Degree of the divisor of solutions of a differential equation on a projective variety

16 Pages·0.17 MB·English

by  Vicente Muñoz

9 DEGREE OF THE DIVISOR OF SOLUTIONS OF A DIFFERENTIAL
9
EQUATION ON A PROJECTIVE VARIETY
9
1

n VICENTE MUN˜OZ AND IGNACIO SOLS
a
J
December, 1998
5
1

]
G Abstract. Using the data schemes from [5] we give a rigorous definition of alge-
A braic differential equations on the complex projective space Pn. For an algebraic
subvariety S ⊆ Pn, we present an explicit formula for the degree of the divisor of
.
h solutions of a differential equation on S and give some examples of applications.
t
a We extend the technique and result to the real case.
m
[

1
1. Introduction
v
4
6 We deal with the problem of finding the degree of the divisor of solutions of a differential
0 equation on a projective variety, which was studied by Halphen in [6] for differential equa-
1 tions on plane curves and on hypersurfaces of Pn. With the notion of infinitesimal data in
0
Pn, introduced in the plane by A. Collino [4], used by S. Colley and G. Kennedy in [2] [3],
9
9 and generalised to higher dimensions in [5], we solve this problem for algebraic subvarieties
/ of Pn (the method of Halphen seems to generalise only to complete intersections). The
h
t plane data of Collino have been rediscovered by Mohamed Belghiti [1] and applied to give a
a
m modern proof of the Halphen formula for plane curves, including some explicit calculation
of the involved invariants of the equation that Halphen was able to obtain in this case.
:
v
i Let Pn the complex projective space of dimension n (although we can work over any
X
algebraically closed field K of zero characteristic). Associated to a set of coordinates
r x ,... ,x ,y ,... ,y on an affine open set U0 = Cn ⊆ Pn, we construct inductively an
a 1 k 1 n−k
open set Ur of the data scheme DrPn, where the partial derivatives of the y with respect
k j
to the x are understood as the canonical coordinates of Ur. In this context, a differential
i
equation
∂y ∂2y ∂ry
j j j
f(x ,y , , ,··· , )= 0,
i j
∂x ∂x ∂x ∂x ···∂x
i i2 i1 ir i1
isunderstoodasaalgebraicequationontheopensetUr. Thedatasatisfyingthedifferential
equation form a divisor DrPn(f) on DrPn defined as the closure of the solutions to f = 0.
k k
Mathematics Subject Classification. Primary: 14N10. Secondary: 14D20, 14M15, 14P05.
1
2 VICENTE MUN˜OZ AND IGNACIO SOLS
For the differential equation f there are naturally well defined enumerative invariants γf,
s
0 ≤ s ≤r.
Let S ⊆Pn be an algebraic subvariety of dimension k, then S(f) will be the locus of the
solutions of the differential equation f = 0 on S. Themain resultof the paper(theorem 13)
gives a closed formula for degS(f) (assuming S(f) is a proper subset of S), expressed as
the scalar product between the γf and the cuspidal numbers γs of S. This reduces to a
s S
simple formula degS(f)= γfγ0+γfγ1 when S is smooth (or normal), where γ0 and γ1 are
0 S 1 S S S
the degree and class of S respectively, and γf and γ1 can be computed directly, either with
0 f
the use of proposition 7 or remark 9 (when f is distinguished in suitable variables, which
is the general case) or by using the formula degS(f) = γfγ0 +γfγ1 applied to a smooth
0 S 1 S
quadric and a smooth cubic subvarieties S ⊆ Pn, and solving the system of linear equations
thus obtained.
Theuseof theorem 13 is two-fold. Ontheonehanditgives thedegree of S(f)if weknow
enough information from f and S. This can be used to understand geometric properties
of S as these are usually described by a suitable differential equation (e.g. flexes of plain
curves, parabolic points). On the other hand the explicit computations of degS(f) in
particular cases may lead to finding the invariants γf of a differential equation f, and the
s
cuspidal numbers γs of a subvariety S. Section 4 is devoted to examples in which these two
S
applications of theorem 13 are worked out. In particular, we find the degree of the divisor
of parabolic points of a subvariety of Pn.
Finally in the last section we extend the results for the real projective space Pn, by
R
studying the behaviour of the constructions involved via the conjugation involution. This
is merely an introduction and not a thorough analysis of the use of data varieties in the
real case. The formula thus obtained in theorem 23 is analogue to the one in the complex
case, but we have to restrict ourselves to coefficients in Z/2Z. We end up with a simple
application on the number of umbilical points of a real surface in P3.
R
2. Differential equations on projective space
To make precise sense of a differential equation in the projective space Pn, we need to
recall from [5] the definition of the smooth compact moduli DrPn of infinitesimal data of
k
dimension k at order r in Pn.
Let Z be a scheme smooth over C. We define schemes DrZ together with embeddings
k
and projections
DrZ ⊂ - Gr TDr−1Z
k k k
(cid:0)
(cid:0)
br (cid:0)
??(cid:9)(cid:0)(cid:9)(cid:0) πr
Dr−1Z
k
DEGREE OF THE DIVISOR OF SOLUTIONS OF A DIFFERENTIAL EQUATION 3
inductively, as follows. We take D0Z = Z, D1Z = Gr TZ and if r ≥ 2 and
k k k
Dr−1Z ⊂ - Gr TDr−2Z
k k k
(cid:0)
(cid:0)
br−1 (cid:0)
??(cid:9)(cid:0)(cid:9)(cid:0) πr−1
Dr−2Z
k
is already defined then DrZ ⊆ Gr TDr−1Z is the Grassmannian Gr F of the (Semple)
k k k k r−1
bundle F ⊆ TDr−1Z obtained as pull-back
r−1 k
Λ ============Λ
r−1 r−1
6 6
6 6
Tb
(1) 0 - TDr−1Z/Dr−2Z - TDr−1Z r−-1 b∗ TDr−2Z - 0
k k k r−1 k
6 6
w
w pull-back
w
w
w
w
w
w ∪ ∪
0 - TDr−1Zww/Dr−2Z - F - Σ - 0
k w k r−1 r−1
where the right hand sequence is the restriction to Dr−1Z of the universal sequence on
k
Gr TDr−2Z.
k k
On each DrZ = Gr F there is a natural divisor CrZ, namely the Schubert special
k k r−1 k
cycle of k-planes of F meeting TDr−1Z/Dr−2Z, whose elements are called cuspidal. The
r−1 k k
elements in the complement are called non-cuspidal.
If Z′ is a subvariety of Z of dimension k and Z′ is the open set of smooth points, then
reg
Z′ ∼= DrZ′ ⊆ DrZ
reg k reg k
by the obvious functoriality of our construction. We then defineDrZ′ ⊆ DrZ as the closure
k k
of this subset.
From now on Z = Pn = P(V) and 1 ≤ k ≤ n − 1. Fix a trivialisation V ∼= Cn+1 or
projective reference, with corresponding hyperplane C0 at infinity and affine part U0 =
Pn−C0 = Cn = Ck ×Cn−k, with affine coordinates which we name
(2) x ,... ,x ,... ,x ,y ,... ,y ,... ,y .
1 i k 1 j n−k
In orderto make senseof adifferential equation, weneed to introduceformalpartial deriva-
tivesymbolsofthevariablesy withrespecttothevariablesx . Foranys ≥ 1,1≤ j ≤ n−k,
j i
and 1≤ i ,... ,i ≤ k we introduce
1 s
∂sy
(3) yj = j .
i1,...,is ∂x ···∂x
is i1
4 VICENTE MUN˜OZ AND IGNACIO SOLS
At the present stage these are to be understood merely as symbols. Later we will identify
them as coordinates of an affine open set of the data scheme DrPn. Note that we have
k
introduced non-commuting derivative symbols (see remark 5 below).
On each point P ∈ U0 we consider the k-space X(P) = P +Ck ⊆ TU0 and the (n−k)-
space Y(P) = P + Cn−k ⊆ TU0 providing subbundles X, Y of TU0 = X ⊕ Y, all of
them trivial. The complement U1 in D1U0 = Gr TU0 of the special Schubert cycle of
k k
k-subbundles of TU0 meeting Y is thus
(4) U1 = Hom(X,Y) = U0×Hom(Ck,Cn−k) = U0×V1,
whereV1 =Hom(Ck,Cn−k)isacartesian powerofCwithcoordinates yj (standingfor ∂yj).
i ∂xi
On U1, the lifted k-subbundle X ⊆ (TU0) provides a split of the universal sequence
U1 U1
0 → Σ → (TU0) → Λ → 0
1 U1 1
on U1. The universal subbundle Σ on U1 becomes then isomorphic to the bundle X ∼=
1 U1
U1×Ck.
We shall denote C1 the closure in D1Pn of the complement of U1 in D1U0. Also for all
k k
r ≥ 2 we let Cr = CrPn be the cuspidal divisor of DrPn. Denote by Cr for all 0 ≤ s ≤ r
k k s
the divisor of DrPn counterimage by
k
(5) b = b ◦···◦b ◦b : DrPn → DsPn
r,s s+1 r−1 r k k
of the divisor Cs of DsPn (in particular Cr = Cr). Denote by Ur the open subset of DrPn
k r k
Ur = DrPn\(Cr ∪...∪Cr ∪...∪Cr)
k 0 s r
consisting of data which are not in Cr (i.e. which are “finite”), are not in Cr (i.e. which are
0 1
nowhere “vertical”) and are not in Cr for any s ≥ 2 (i.e. which are “nonsingular”). It is
s
important to note that Cr is intrinsically defined for s ≥ 2, but is dependent on the choice
s
of trivialisation for s = 0,1. The data in Ur are the data for which the symbols (3), for
s = 1,... ,r, are going to acquire a precise meaning.
Lemma 1. Cr,Cr,... ,Cr,... ,Cr is a basis for A1DrPn.
0 1 s r k
Proof. We can see this by induction on r ≥ 0. For r = 0 it is obvious. In general, we have
to show that the natural map
(6) PicDr−1Pn×ZCr → PicDrPn = PicGr (F )
k k k r−1
is an isomorphism. We need only to remark that for any point P ∈ Dr−1Pn, (Cr) is a
k P
basisofthePicardgroupoftheGrassmannianGr ((F ) ). Surjectivity of(6)nowfollows
k r−1 P
easily from [7, Ex. III.12.4] and injectivity by restricting to a fibre.
Alternatively, the lemma follows from the explicit descriptions given below. The com-
plement of the union of all Cr is a cartesian power of C (proposition 2) and Cr are all
s s
irreducible, therefore they generate A1DrPn. On the other hand, by the proof of step 2 of
k
proposition 7, they are linearly independent. So they form a basis for A1DrPn.
k
DEGREE OF THE DIVISOR OF SOLUTIONS OF A DIFFERENTIAL EQUATION 5
Proposition 2. The given trivialisation (2) of U0 induces trivialisations
Ur ∼= Ur−1×Hom((Ck)⊗r,Cn−k)
of Ur withcartesian powers of C. Thecoordinates ofUr aregiven by(2) and(3), 1≤ s ≤ r.
Proof. Starting from the trivialisation of U0, we get an induced trivialisation (4) of U1.
Suppose we have already trivialisations Us = Us−1 × Vs with Vs ∼= Hom((Ck)⊗s,Cn−k),
which is isomorphic to a cartesian power of C, for 1 ≤ s ≤ r. Then (Σ ) ∼= (Σ ) ∼=
r Ur r−1 Ur
··· ∼= (Σ ) ∼= X ∼= Ur ×Ck. The Semple sequence on Ur is split by
1 Ur Ur
F ֒→ TUr ։ TUr/Ur−1
r
for all r, so that F = Σ ⊕TUr/Ur−1. Consequently,
r r
Ur+1 ∼= Hom((Σ ) ,TUr/Ur−1) = Ur ×Hom(Ck,Vr),
r Ur
i.e. Ur+1 ∼= Ur ×Vr+1 with Vr+1 ∼= Hom((Ck)⊗(r+1),Cn−k), as required.
As for naming the coordinates corresponding to this chart Ur of DrPn, these are given
k
by (2) for U0, yj for V1, and in general, assuming yj are coordinates for Vr, then
i i1,...,ir
the space Vr+1 ∼= Hom(Ck,Vr) is described by coordinates which we denote ∂yij1,...,ir, i.e.
∂xir+1
yj .
i1,...,ir,ir+1
For later use, there is a zero section 0 : Ur−1 → Ur of b : Ur → Ur−1 and composing,
r r
a zero section 0 : Us → Ur of b , for all s ≤ r. Now we are in the position of defining
s,r r,s
differential equations on Pn.
Definition 3. A differential equation on Pn relative to a reference x ,y is a non-zero
i j
algebraic equation
f(x ,y ,yj,yj ,... ,yj ) = 0,
i j i i1,i2 i1,...,ir
on Ur, for some r ≥ 1.
Let Ur(f) be the divisor of Ur defined by the differential equation f, and let H =
f
DrPn(f) denote its closure in DrPn, i.e. the data satisfying the differential equation. In
k k
this way, any differential equation f gives a hypersurface H in DrPn not containing any of
f k
Cr,... ,Cr. Conversely, given a hypersurface H in DrPn not containing any of Cr,... ,Cr,
2 r k 2 r
we can find a suitable reference (2) such that H does not contain Cr and Cr, thus being
0 1
defined as DrPn(f) for a suitable polynomial f on Ur. So we can give a more intrinsic
k
definition
Definition 4. A differential equation on Pn is a hypersurface H ⊆ DrPn not containing
k
any of Cr,... ,Cr.
2 r
This second viewpoint allows us to forget coordinates. Nonetheless, in practice, we
need to work with coordinates. Whenever we say that f is a differential equation on the
projective space, we shall understand that there is a projective reference implicit. There
are two caveats. First, if we change the reference, the polynomial f will change in general.
6 VICENTE MUN˜OZ AND IGNACIO SOLS
Second, there might be some forbidden references that we cannot choose (namely, those in
which H contains either of Cr, Cr).
f 0 1
Remark 5. Theformalpartialderivativesymbolswehavedefinedarenon-commuting. This
simply means that, for instance, yj and yj (i 6= i ) are independent coordinates. We
i1,i2 i2,i1 1 2
can solve this difficulty by restricting to the subset of Ur given by the equations yj =
i1,...,is
yj , 1 ≤ j ≤ n−k, 1 ≤ s ≤ r, σ any permutation of indices, and considering the
clσo(si1u),r..e.,σo(fiss)uch subset in DrPn.
k
More intrinsically, this subvariety of symmetric data SrZ ⊆ DrZ has been extracted
k k
in [5], for any smooth scheme Z. In our case, dealing with DrZ is simpler and will suffice.
k
Associated to a differential equation we have naturally defined enumerative invariants,
thanks to lemma 1.
Definition 6. For a differential equation f on Pn, define γf,... ,γf ∈ Z by
0 r
[DrPn(f)]= γfCr +···+γfCr
k 0 0 r r
in A1DrPn.
k
It is important to note that γf do not depend on the trivialisation, as the divisor classes
s
of Cr and Cr are independent of the trivialisation as well. The following proposition allows
0 1
us to compute some of γf in the general case.
s
Proposition 7. Let f be a differential equation on Pn relative to a reference x ,y .
i j
• Suppose f is distinguished in the variable y1 (i.e. for d = deg(f) the monomial
1,(.r..),1
(y1 )d appears in f with nonzero coefficient), then the leading γf is
1,(.r..),1 r
γf = degf(0,... ,y1 ,... ,0).
r 1,(.r..),1
• Suppose f is distinguished in the variable y1, then
1
γf = degf(0,... ,y1,... ,0).
1 1
• Suppose f is distinguished in the variable x , then
1
γf = degf(x ,... ,0).
0 1
Proof. Step 1. We need first to exhibit a suitable basis cr,... ,cr,... ,cr of A DrPn (we
0 s r 1 k
name equally closed subsets, their associated cycles, and their rational classes). These will
be the closure in DrPn of the one dimensional subschemes c◦r ⊆ Ur defined by the vanishing
k s
of all coordinates but y1 (if s = 0, this is the coordinate x ). That they form a basis
1,(.s..),1 1
will be proved in Step 2.
Togive anintrinsicdescriptionofc◦r,wedefinethesubspaceCk−1 ⊆ Ck inthegiven affine
s
plane Cn ⊆ U0 as the space corresponding to coordinates x ,... ,x , and the rank (k−1)-
2 k
subbundle X˜ of the bundle X on this plane by X˜(P) = P +Ck−1, and correspondingly the
trivial rank (k−1)-subbundle (Σ˜ ) ∼= X˜ of (Σ ) ∼= X . Analogously, if C ⊆ Cn−k is
r Ur Ur r Ur Ur
the y -axis, we define the rank 1 subbundle Y˜ of Y by Y˜(P) = P +C.
1
DEGREE OF THE DIVISOR OF SOLUTIONS OF A DIFFERENTIAL EQUATION 7
Furthermore, we define inductively U˜r = U˜r−1 × C inside Ur = Ur−1 × Vr by taking
U˜0 = U0,
U˜1 = Hom(X/X˜,Y˜) ⊆ Hom(X,Y) = U1,
and assuming that U˜r is already defined, by taking
U˜r+1 = Hom((Σ ) /(Σ˜ ) ,TU˜r/Ur−1) = Hom(Ur ×C,U˜r ×C)= U˜r ×C
r Ur r Ur
inside Ur+1 = Hom((Σ ) ,TUr/Ur−1).
r Ur
Let 0 be the origin of Cn, and let 0 = 0 (0) be the origin of Us−1. Note that
s−1 0,s−1
0 ∈ U˜ . Define c◦0 = 0 + C ⊆ U0, where C ⊆ Ck is the x -axis. For s ≥ 1, let
s−1 s−1 1
c◦s ⊆ Us be the 1-dimensional subvariety U˜s ∩b−1(0 ) and cs the closure of c◦s in DsPn.
s s−1 k
◦ ◦ ◦ ◦
For 0 ≤ s ≤ r, the above cr ⊆ Ur is just 0 (cs), thus b (cr)= cs and b (cr)= cs.
s s,r r,s s r,s s
Step 2. Now we want to relate the elements cr,... ,cr,... ,cr of A DrPn with the
0 s r 1 k
elements Cr,... ,Cr,... ,Cr ofA1DrPn. Onehasc◦r ∼= Candthereforecr∩(Cr∪...∪Cr)=
0 s r k s s 0 r
cr\Ur = cr\c◦r ∼= P\C consists, set-theoretically, of just one point, which we want now to
s s s
show not to be in any Cr for t < s and that this point is in fact the schematic intersection
t
cr∩Cr, i.e. that cr ·Cr = 1, as rational classes. On the one hand, this gives an alternative
s s s s
proof that the Cr are linearly independent in lemma 1. On the other hand, this proves that
s
cr are a basis for A DrPn, since the intersection matrix (Cr ·cr) is lower-triangular.
s 1 k t s
Let 0 ≤ t < s ≤ r. Since Cr = b−1(Ct), in order to show that cr ∩Cr = ø it is enough
t r,t s t
◦
to show that b (cr)∩Ct is empty. By construction, b (cr) = 0 ∈ Ut, since t < s, thus
r,t s r,t s
b (cr) = 0 is disjoint with Ct. This proves that, as rational classes, cr ·Cr = 0.
r,t s s t
Again, since Cr = b−1(Cs), in order to show that cr · Cr = 1, it is enough to show
s r,s s s
that b (cr)·Cs = cs ·Cs = 1. For s = 0 this is true since c0 ⊆ Pn is a line and C0 is
r,s s
the hyperplane. For s ≥ 1, we note that since cs ⊆ b−1(0 ), it is enough to show for
s s−1
Cs = Cs∩b−1(0 ) that
0s−1 s s−1
cs·Cs = 1
0s−1
in the Grassmannian b−1(0 ). Suppose now that s ≥ 2 (the case s = 1 is similar and
s s−1
is left to the reader). Then b−1(0 ) = Gr ((F ) ), where (F ) is the Semple
s s−1 k s−1 0s−1 s−1 0s−1
vector space
(F ) = (TUs−1/Us−2) ⊕(Σ ) .
s−1 0s−1 0s−1 s−1 0s−1
In this Grassmannian, Cs is the Schubert cycle of k-spaces meeting (TUs−1/Us−2) ,
0s−1 0s−1
i.e. the base of its Picard group A1. On the other hand, the 1-dimensional subvariety
c◦s = Hom((Σ ) /(Σ˜ ) ,(TU˜s−1/Us−2) )
s−1 0s−1 s−1 0s−1 0s−1
of the open subset Hom((Σ ) ,(TUs−1/Us−2) ) of Gr ((F ) ) is an open subset
s−1 0s−1 0s−1 k s−1 0s−1
oftheGrassmannianofk-subspacesofthe(k+1)-subspace(TU˜s−1/Us−2) ⊕(Σ ) ⊆
0s−1 s−1 0s−1
(F ) which contain the (k−1)-subspace (Σ˜ ) ⊆ (Σ ) ⊆ (F ) . Thus
s−1 0s−1 s−1 0s−1 s−1 0s−1 s−1 0s−1
cs is the Grassmannian of such k-spaces, i.e. the Schubert cycle base of A dual to the base
1
Cs of A1, thus cs·Cs = 1.
0s−1 0s−1
8 VICENTE MUN˜OZ AND IGNACIO SOLS
Step 3. Now we can find the coefficient γf in the expression of definition 6. By the
r
above, this is
γf = [DrPn(f)]·cr =length Ur(f)∩c◦r = degf(0,... ,y1 ,... ,0).
r k r r 1,(.r..),1
Indeed, the second equality is due to
(DrPn(f)\Ur(f))∩cr = DrPn(f)∩(Cr ∪···∪Cr)∩cr = ø
k r k 0 r r
since (Cr ∪ ··· ∪ Cr) ∩ cr is schematically one point, say the point (0,... ,∞,... ,0) at
0 r r
infinity of c◦r = {(0,... ,y1 ,... ,0)} ∼= C, and this point is not in DrPn(f), i.e.
r 1,(.r..),1 k
0 6= f(0,... ,∞,... ,0) = lim f(0,... ,y1 ,... ,0),
y1 →∞ 1,(.r..),1
1...1
because f is distinguished in the variable y1 . This finishes the proof of the first item in
1,(.r..),1
the statement of the proposition.
Step 4. To get the coefficient γf, we prove first that cr ·Cr = 0 for s > 1. For this we
1 1 s
consider the (flat) family of subvarieties {S } of Pn such that for λ ∈ C, S is given by
λ λ∈P1 λ
the equations y −λx = 0, y = 0, ..., y = 0, and S has equations x = 0, y = 0,
1 1 2 n−k ∞ 1 2
..., y = 0. Clearly the points of c◦r are parametrized byDrS ∩b−1(0), λ ∈C. Therefore
n−k 1 k λ r,0
cr\c◦r is the point DrS ∩b−1(0). But S is smooth, so DrS is disjoint from Cr = 0 for
1 1 k ∞ r,0 ∞ k ∞ s
s > 1 (see remark 11). This proves that cr ·Cr = 0 for s > 1.
1 s
Now γf = [DrPn(f)]·cr = degf(0,... ,y1,... ,0), whenf is distinguished in thevariable
1 k 1 1
y1, arguing as in step 3.
1
Step 5. To get the coefficient γf, we need to prove first that cr · Cr = 0 for s > 0.
0 0 s
This time we fix the subvariety S with equations y = 0, y = 0, ..., y = 0. Recall
1 2 n−k
c◦0 = 0+C ⊆ U0, where C ⊆ Ck is the x -axis and c0 is its closure in Pn. The points of cr
1 0
are parametrized by DrS∩b−1(λ), λ ∈ c0 ⊆ Pn. Now DrS is disjoint from Cr, for s ≥ 1, so
k r,0 k s
cr ·Cr = 0 for s> 0. The rest of the argument is as in steps 3 and 4.
0 s
Remark 8. Theexplicit computation of the other γf, 2≤ s < r, requires the full knowledge
s
oftheintersection matrixcr·Cr,whichisamoredelicate issue(seeremark14). Nonetheless
s t
if we are only interested in applications of differential equations on smooth varieties, γf and
0
γf will suffice (see corollary 15).
1
Remark 9. Suppose that f(x ,y ,yj,yj ,··· ,yj ) is a differential equation on Pn rel-
i j i i1,i2 i1,...,ir
ative to a reference x ,y with d= deg(f). Suppose that
i j
degf(0,... ,0,y1 ,... ,yn−k )= d.
1,(.r..),1 k,(.r..),k
Then γf = d. The proof is similar to steps 2 and 3 in the proof of proposition 7. Choose a
r
◦ ◦
generic line l ⊆ Vr and identify it with 0 × l ⊆ Ur = Ur−1 ×Vr. Let l ⊆ DrPn be its
r−1 k
closure. Then it is easy to see that l·Cr = 0 for s < r and l·Cr = 1. Now
s r
γf = [DrPn(f)]·l = degf(0,... ,0,y1 ,... ,yn−k ) = d,
r k 1,(.r..),1 k,(.r..),k
as in step 3 in the proof of proposition 7.
DEGREE OF THE DIVISOR OF SOLUTIONS OF A DIFFERENTIAL EQUATION 9
3. Degree of the divisor of solutions
Let S be a subvariety of Pn of dimension k. We have defined DrS ⊆ DrPn as the closure
k k
of DrS ⊆ DrPn. There is a natural map
k reg k
DrS → S,
k
which is an isomorphism over the smooth part. Therefore if S is smooth then S ∼= DrS. In
k
general, we have the following invariants associated to S
Definition 10. The cuspidal numbers γs, s ≥ 0, of S are defined as
S
• γ0 is the degree of S ⊆ Pn.
S
• γ1 is the class of S, i.e. the degree of the divisor of S consisting of the points of
S
tangency of tangent k-planesto S thatcanbedrawnfromageneric (n−k−1)-plane
of Pn.
• For s ≥ 2, γs is thedegreeof thepush-forwardb (CsS) ⊆ Pn of CsS = DsS∩Cs ⊆
S s,0 k
DsPn under b :DsPn → Pn.
k s,0 k
Remark 11. As remarked in [5], if S is smooth then γs = 0 for s ≥ 2. Indeed, in this case,
S
DrS is disjoint from the cuspidal divisors Cr, s ≥ 2. Moreover if S has singularities only
k s
in codimensions 2 or more (e.g. S normal), the cycle b (CsS) ⊆ Pn, s ≥ 2, has dimension
s,0
less or equal than k−2 and hence γs = 0 for s ≥ 2. If S has no cuspidal singularities (for
S
instance, if the only singularities are normal double crossings along a smooth subvariety)
thenγs = 0fors ≥ 2aswell. Thecuspidalnumbersmeasureinsomesensehowcomplicated
S
the singularities (on codimension 1) of S are.
Lemma 12. For any 0 ≤ s ≤ r, it is γs = DrS ∩Cr ∩Hk−1, where H = Cr = b−1(H) is
S k s 0 r,0
the hyperplane in DrPn.
k
Proof. Using that DrS ∩Cr ∩b−1(Hk−1) = DsS ∩Cs ∩b−1(Hk−1), we reduce to the case
k s r,0 k s,0
r = s. Now for r = 0 is obvious, for r = 1 follows from the definition, and for r ≥ 2,
DrS ∩Cr ∩b−1(Hk−1)= CrS ·b−1(Hk−1) = b (CrS)·Hk−1 = γr.
k r,0 r,0 r,0 S
Given a differential equation f on Pn, we define the divisor of solutions of f on S as
follows. First, DrS(f) is the schematic intersection in DrPn
k k
DrS(f)= DrS ∩DrPn(f).
k k k
Unless DrS(f) = DrS, we have that DrS(f) is a divisor in DrS. The divisor of solutions
k k k k
S(f) ⊂ S of f on S is the push-forward of DrS(f) under b : DrPn → Pn, i.e. S(f) =
k r,0 k
b (DrS(f)).
r,0 k
Theorem 13. Let f be a differential equation on Pn with γf, 0 ≤ s ≤ r. Let S be a
s
subvariety of Pn of dimension k. Suppose that S(f) is a proper subset of S. Then the
degree of S(f) is
γfγ0 +···+γfγs +···+γfγr.
0 S s S r S
10 VICENTE MUN˜OZ AND IGNACIO SOLS
Proof. The numbers γf are defined by the condition
s
[DrPn(f)]= γfCr +···+γfCr +···+γfCr
k 0 0 s s r r
in A1DrPn. Now the degree of S(f) is S(f)·Hk−1 i.e.
k
DrS(f)·b−1(Hk−1) =DrS ∩DrPn(f)∩Hk−1 = [DrS ∩Hk−1]·[DrPn(f)].
k r,0 k k k k
Lemma12saysthatthesth-cuspidaldegreeofS isgivenasγs = [DrS∩Hk−1]·Cr. Therefore
S k s
we have
degS(f) = γfγ0 +···+γfγs +···+γfγr.
0 S s S r S
Remark 14. Theorem 13 can be used for computing γf for a given differential equation
s
f, by using a standard set of subvarieties whose cuspidal numbers are known (or easily
obtainable). This method is used in the examples of section 4.
In the smooth case we get the following
Corollary 15. Letf beadifferentialequationonPn andletS beasmooth(orjustnormal)
subvariety of Pn of dimension k. Suppose that S(f) is a proper subset of S. Then
degS(f) =γfγ0 +γfγ1.
0 S 1 S
Let us work out the values of γ0 and γ1 for a smooth S. In the case k = 1, S ⊆ Pn is a
S S
smooth curve. If d is its degree and g is genus, then γ0 = d and γ1 = 2g −2+2d. Note
S S
that when n = 2, i.e. S ⊆ P2 is a smooth plane curve, the class of S is γ1 = d(d−1), which
S
can be obtained by using the adjunction formula 2g−2= d(d−3).
For k > 1, let S ⊆ Pn be a smooth (or just normal) subvariety of degree d and let g be
the genus of the generic section C = S∩Hk−1 (which is a smooth curve). Then the degree
of C is d and its class is γ1 = 2g−2+2d. Now it is easy to see that the class of S equals
C
that of C, γ1 = γ1 (for instance take a reference x ,y such that the Hk−1 has equations
S C i j
y = 0,... ,y = 0,x =0 and use lemma 12).
1 n−k 1
Corollary 16. Let f be a differential equation on Pn and let S ⊆ Pn be a smooth (or just
normal) subvariety of degree d whose generic section S ∩Hk−1 has genus g. Suppose that
S(f) is a proper subset of S. Then degS(f) = γf d+γf (2g−2+2d). Furthermore, if S is
0 1
a hypersurface, then the formula is reduced to degS(f) = γf d+γf d(d−1).
0 1
4. Examples
Theorem 13 can be used in two directions, either to compute the degree of the divisor of
solutions of a differential equations on a subvariety of Pn or to extract information about a
differential equation f and a subvariety S ⊆ Pn once we have computed the degree of the
divisor of solutions of f on S.
Computationof deg S(f). Givenadifferentialequationf andak-dimensionalsubvariety
S ⊆ Pn, to find the divisor S(f) we proceed as follows. Take a reference x ,y . When S is
i j
smooth we find explicitly Ur(f)∩DrS by considering the ideal generated by the equations
k