the intersectionV(f)∩V(g) inU, we can assume (after shrinkingU if necessary) that it is, indeed, the unique intersection point of the curves C andD inU. It remains to deduce (3.2.4), maybe again after shrinkingU and

W. To do so, we apply the principle of conservation of numbers (Theorem 1.81).

Let X⊂U×W be given by the ideal sheaf J :=F, G, and consider the map π:X →W induced by the natural projection U×W →W. The structure sheafOX =OU×W/J is coherent (Corollary 1.64) and satisfies

z∈π⁻¹(t)

dim_COX,z

m_W,tOX,z=

z∈U

dim_COU,z

F_t, G_t=

z∈U

i_z(F_t, G_t).

Hence, it only remains to show thatπis a flat morphism.

Since (C,0) and (D,0) have no common component, we can assume by Proposition 1.70 that π is finite. Hence, due to Theorem 1.78, the flatness ofπis equivalent to the local freeness ofπ_∗OX, and for our needs it is even sufficient to show that (π_∗OX)0∼=C{x, y, t}/F, Gis a freeC{t}-module (us- ing Theorem 1.80 (1)). ButC{x, y, t}/F, Gis a complete intersection, hence Cohen-Macaulay (Corollary B.8.10) and, hence, free (Corollary B.8.12).

A direct proof of the freeness goes as follows: since C{t} is a principal ideal domain, and sinceC{x, y, t}/F, Gis a finitely generatedC{t}-module, it suffices to show thatC{x, y, t}/F, Gis torsion free (cf. [Lan, Thm. III.7.3]), or, equivalently, that for anyH ∈C{x, y, t}andk≥1 we have the implication

t^k·H ∈ F, G ⇐⇒ H ∈ F, G.

Assume that t^kH =AF+BG with A, B∈C{x, y, t}. Setting t= 0 gives A(x, y,0)·f +B(x, y,0)·g= 0, which implies

A(x, y,0) =h·g , B(x, y,0) =−h·f , (3.2.5) for someh∈C{x, y} (since C{x, y} is a UFD and gcd(f, g) = 1). Moreover, obviously,

t^kH = AF+BG = (A−hG)F+ (B+hF)G ,

and (3.2.5) implies that the power seriesA−hGandB+hF are both divis- ible byt. It follows thatt^k−1H ∈ F, G, and, by induction,H ∈ F, G. There exists another, purely topological, characterization of the (local) total intersection number of two plane curves germs without common components:

Proposition 3.15.Let f, g∈C{x, y} be reduced and without common fac- tors, and letB be a closed ball centred at the origin such thatf, g converge on B and(f(z), g(z))= (0,0) asz∈∂B. TheniB(f, g)is equal to the (topolog- ical) degree of the map

Φ:∂B−→S³, z −→ (f(z), g(z)) |f(z)|²+|g(z)|².

Proof. See [Mil1, Lemma B.2].

We close this section by a computational remark. If we want to compute the intersection multiplicity of two polynomials (or, power series) f and g in a computer algebra system asSingular, we may either use a parametrization as in the definition of the intersection multiplicity or use the formula (3.2.2) expressing the intersection multiplicity as codimension of an ideal (which can be computed then by using standard bases, see, e.g., [GrP, Cor. 7.5.6]).

Example 3.15.1.To compute the intersection of f = (x³−y⁴)(x²−y²−y³) with g= (x³+y⁴)(x²−y²−y³+y¹⁰) in Singular, we may either start by computing a (sufficiently high¹⁸jet of a) parametrization for each irreducible factor off,

LIB "hnoether.lib";

ring r = 0,(x,y),ls; // we have to use a local ordering poly f = (x3-y4)*(x2-y2-y3);

poly g = (x3+y4)*(x2-y2-y3+y10);

list L = hnexpansion(f); // result is a list of rings def R = L[1];

setring R; // contains list hne = HNE of f // computing higher jets of the HNE (where needed):

for (int i=1; i<=size(hne); i++) {

if (hne[i][4]<>0) { hne[i]=extdevelop(hne[i],10) };

};

// deducing a parametrization:

list P = parametrisation(hne);

// substituting the parametrization for x,y

for (i=1; i<=size(P); i++) { map phi(i) = r,P[i]; };

ord(phi(1)(g))+ord(phi(2)(g))+ord(phi(3)(g));

//-> 44

or we may compute the codimension of the (complete intersection) ideal gen- erated by f, g:

setring r; // we have to change from R back to r ideal I=f,g;

vdim(std(I));

//-> 44

18 A priori, it is clear, that forN sufficiently large, theN-jet of the parametrization is sufficient for the computation of the intersection number. The problem is to have a good lower bound forN. To get such a bound with Singular, one may compute a system of Hamburger-Noether expansionshne for the productf·g, and computelist P = parametrisation(hne,0);. Then the maximal integer in the entriesP[i][2],i= 1, . . . ,size(P), gives an appropriate lower bound forN. In the example, this maximal entry is 10.

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.

No documento Introduction to (páginas 185-192)