Annales de la Faculté des Sciences de Toulouse

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Vik.S. Kulikov

ANNALES
DE LA FACULTÉ
DES SCIENCES
Mathématiques
VIK.S. KULIKOV
A Remark on Classical Pluecker’s formulae
Tome XXV, no 5 (2016), p. 959-967.
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Annales de la Faculté des Sciences de Toulouse Vol. XXV, n◦ 5, 2016
pp. 959-967
A Remark on Classical Pluecker’s formulae
Vik.S. Kulikov(1)
RÉSUMÉ. — Pour toute courbe réduite C ⊂ P2, on introduit la notion de
nombre des points de rebroussement (cusps) virtuels cv et celle de nombre
des points doubles ordinaires (nodes) virtuels nv . Ces deux nombres sont
positifs ou nuls et ils cöıncident avec le nombre des points singuliers du
type respectif lorsque ce sont les seules singularités de la courbe. De plus, si
Ĉ est la courbe duale d’une courbe irréducible C, et si n̂v et ĉv designent le
nombre de singularités virtuelles de Ĉ du type respectif, alors les nombres
entiers cv , nv , ĉv , n̂v vérifient les formules de Plücker classiques.
ABSTRACT. — For any reduced curve C ⊂ P2, we introduce the notions
of the number of its virtual cusps cv and the number of its virtual nodes
nv . We prove that the numbers cv and nv are non-negative and if C is a
curve with only ordinary cusps and nodes as its singular points, then cv
is the number of its ordinary cusps and nv is the number of its ordinary
nodes. In addition, if Ĉ is the dual curve of an irreducible curve C and
n̂v and ĉv are the numbers of its virtual nodes and virtual cusps, then
the integers cv , nv , ĉv , n̂v satisfy classical Plücker’s formulae.
Introduction
Let C ⊂ P2 be a reduced curve defined over the field of complex numbers
C. A curve C is called cuspidal if the singular points of C are only the
ordinary cusps and nodes.
In modern textbooks on algebraic geometry, classical Plücker’s formulae
are stated as follows (see, for example, [1], [3]).
(∗) Reçu le 08/07/2015, accepté le 26/10/2015
(1) Steklov Mathematical Institute
kulikov@mi.ras.ru
This research was partially supported by grants of RFBR 14-01-00160 and 15-01-02158.
Article proposé par Stepan Orevkov.
– 959 –
Vik.S. Kulikov
Classical Plücker’s formulae. — Let C ⊂ P2 be an irreducible cusp-
idal curve of genus g, degree d  2, having c ordinary cusps and n nodes
Assume that the dual curve Ĉ of C is also a cuspidal curve. Then
d̂ = d(d− 1)− 3c− 2n; (0.1)
g =
(d− 1)(d− 2)
2
− c− n; (0.2)
d = d̂(d̂− 1)− 3ĉ− 2n̂; (0.3)
g =
(d̂− 1)(d̂− 2)
2
− ĉ− n̂, (0.4)
where ĉ and n̂ are the numbers of ordinary cusps and nodes of Ĉ and d̂ =
deg Ĉ.
Denote by V (d, c, n) ⊂ P d(d+3)2 the variety parametrizing the irreducible
cuspidal curves of degree d with c ordinary cusps and n nodes. Very often,
if for given d, c, and n one of the invariants ĉ or n̂, obtained as the solution
of (0.1) – (0.4), is negative, then it is claimed that this is sufficient for the
”proof” of the emptiness of V (d, c, n). However, the correctness of the fol-
lowing statement is unknown: ”the dual curve Ĉ of a curve C corresponding
to a generic point of V (d, c, n) is cuspidal”. Therefore, in general case, it
is impossible to conclude the non-existence of cuspidal curve C if ĉ or n̂ is
negative. Of course, to avoid this problem, one can use generalized Plücker’s
formulae including the numbers of all possible types of singular points of
Ĉ. But, we again have a difficulty, namely, in this case we must take into
account too many unknown variables.
To obviate the arising difficulty, in Section 1 for any reduced plane curve
C we define the notions of the number of its virtual cusps cv and the number
of its virtual nodes nv which are non-negative, coincide respectively with the
numbers of ordinary cusps and nodes in the case of cuspidal curves, and if
the dual curve Ĉ of an irreducible curve C has n̂v virtual nodes and ĉv
virtual cusps, then the integers cv, nv, ĉv, and n̂v satisfy Classical Plücker’s
formulae.
In Section 2, we investigate the behaviour of the Hessian curve HC of a
cuspidal curve C at cusps and nodes of C, and in Section 3, we generalize to
the case of arbitrary irreducible plane curve the inequalities for the numbers
of cusps and nodes of plane cuspidal curves of degree d which was obtained
early in [5] under additional assumption that the dual curve of a generic
cuspidal curve is also cuspidal.
– 960 –
A Remark on Classical Pluecker’s formulae
1. The numbers of virtual cusps and nodes
Let (C, p) ⊂ (P2, p) be a germ of a reduced plane singularity. It splits
into several irreducible germs: (C, p) = (C1, p)∪ . . .∪ (Ck, p). Denote by mj
the multiplicity of the singularity (Cj , p) at the point p and let δp be the
δ-invariant of the singularity (C, p). By definition, the integers
cv,p :=
k∑
i=1
(mi − 1)
and
nv,p := δp −
k∑
i=1
(mi − 1)
are called respectively the numbers of virtual cusps and virtual nodes of the
singularity (C, p). Note that in [2] it was shown that any reduced singular
curve germ can be deformed into a germ with exactly cv,p cusps and nv,p
nodes.
We have δp = cv,p + nv,p.
Lemma 1.1. — Let (C, p) ⊂ (P2, p) be a germ of a reduced plane singu-
larity, cv,p be the number of its virtual cusps and nv,p be the number of its
virtual nodes. Then
(i) cv,p  0, nv,p  0;
(ii) if (C, p) is an ordinary cusp, then cv,p = 1 and nv,p = 0;
(iii) if (C, p) is an ordinary node, then cv,p = 0 and nv,p = 1.
Proof. — We prove only the inequality nv  0, since all the other claims
of Lemma 1.1 are obvious. Let (C, p) = (C1, p)∪ . . .∪ (Ck, p) and mi be the
multiplicity of its irreducible branch (Ci, p). Then the multiplicity of (C, p)
at p is equal to mp =
k∑
i=1
mi and we have
nv,p = δp −
k∑
i=1
(mi − 1)  δp −
k∑
i=1
mi + 1 =
δp − (mp − 1)  δp −
mp(mp − 1)
2
 0,
since mp  2 for singular points and δp 
mp(mp − 1)
2
. Therefore, we have
nv =
∑
p∈SingC
nv,p  0.
– 961 –
Vik.S. Kulikov
Let C ⊂ P2 be a reduced curve. Denote by SingC the set of its singular
points. By definition, we put
cv :=
∑
p∈SingC
cv,p,
nv :=
∑
p∈SingC
nv,p
and call these integers respectively the number of virtual cusps and the
number virtual nodes of the curve C. If C is an irreducible curve of degree
d and geometric genus g, then we have g =
(d− 1)(d− 2)
2
− δC , where
δC =
∑
p∈SingC
δp is the δ-invariant of C. Therefore, we have
g =
(d− 1)(d− 2)
2
− cv − nv. (1.5)
The following proposition is a corollary of Lemma 1.1.
Proposition 1.2. — Let cv be the number of virtual cusps and nv be
the number of virtual nodes of a reduced curve C ⊂ P2. We have
(i) cv  0 and nv  0,
(ii) if C is a cuspidal curve, then cv and nv are equal respectively to the
number c of cusps and the number n of nodes of C.
Theorem 1.3. — (Plücker’s formulae). Let C and Ĉ be irreducible dual
curves of genus g, degC = d  2, deg Ĉ = d̂, and cv, nv, ĉv, n̂v are the
numbers of their virtual cusps and nodes, respectively. Then we have the
following equalities:
d̂ = d(d− 1)− 3cv − 2nv; (1.6)
2g = (d− 1)(d− 2)− 2cv − 2nv; (1.7)
d = d̂(d̂− 1)− 3ĉv − 2n̂v; (1.8)
2g = (d̂− 1)(d̂− 2)− 2ĉv − 2n̂v. (1.9)
Proof. — To prove Plücker’s formulae, we need the following
– 962 –
A Remark on Classical Pluecker’s formulae
Lemma 1.4. — For an irreducible plane curve C we have
d̂ = 2d+ 2(g − 1)− cv, (1.10)
ĉv = 3d+ 6(g − 1)− 2cv, (1.11)
d = 2d̂+ 2(g − 1)− ĉv, (1.12)
cv = 3d̂+ 6(g − 1)− 2ĉv. (1.13)
Proof. — Denote by ν : C → C and ν̂ : C → Ĉ the normalization
morphisms, consider generic (with respect to C and Ĉ) linear projections
pr : P2 → P1 and p̂r : P̂2 → P1, and put π = pr ◦ ν and π̂ = p̂r ◦ ν̂. We have
deg π = d and deg π̂ = d̂.
Let ν−1(xi) = {yi,1, . . . , yi,mi} for xi ∈ SingC. For each point yi,j denote
by ri,j the ramification index of π at yi,j . It is easy to see that ri,j coincides
with the multiplicity mi,j at xi of the irreducible germ (Ci,j , xi) ⊂ (C, xi)
corresponding to the point yi,j . Therefore, we have
cv =
∑
i,j
(ri,j − 1).
Applying Hurwitz formula to π and π̂, we obtain
2(g − 1) = −2d+ cv + d̂ (1.14)
and
2(g − 1) = −2d̂+ ĉv + d (1.15)
which give formulae (1.10) and (1.12).
To prove (1.11), note that ĉv = 2d̂+ 2(g − 1)− d by (1.12). Therefore
ĉv = 2(2d+ 2(g − 1)− cv) + 2(g − 1)− d
by (1.10), that is, ĉv = 3d + 6(g − 1) − 2cv. Formula (1.13) is obtained
similarly. 
It follows from (1.5) that
2(g− 1)+2cv +2nv = d(d− 3), 2(g− 1)+2ĉv +2n̂v = d̂(d̂− 3) (1.16)
which are equivalent to (1.7) and (1.9). To complete the proof of Plücker’s
formulae, notice that formulae (1.6) and (1.8) easily follow from equations
(1.10) – (1.13) and (1.16). 
– 963 –
Vik.S. Kulikov
2. On the Hessian curve of a cuspidal curve
Let C ⊂ P2 be an irreducible curve of degree d with cv virtual cusps and
nv virtual nodes. It follows from (1.7) and (1.11) that
8cv + 6nv + ĉv = 3d(d− 2). (2.1)
If C is a cuspidal curve then quality (2.1) has a natural geometric mean-
ing. To explain it, let the curve C is given by equation F (x0, x1, x2) = 0,
where x0, x1, x2 are homogeneous coordinates in P2. Consider the Hessian
curve HC ⊂ P2 of the curve C. It is given by equation det( ∂
2F
∂xi∂xj
) = 0.
We have degHC = 3(d−2). Therefore the intersection number (C,HC)P2 is
equal to 3d(d−2). On the other hand, it is well-known (see, for example, [1])
that the curves C and HC meet at the singular points and at the inflection
points of the curve C. Therefore we have
Σ′(C,HC)p + Σ
′′(C,HC)p + Σ
′′′(C,HC)p = (C,HC)P2 = 3d(d− 2), (2.2)
where (C,HC)p is the intersection number of the curves C and HC at a
point p ∈ C and the sum ∑′ is taken over all cusps of C, the sum ∑′′ is
taken over all nodes of C, and the sum
∑′′′
is taken over all inflection points
of C.
Let us show that the coefficients involving in equation (2.1) have the
following geometric meaning: equality (2.1) is the same as equality (2.2),
that is, the coefficient 8 in (2.1) is the intersection number (C,HC)p at a
cusp p ∈ C, the coefficient 6 is the intersection number (C,HC)p at a node
p ∈ C, and ĉv =
∑′′′
(C,HC)p. Note that the following computations are
classical (see, for example, [6]), but we give them here briefly.
Let p be a cusp of C. Without loss of generality, we can assume that
p = (0, 0, 1) and
F (x0, x1, x2) = x
2
0U(x0, x1, x2) + x0x
2
1V (x0, x1, x2) + x
3
1W (x0, x1, x2),
where U is a homogeneous polynomial of degree d−2 such that U(0, 0, 1) = 1
and V and W are homogeneous polynomials of degree d − 3 such that
W (0, 0, 1) = 1. Put a = V (0, 0, 1), then in non-homogeneous coordinates
x = x0x2 , y =
x1
x2
we have p = (0, 0), the curve C is given by equation of the
form
x2 + y3 + axy2 + bx2y + cx3 + terms of higher degree = 0,
and the curve HC is given by equation of the form
x2(6y + 2ax) + terms of higher degree = 0.
– 964 –
A Remark on Classical Pluecker’s formulae
Easy computation (applying σ-process with center at p) gives the following
inequality:
(C,HC)p  8 (2.3)
if p is a cusp of C.
Let p be a node of C. Again, without loss of generality, we can assume
that p = (0, 0, 1) and
F (x0, x1, x2) = x0x1U(x0, x1, x2) + V (x0, x1)W (x0, x1, x2),
where U is a homogeneous polynomial of degree d−2 such that U(0, 0, 1) =
1, V is a homogeneous polynomial of degree 3, and W is a homogeneous
polynomial of degree d−3. In non-homogeneous coordinates x = x0x2 , y =
x1
x2
we have p = (0, 0), the curve C is given by equation of the form
xy + terms of higher degree = 0,
and the curve HC is given by equation of the same form
xy + terms of higher degree = 0.
Easy computation (applying σ-process with center at p) gives the following
inequality:
(C,HC)p  6 (2.4)
if p is a node of C.
If p is an r-tuple inflection point of C (that is, (C,Lp)p = r + 2, where
the line Lp is tangent to C at p), then by Theorem 1 on page 289 in [1], we
have (C,HC)p = r. On the other hand, the branch (Ĉi, p̂) of the dual curve
Ĉ, corresponding to an irreducible branch (Ci, p) ⊂ (C, p) at a point p of a
cuspidal curve C, is singular if and only if p is an inflection point of C; and
the branch (Ĉ, p̂), corresponding to the branch (C, p) at r-tuple inflection
point p ∈ C, has a singularity of type ur+1− vr+2 = 0. The multiplicity mp̂
of this singularity is equal to r + 1. Therefore, we have
Σ′′′(C,HC)p =
∑
(Ĉ,p̂)
(mp̂ − 1) = ĉv. (2.5)
Finally, it follows from (2.1) – (2.5) that inequalities (2.3) and (2.4) are
the equalities in the case of cuspidal curves.
– 965 –
Vik.S. Kulikov
3. Lefschetz’s inequalities
In [5], assuming that for a generic cuspidal curve with given numerical
invariants the dual curve is also cuspidal, Lefschetz proved the following
inequalities
c 
3
2
d+ 3(g − 1) (3.1)
if d is even and
c 
3d− 1
2
+ 3(g − 1) (3.2)
if d is odd. It follows from (1.11) that these inequalities occur for any ir-
reducible plane curve, since ĉv is a non-negative integer. Namely, for any
irreducible plane curve we have
cv 
3
2
d+ 3(g − 1) (3.3)
if d is even and
cv 
3d− 1
2
+ 3(g − 1) (3.4)
if d is odd, and equality (2.1) gives rise also to the following inequality:
8cv + 6nv  3d(d− 2)−
1− (−1)d
2
. (3.5)
Remark 3.1. — One can show that for any d = 2k, k  3, and for any
g  0 such that 2  3g  k − 4 or g  1, there exist a cuspidal curve of
degree d having c = 3(k + g − 1) cusps and n = 2(k − 1)(k − 2)− 4g nodes
for which inequality (3.5) becomes the equality. If d = 2k + 1, k  3, then
for any g such that 2  3g  k − 4 or g  1, there exist a cuspidal curve of
degree d having c = 3(k + g) − 2 cusps, n = 2(k − 1)2 − 4g nodes, and for
which inequality (3.5) becomes the equality. The proof of these statements
follows from the fact that the genus of such curves C is equal to g and for
these curves the dual curves Ĉ have degree d̂ = 2(g − 1) + 7 + 1−(−1)
d
2 and
the number of virtual cusps ĉv =
1−(−1)d
2 . Therefore in the case of even d
(resp., odd d) such curves can be obtained as the images of generic (resp., as
the images of almost generic, that is, belonging to a codimension one variety
in the space of linear projections) linear projections to P2 of a smooth curve
C ⊂ Pd̂−g of degree d̂ birationally isomorphic to C. Standard computations
of codimension of the locus of ”bad” projections (which we leave to the
reader) show that in these cases there are linear projections pr : Pd̂−g → P2
such that pr(C) = Ĉ are cuspidal curves with ĉ = 1−(−1)
d
2 and their dual
curves C are also cuspidal.
– 966 –
A Remark on Classical Pluecker’s formulae
For completeness, let me notice that there are also the following inequal-
ities (well-known in the case of cuspidal curves) which we have for any plane
irreducible curve:
3cv + 2nv < d(d− 1)−
√
d, (3.6)
2cv + 2nv  (d− 1)(d− 2), (3.7)
d(d− 2)(d2− 9) + (3cv + 2nv)2 + 27cv + 20nv  2d(d− 1)(3cv + 2nv) (3.8)
and which are consequences of equalities (1.6) – (1.9) and the inequalities
d̂ 
√
d, g  0, n̂v  0.
Also, let me mention that the following inequality ([4]):
16c+ 9n  d(5d− 6) (3.9)
holds for any (not necessary irreducible) cuspidal curve C of even degree d.
It follows from inequality (3.9) that
16c+ 9n  5d(d− 1)− 1 (3.10)
for any cuspidal curve C of odd degree d > 1. To show this, it suffices to
add a line L in general position with respect to C and apply inequality
(3.9) to C ∪L. But, up to now it is unknown whether inequality (3.9) (and
respectively (3.10)) holds for any reduced plane curve C of even degree
(respectively, of odd degree) if we substitute nv and cv in (3.9) instead of n
and c.
Bibliography
[1] Brieskorn (E.), Knörrer (H.). — Plane algebraic curves, Birkhäuser-Verlag,
Basel-Boston-Stuttgart, (1986).
[2] Diaz (D.) and Haris (J.). — Ideals associated to deformations of singular plane
curves, Transactions of Am. Math. Soc. 309, p. 433-468 (1988).
[3] Griffiths (P.) and Harris (J.). — Principles of algebraic geometry. A Wiley-
Interscience Publication, John Wiley & Sons, New York Chichester Brisbane
Toronto (1978).
[4] Hirzebruch (F.). — Singularities of algebraic surfaces and characteristic numbers.
Algebraic geometry, Proc. Lefschetz Centen. Conf., Mexico City/Mex. 1984, Part I,
Contemp. Math., 58, I, p. 141-155 (1986).
[5] Lefschetz (S.). — On the existence of loci with given singularities, Transactions of
Am. Math. Soc. 14, p. 23-41 (1913).
[6] Van der Waerden (B.L.). — Einführung in die algebraische Geometrie, Springer
(1939).
– 967 –


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