manuscripta math. 106, 101 – 116 (2001)
© Springer-Verlag 2001
Pietro De Poi
On first order congruences of lines of P4 with a
fundamental curve
Received: 2 August 2000 / Revised version: 11 July 2001
Abstract. We discuss projective families of lines of Pn , and in particular congruences of
order one. After giving general results, we obtain a complete classification of the case of P4
in which there is a fundamental curve.
1. Introduction
By definition a congruence of lines in Pn is a family of lines of dimension n − 1.
The order of a congruence is the number of lines passing through a general point
of Pn . Here we give first of all a general classification and general formulae, then
we are interested in classifying congruences of order one in P4 with a fundamental
curve.
The starting point of the study of the congruences of P4 is an Ascione’s work
[4], whose aim is to classify the projective surfaces of P4 with only one apparent
triple point, i.e. he wants to classify the general surfaces whose family of trisecant
lines generates a first order congruence of P4 . Unfortunately, his proof of this
classification had a gap, which was filled up by Severi in [17], who found a surface
which did not appear in Ascione’s classification. The congruences of lines in P4
are considered by G. Marletta whose point of view in [15] and [14] is different:
he is interested in classifying congruences of lines of P4 of order one. He studies
in particular the focal loci of congruences and classifies them according to the
dimension and number of the irreducible components of this locus. It has, in general,
dimension two and the case of an irreducible surface is the one of Ascione and
Severi.
This paper studies congruences in the spirit of Marletta’s work by a modern
point of view: we apply the technique of focal diagrams of a projective flat family
and the Schubert calculus to the congruences of lines of order one. From this, we
deduce a complete classification of congruences of P4 with a fundamental curve.
The article is divided in two parts: the first one, which comprehends the first
three sections, contains some general results about (first order) congruences, which
P. De Poi: Matematisk Institutt, Universitetet i Oslo, P.O.Box 1053, Blindern, N-0316 Oslo,
Norway and Dipartimento di Scienze Matematiche, Università degli Studî di Trieste, Via
Valerio, 12/b, 34127 Trieste, Italy. e-mail: [email protected]
Mathematics Subject Classification (2000): Primary 14J40, 14M15, 14N05; Secondary
51N35
102
P. De Poi
will be – hopefully – useful for further works on the subject; we apply these results
to the second one, in Sect. 5, to obtain the complete classification of the congruences
of P4 with a fundamental curve.
More precisely, in Sect. 2, we give the basic definitions and notations which
are used throughout the article. We note in particular that for us the focal locus will
always be considered with its natural scheme structure, in order to use formulae on
foci in their whole generality, taking account of the multiple structures. The section
ends with the proof of the known fact that on a general line of a congruence of Pn
there are n − 1 foci, in Proposition 1.
Sect. 3 treats general facts about congruences; the main original results in it are
the definition of d-loci (Definition 4) and their characterisation in Proposition 2.
We use these results in Sect. 4, which is about first order congruences: the d-loci
are the key ingredient of the Classification Theorem (Theorem 5), which gives a
classification of the first order congruences from the point of view of their focal
loci. Then, in Theorem 6, we investigate what can be the dimension of the focal
locus of a first order congruence. From this, we get the explicit classification of first
order congruences of P4 from the point of view of the splitting of the fundamental
d-loci, in Example 1. The section ends with the proof of the rationality of some
particular d-loci, in Theorem 7, from which we easily obtain the rationality of all
the reduced components of the focal locus in the case of P4 with a fundamental
curve.
Finally, in Sect. 5, we obtain the complete classification of the case of P4
with a fundamental curve. We will use for this case the following notations: the
fundamental 2-locus will be denoted by F1 and the fundamental 1-locus by C1 ;
moreover, C := (C1 )red , deg(F1 ) := m1 , and deg(C) := m2 . Finally, if C ⊂ F1 ,
we set c := length(F1 ∩ C).
After the classification of the proper intersections in Proposition 4 and all the
cases in which F is set-theoretically linear in Theorem 8, we prove the following:
Theorem 1. If the fundamental locus of a first order congruence of lines of P4 has,
as irreducible components, a fundamental 2-locus F1 and a fundamental 1-locus
C1 , then the congruence is given by the lines meeting both F1 and C; we have that
1. either C ⊂ F1 and:
(a) F is a plane and C is a rational curve (see Theorem 8 for more details); or
(b) C is a line and F1 is a rational surface with sectional (arithmetic) genus
m1 − 2; or
(c) C is a conic and F1 is a projection of a rational normal scroll of type
Sm1 −2k,2k , with m1 ≥ 3, one of its unisecant curves is C and a general
hyperplane through C intersects F1 in C with algebraic multiplicity k and
in a line;
2. or C ⊂ F1 , and we have:
(a) F1 is a plane and C is a rational curve such that c = m2 − 1; or
(b) F1 is a projection of a rational normal scroll and C is an (m1 − 1)-secant
line of it; or
(c) F1 is a (projected) Veronese surface and C is a trisecant line of it; or
On first order congruences of lines of P4 with a fundamental curve
103
(d) C is a rational plane curve with a point P of multiplicity m2 − 1 and F1 is a
cone with vertex in P and basis a rational curve, and the intersection of F1
with the plane of C is given by m1 − 1 lines (and so, c ≥ (m1 − 1)m2 ).
Vice versa the lines meeting one of the above surfaces F1 and curves C generate a
first order congruence.
If we suppose that C and F1 are smooth, we get a finite list of possibilities:
Theorem 2. If the fundamental locus of a first order congruence of lines of P4 has,
as irreducible reduced components, a smooth surface F1 and a smooth curve C,
then the congruence is given by the lines meeting both F1 and C; we have that
1. either C ⊂ F1 and:
(a) F1 is a plane and C is either a line or an irreducible conic curve (and the
congruence is constructed as in Theorem 8); or
(b) F1 is S1,2 and C is a unisecant conic;
(c) C is a line and F1 is a speciality one rational surface, i.e. we have the
following (effective) possibilities (see [2] for further details) 8 ≤ deg(F1 ) ≤
10 and “possibly1 ” deg(F1 ) = 11;
2. or C ⊂ F1 and:
(a) F1 is a plane and C a conic meeting in one point, i.e. c = 1; or
(b) F1 is S1,2 and C is a secant line of it, i.e. c = 2; or
(c) F1 is a (projected) Veronese surface and C is a trisecant line of it, i.e. c = 3.
Vice versa the lines meeting one of the above surfaces F1 and curves C generate a
first order congruence.
Besides, using results of [3], we get all the possible smooth congruences from
the list of Theorem 1 as subvarieties of the Grassmannian:
Theorem 3. The smooth congruences B of P4 with a fundamental curve are only
the following:
1. B is given by the lines meeting a plane F1 and a line C not intersecting, i.e. the
case of Proposition 4;
2. the congruence is given by the lines meeting a plane F1 and a conic C meeting
in a point (out of the point in common), i.e. we are in the case (1) of Theorem 1
with m1 = 1 and m2 = 2. For this congruence, we have that its class is two and
c = 1.
2. Notations and definitions
We will work with schemes and varieties over the complex field C. By variety we
mean a reduced and irreducible algebraic C-scheme.
1 possibly, since it is not known if speciality one rational surfaces of P4 of degree 11 do
exist
104
P. De Poi
We denote by G(1, n) the Grassmannian of lines of Pn . Besides, we will use
the notations of Griffiths and Harris’ book [11] for the Schubert cycles. We refer to
[7] for more general results and references about families of lines, focal diagrams
and congruences. We recall that
– A congruence of lines of Pn is a flat family (, B, p) of lines of Pn obtained
by the desingularization of a subscheme (which will be supposed, for simplicity,
irreducible) B of dimension n − 1 of the Grassmannian G(1, n). p is the restriction of the projection p1 : B ×Pn → B to , while we will denote the restriction
of p2 : B × Pn → P4 by f . b := p−1 (b), (b ∈ B) will be an element of the
family and f (b ) =: (b) is a line of Pn .
– A point y ∈ Pn is called fundamental if its fibre has dimension greater than the
dimension of the general one. The fundamental locus is the set of the fundamental
points.
– The subscheme of the foci of the first order V ⊂ is the scheme of the ramification points of f .
– The locus of the first order foci, or – simply – the focal locus F = f (V ) ⊂ Pn ,
is the set of the branch points of f . In this article, we will endow this locus with
the scheme structure given by considering it as the scheme-theoretic image of V
under f (see, for example, [12]).
– The fundamental locus is contained in the focal locus.
For fixing some notations useful in the paper, we recall the focal diagram of
[6], which is the commutative and exact diagram of coherent sheaves on :
0


λ
T(B×Pn /Pn )| −−−−→ N/B×Pn




=
0 −−−−→
T


df f ∗ TPn
−−−−→
TB×Pn |


−−−−→ N/B×Pn −−−−→ 0
p2∗ TPn |


0
– the row is the exact sequence defining the normal sheaf of in B × Pn and the
column is the exact sequence defining the relative tangent sheaf of B × Pn with
respect to Pn restricted to .
The homomorphism λ is defined by the commutativity of the diagram, while
df is the differential of the map f .
Definition 1. The map λ : T(B×Pn /Pn )| → N/B×Pn is called the global characteristic map for the family .
On first order congruences of lines of P4 with a fundamental curve
105
Remark 1. From the diagram (if we think of T(B×Pn /Pn )| and T as subsheaves of
TB×Pn | ), it is easy to see that
ker df = ker λ.
(1)
So, we can think of V as the degeneracy locus of λ.
If we restrict the map λ to a fibre b , we obtain a morphism called the characteristic map of the family relative to b:
λ(b) : TB,b ⊗ O(b) −−−−→

∼
=
n−1
O(b)
N(b)/Pn

∼
=
−−−−→ O(b) (1)n−1 .
From this description, we obtain
Proposition 1. On every fibre b of the family, (respectively, on every line of the
family (b)), the subscheme of the first order foci V (the locus of the first order
foci F ) either coincides with the whole line b (with the line (b)) or is a zero
dimensional scheme of b (of (b)) of length n − 1 (of length at least n − 1).
Proof. From the preceding isomorphisms, we obtain that the map λ(b) can be seen
as an (n − 1) × (n − 1)-matrix with linear entries on (b); so the scheme of foci
on b is given by the vanishing of the determinant of this matrix, and our claim
follows. Definition 2. If the line b (respectively, (b)) is contained in the scheme of the
first order foci V (in the locus of first order foci F ), it is called focal line.
Remark 2. In the preceding proposition we said that, if the intersection F ∩ (b)
is proper, it can have length greater than n − 1. This is due to the fact that (b)
can contain points which are focal for other lines, i.e. points which are images of
points contained in some V ∩ b , with b = b .
Moreover, a fundamental point P is a focal point for every line (b) which
contains it, since the characteristic map relative to it, λ(b), drops rank in P .
3. Congruences of lines in Pn
Since for the low dimensional projective spaces there is not much to say, we will
suppose, in what follows, that n ≥ 3.
If we identify the basis B of a congruence with its class in the analytic cohomology of the Grassmannian, we get that n − 1 = dim B = codim B, and therefore
we can write,
[B] =
where we put ν :=
n−1 2
ν
i=0
.
ai σ(n−1−i)i ,
(2)
106
P. De Poi
Definition 3. We say that the congruence B has the sequence of degrees
(a0 , . . . , aν ) if equation (2) holds.
Remark 3. The first thing one can say is to explain the geometrical meaning of the
sequence of degrees: it is an easy computation with the Schubert calculus that aj
is the number of lines intersecting a general j -plane and contained in a general
(n − j )-plane of Pn .
Definitions 1. Classically, a0 – i.e. the number of straight lines passing through a
general point P ∈ Pn – is called the order of the congruence, and aν – the number of
lines contained in a general (n − ν)-plane which meet a general ν-plane contained
in it – its class. So a first order congruence is a congruence with sequence of degrees
(1, a1 , . . . , aν ).
Remark 4. By Proposition 1, the “expected” dimension for F is n − 1, and on a
general line of the congruence the scheme of foci has length n − 1; so, in general,
the “expected” dimension for the fundamental locus is ≤ n − 2, since “in general”
it has dimension less than the focal locus. Then, it can happen that there exist
d-(pure) dimensional schemes, with 0 ≤ d ≤ n − 2 which intersect every line
of the congruence in at most n − 1 points. These are clearly subschemes of the
fundamental locus of the family. Besides, “in general”, F does not contain every
line of the congruence. This obviously happens if the order is different from zero.
Since we will consider mainly first order congruences, we will suppose in what
follows that a0 > 0.
Definition 4. The fundamental d-locus, with 0 ≤ d ≤ n − 2, is the subscheme of
the fundamental locus of pure dimension d which is met by the general line of the
congruence, and with the scheme structure given in the following way: the closed
set
Sd := {((b), P ) ∈ | rk(df((b),P ) ) ≤ d}
has a natural subscheme structure, which is defined by a Fitting ideal, i.e. the ideal
generated by the (d + 1)-minors of df (or, by the Fitting lemma, see [9], by the
d-minors of λ), see [13]; in particular, Sn−1 = V . Let us define
Dd+1 := Sd+1 \ Sd
with the scheme structure induced by Sd+1 . Finally, we consider the schemetheoretic image d of Dd+1 in Pn under f . The component of d of pure dimension
d, with the scheme structure induced by d , which is met by the general line of the
congruence, is the fundamental d-locus.
In what follows, we will denote by Ci the fundamental (n − 1 − i)-locus, and
with Ci,j its irreducible components (with the scheme structure induced by Ci ).
Moreover, if (b) is a line of the congruence not contained in Ci,j , we will set
Ci,j ∩ (b) := Zi,j = P1 ∪ · · · ∪ Pki,j , where Pa , a ∈ {1, . . . , ki,j } is a fat point
of length ha .
Proposition 2. Let Ci,j , ki,j and ha be as above; then ha ≥ i, ∀a ∈ {1, . . . , ki,j }.
On first order congruences of lines of P4 with a fundamental curve
107
Proof. By definition of fundamental i-locus, the matrix associated to the characteristic map relative to a general element of the family b, λ(b), restricted to Zi,j –
the zero-dimensional scheme defined above – has rank at most (n − 1 − i). This
means that the determinant of this matrix has at least i roots concentrated in the
point P ∈ Zi,j ; in fact, the characteristic polynomial of the matrix λ(b)(P ) (i.e.
the matrix λ(b) calculated in P ), has zero as root of multiplicity at least i, since the
kernel has dimension i, and our thesis follows. Corollary 1. If P is a fundamental 0-locus for a congruence (, B, p), then
(, B, p) is the star of lines through P .
Proof. Since P intersects every line of the congruence, we must have
dim f −1 (P ) = n − 1,
so B = p(f −1 (P )); but will be contained in the star of lines through P , which
has dimension n − 1, so our thesis follows. We finish this section recalling the main result of [8], i.e. the classification of
first order congruences of P3 from the point of view of their focal locus, since it
will be used frequently in Sect. 5. For this we need more notations: first of all, "
will be a fixed line of P3 and P1" the set of the planes containing ". Besides, we will
denote by φ a general nonconstant map from P1" to " and by $ a general element
of P1" . Finally, P1$ is the star of lines through the point φ($) and contained in $.
The result is the following:
Theorem 4. The focal locus of a congruence of lines of P3 of order one can be:
1. an irreducible curve, which can be one of the following:
(a) a rational normal curve C of P3 , in which case the congruence is given by
the secant lines of C;
(b) a line ", and the congruence is – using the above notations – ∪$∈P1 P1$ ;
"
2. a reducible curve, union of a curve C1 of degree m1 and a line C2 such that
length(C1 ∩ C2 ) = m1 − 1; in this case the congruence is given by the lines
meeting C1 and C2 ;
3. a (fat) point, i.e. the congruence is a star of lines.
4. First order congruences of lines of Pn
From now on we will consider first order congruences; in any case, some of the
results we will obtain will be valid without this hypothesis.
As we have seen in Definition 3, a first order congruence is a congruence with
sequence of degrees (1, a1 , . . . , aν ), i.e. through a general point of Pn there passes
only one line of the congruence.
The first observation, due to C. Segre in [16] is the following:
Proposition 3. The fundamental locus of a first order congruence coincides (settheoretically) with the focal locus and has dimension at most n − 2.
108
P. De Poi
Proof. The fact that the two loci coincide is a straightforward consequence of the
fact that the map f is generically (1 : 1). Then, the fundamental locus F cannot
have dimension n − 1; otherwise the subscheme of the first order foci V would
coincide with , and this would contradict the fact that we have a (1 : 1) map. Therefore, in what follows, we will identify fundamental and focal loci (with
the structure scheme of the focal locus).
Remark 5. As a corollary of the preceding proposition we obtain that a line (b) of a
first order congruence either is a focal line or it contains exactly n−1 foci (counting
multiplicities), since, as we observed after Proposition 1, the characteristic map
relative to b drops rank at every fundamental point contained in it.
Then, we can collect in the next theorem – which will be called “Classification
Theorem” – the results of Section 3, in the case of a first order congruence of Pn :
Theorem 5. Let be a first order congruence of Pn . Then, using the notations
of the preceding section, the congruence is given by the lines meeting Ci,j in a
zero-dimensional scheme Zi,j , for every i = 1, . . . , n − 1, and j = 1, · · · , j (i)
and the following relation holds:
n−1=
ki,j
j (i) n−1 ( (
ha )),
(3)
i=1 j =1 a=1
and ha ≥ i.
Proof. It is a corollary of Proposition 2. Theorem 6. Let be a first order congruence of Pn . If Cn−1−i is the component
of the focal locus of maximal dimension i > 0, then n−1
2 ≤ i ≤ n − 2. Besides, if
n−1
i = 2 , |F | = |Cn−1−i | and |F | is irreducible, then Cn−1−i is – set-theoretically
– an i-plane.
Proof. Given i + 1 general hyperplanes H0 , . . . , Hi of Pn and the corresponding
hyperplane sections of Cn−1−i , D0 , . . . , Di , the lines of which meet Dj form
a family of dimension n − 2, j = 0, . . . , i which will generate a hypersurface
MDj of Pn . Since D0 ∩ · · · ∩ Di = ∅, we get that MD0 ∩ · · · ∩ MDi ⊂ F .
Then, since dim(MD0 ∩ · · · ∩ MDi ) ≥ n − 1 − i, we have 2i ≥ n − 1. If the
equality holds and |F | = |Cn−1−i | is irreducible, we obtain that – set-theoretically
– Cn−1−i = H0 ∩ · · · ∩ Hi . Corollary 2. Let be a first order congruence of Pn and Cn−1−i the component
of the focal locus of maximal dimension i > 0; if Cn−1−k is a fundamental k-locus
contained in Cn−1−i , then n − 1 − i ≤ k ≤ i. In particular, if i = n−1
2 we have
only the i-fundamental locus Cn−1−i .
Proof. With notations as in the proof of the preceding theorem, we consider the
hyperplane sections of Cn−1−k , D0 , . . . , Dk , for which we have D0 ∩· · ·∩Dk = ∅,
and so MD0 ∩ · · · ∩ MDk ⊂ F . Therefore our thesis easily follows. On first order congruences of lines of P4 with a fundamental curve
109
Example 1. To explain Theorem 5, we use it to give a first classification of congruences of P4 ; in this case, n − 1 = 3. Let be such a congruence; then we can have
a fundamental 2-locus C1 , a fundamental 1-locus C2 and a fundamental 0-locus
C3 , with the following possibilities:
1. there is only an irreducible fundamental 2-locus C1 and
(a) k1,1 = 3, h1 = h2 = h3 = 1, i.e. the congruence is given by the trisecant
lines of the surface C1 ; or
(b) k1,1 = 2, h1 = 2, h2 = 1, and the lines of the congruence are particular
secant lines of (C1 )red ; or
(c) k1,1 = 1, h1 = 3, and therefore the lines of the congruence meet C1 in only
a fat point;
2. there is only a fundamental 2-locus C1 with two irreducible components, C1,1
and C1,2 , and
(a) either k1,1 = 2 for which h1 = h2 = 1 and k1,2 = 1 with h1 = 1, i.e. the
congruence is given by the secant lines of the surface C1,1 which meet the
surface C1,2 also; or
(b) k1,1 = 1, for which h1 = 2 and k1,2 = 1, with h1 = 1, therefore the lines of
the congruence are a subfamily of the lines of the join of (C1,1 )red and C1,2 ;
2. there is only a fundamental 2-locus C1 , union of three irreducible components
C1,1 , C1,2 and C1,3 and k1,1 = k1,2 = k1,3 = 1 always with h1 = 1, i.e. the
congruence is given by the lines meeting once each of the three surfaces C1,1 ,
C1,2 and C1,3 ;
4. we have a fundamental 2-locus C1 and a fundamental 1-locus C2 , both irreducible
with k1,1 = 1 for which h1 = 1, and k2,1 = 1 with h1 = 2, i.e. the congruence
is given by the lines which meet the surface C1 and the curve (C2 )red ;
5. we have only a fundamental 0-locus C3 and is a star of lines.
We observe that in cases (1b), (1c), (2b), (4) and (5), at least a component of the
fundamental locus is non-reduced.
Example 2. Let us consider a surface S of P4 whose trisecant lines generate a first
order congruence. Besides, we suppose that the surface S contains a plane curve
C of degree at least three. Let η be the plane of C. Then, every line of η is a line
of the congruence, all points of η are fundamental points and all the lines of η are
focal lines. So η is a component of the fundamental locus of dimension two which
is not a fundamental 2-locus.
4.1. Rationality
Concerning the rationality of a first order congruence, we have that B and are
rational.
Theorem 7. If (, B, p) is a first order congruence of lines, then B and are
rational.
Besides, if Cn−1−i is a fundamental i-locus such that the general line of B meets
Cn−1−i in only one point, then Cn−1−i is rational.
110
P. De Poi
Proof. Fix a general (n − 1)-plane $ of Pn ; by its generality, through a general
point P ∈ $ there passes only one line "(bP ) of the congruence. Then the maps
φ : $ B
ϕ : $ Cn−1−i
defined, respectively, by φ(P ) := bP and ϕ(P ) := "(bP ) ∩ Cn−1−i are – where
defined – injective, hence birational. Besides, the map p : → B defines on a
P1 -bundle structure on B. 5. First order congruences of P4 with a fundamental curve
In this section, we are interested in classifying the congruences of Example 1,
case (1), i.e. the first order congruences of P4 whose irreducible components are a
fundamental 1-locus and a fundamental 2-locus and the congruence is given by the
lines meeting both of them; as we said in the introduction, F1 denotes the fundamental 2-locus and the irreducible curve C the reduced locus of the fundamental
1-locus of the congruence of lines which are considering. m1 is the degree of F1 ,
and m2 the degree of C.
Remark 6. We observe first of all that, by Theorem 7, both F1 and C are rational.
A standard computation with the Schubert cycles gives
Proposition 4. If F1 and C meet properly, i.e. F1 ∩ C = ∅, then F is a plane and
C is a line. The congruence has bidegree (1, 1), F1 is the fundamental 2-locus and
C is the reduced locus of the fundamental 1-locus.
Remark 7. It is easy to see that in this case the congruence, as a subscheme of the
Grassmannian, is the intersection of a hyperplane section (i.e. σ1 ) and a codimension
two section (i.e. σ2 ). In particular, we see that this is a limit case of the intersection
of three hyperplane sections of the Grassmannian, i.e. if the corresponding planes
meet in a line. See also [5].
The following theorem characterises the first order congruences of P4 whose
focal locus is – set-theoretically – a plane; from this, we obtain also the classification
of the cases for which the reduced locus of the fundamental 1-locus is contained in
a linear fundamental 2-locus. We need the following notations: first of all, P1F1 is the
set of hyperplanes containing the plane F1 . Besides, we will denote by F1∗ the set of
all the lines of F1 , by φ2 (respectively, by φ2∗ ) a map from P1F1 to F1 (respectively,
F1∗ ) and by $3 a general element of P1F1 . P1$3 is the pencil of planes contained in
a fixed $3 and containing φ2∗ ($3 ); φ1 is a general map from P1$3 to φ2∗ ($3 ) and
$2 is a general element of P1$3 . P2$3 will be the star of lines through φ2 ($3 ) and
contained in $3 . P1$2 is the pencil of lines through φ1 ($2 ) and contained in $2 .
If φ2 is nonconstant, we set C := Im(φ2 ), m2 := deg(C) and di := deg(φi ),
with i = 1, 2. If φ2∗ is constant, we set "1 := Im(φ2∗ ); if instead φ2∗ is nonconstant,
we set d2∗ := deg(φ2∗ ) and m := deg(Im(φ2∗ )). Finally, r will be a general line
contained in F1 .
On first order congruences of lines of P4 with a fundamental curve
111
Theorem 8. If the focal surface is set-theoretically a plane F1 , then the congruence
is obtained – using the above notations – in one of the following ways:
1. we have a nonconstant map φ2 and the congruence is ∪$3 ∈P1 P2$3 ; besides, the
F1
bidegree of is (1, d2 m2 ) and deg(f −1 (r)) = d2 m2 ; F1 is the support of the
fundamental 2-locus and C is the support of the fundamental 1-locus;
2. we have a constant map φ2∗ , ∀$3 ∈ P1F1 the corresponding map φ1 is nonconstant and the congruence is ∪$3 ∈P1 ∪$2 ∈P1 P1$2 ; besides, the bidegree of is
$3
F1
(1, d1 ), and deg(f −1 (r)) = d1 ; moreover, the reduced locus of the fundamental
2-locus is F1 and "1 is the reduced locus of the fundamental 1-locus;
3. we have a nonconstant map φ2∗ , ∀$3 ∈ P1F1 the corresponding map φ1 is nonconstant and the congruence is ∪$3 ∈P1 ∪$2 ∈P1 P1$2 ; besides, the bidegree of
F1
$3
is (1, d1 d2∗ m − 1), and deg(f −1 (r)) = d1 d2∗ m; moreover, we have only a
fundamental 2-locus whose reduced locus is F1 .
Proof. First of all we note that, if we consider a general point Q ∈ P4 , this determines a general hyperplane $3 of P1F1 , and the lines of the congruence contained
in this hyperplane form a first order congruence, since for the general point of $3
there passes only one line of .
Besides, we note that the focal locus of |$3 is contained in F1 ; in fact it can
be either – set-theoretically – a point P ∈ F1 or a line " ⊂ F1 , by Corollary 1 and
from what is known about first order congruences of P3 (see Theorem 4). Then, we
have the following possibilities:
1. If for almost every $3 we have that the congruence induces a star of lines, we
can construct a map
φ2 : P1F1 → F1
which associates to each hyperplane $3 the support P$3 of its star of lines. The
map φ2 cannot be constant since otherwise we have a star of lines of P4 . Then,
we apply Riemann-Hurwitz’s theorem getting that the image C of φ2 is rational
and we obtain case (1).
If deg(C) = m2 and deg(φ2 ) = d2 , we can in fact calculate the bidegree of
the congruence: the second degree is the number of lines of the congruence
contained in a hyperplane H and meeting a line rH ⊂ H . But H ∩ F1 is a line
"H , which meets C in m2 points and so the thesis follows.
2. If for almost every $3 the congruence induces a congruence of P3 with a focal
line only, this means that the congruence |$3 is given as in Theorem 4, 1b; besides, we have two possibilities: either the line is the same for all the hyperplanes
containing F1 or it changes:
(a) If the line is the same for all the planes of P1F1 , by Theorem 4, 1b, we are
in case (2). The second degree is the number of lines of the congruence
contained in a hyperplane H and meeting a line rH ⊂ H and, as before
H ∩ F1 = "H . "H meets "1 in one point P ; then we have to find how
many lines of the pencil of lines through P and contained in P rH are in the
congruence. But the lines of the congruence through P form a cone CP of
112
P. De Poi
dimension three and degree d1 , since CP ∩ $3 , with $3 ∈ P1F1 is given by
d1 planes.
(b) If the lines vary, we set the following map
φ2∗ : P1F1 → F1∗
which associates the line "$3 to the hyperplane $3 , centre of the pencil of
planes defining the congruence. Then, as before, using Riemann-Hurwitz’s
theorem, we conclude – by Theorem 4 1b – that we are in case (3).
For calculating the second degree a, let us consider H and "H as before.
Through a point P of "H there will pass m lines of Im(φ2∗ ), and through
each of these lines, d1 d2∗ pencils of the congruence. Besides, the degree of
the scroll V$ of the lines of which meet a plane $ is – as can be shown
by the Schubert calculus – a + 1 and F1 has algebraic multiplicity d1 d2∗ m
in V$ . Since a general line of intersects V$ in a point of F1 , the thesis
follows. First we will consider the case in which C is contained in F1 and then the other
case.
5.1. The case C ⊂ F1
We have analysed in Theorem 8, 1 and 2 the case in which F1 is a plane, so in the
following we will suppose that F1 is not a plane. We start with the following
Lemma 1. If C ⊂ F1 , then C must be a plane curve.
Proof. Let us prove the lemma ab absurdo: if " = P Q is a general secant line of
C, we get that deg f −1 (") = 2, since P and Q are points both of C and F1 ; i.e. "
is a focal line. Therefore the (embedded) secant variety of C, which has dimension
three because C is not plane, is contained in the focal locus, which is absurd. A standard fact is the following
Proposition 5. Let S ⊂ PN be an irreducible surface with rational sections; then
S is a projection of either a rational normal scroll or the Veronese surface.
For a proof of this, see [10]. We are now able to give the following
Proof of Theorem 1, case (1). Let η be a plane containing C and P1η the pencil of 3planes containing η. If H is a general element of P1η , |H is a first order congruence
of H ∼
= P3 ; in particular it is a congruence as in Theorem 4, 2.
Besides, since C is a plane curve, and F1 is not a plane, C can either be a
line or a conic. In fact, if m2 > 2, H ∩ F1 is given by C and a line "H , and
length(C ∩ "H ) = m2 − 1 > 1, by Theorem 4, 2. It cannot be "H ⊂ H , because
otherwise length(C ∩ "H ) = m2 . So, C must have a (m2 − 1)-multiple point P and
"H passes through it. So, varying H in P1η , we obtain that F1 is a cone with vertex
in P . But the general hyperplane section through P must be a line only out of the
plane η, and so F1 should be the union of η and a plane, which is a contradiction.
On first order congruences of lines of P4 with a fundamental curve
113
1. If C is a line, this is a fundamental line of Theorem 4; besides, H ∩ F1 is a
rational curve CH of degree m1 such that length(CH ∩ C) = m1 − 1. Besides,
the sectional (arithmetic) genus of F1 is obtained by adjunction.
2. If C is a conic, H ∩ F1 must be a line " and length(" ∩ C) = 1, by Theorem 4,
and the thesis follows from Proposition 5.
Vice versa, it is easy to see that all these cases are effective, since in the general
hyperplane containing the plane of the curve C we have a first order congruence of
P3 . From this result, we can obtain also
Proof of Theorem 2, case (1).
An easy corollary of the preceding proof is that, if we are in case (1c) of
Theorem 1, F1 is smooth if and only if is the rational normal scroll S1,2 of degree
three linearly normal in P4 , i.e. for m1 = 3, and C is a unisecant conic (clearly
with algebraic multiplicity k = 1 in the scroll).
Let us consider case (1b) of Theorem 1: if F1 is smooth, its hyperplane section
is a (smooth) curve CH ⊂ H ∼
= P3 of degree m1 and genus m1 − 2; by [12], the
hyperplane section of CH is special, since it cannot be a plane curve. From this we
get that g ≥ 21 d + 1 and d ≥ 6. By Riemann–Roch we have
χ (OCH (1)) = 3
(4)
and, since CH is not a plane curve, we have that h0 (OCH (1)) ≥ 4.
Let us consider then the exact sequence of coherent sheaves on F1 defining CH :
·H
0 → OF1 −→ OF1 (1) → OCH (1) → 0
(5)
and the corresponding long exact sequence of cohomology; from the fact that F1
is a rational surface distinct from the Veronese surface – since its sectional genus
is not zero – we get that it is linearly normal; from this, we deduce
h0 (OF1 ) = 1,
(6)
h0 (OF1 (1)) = 5,
(7)
h (OF1 ) = 0,
(8)
h (OF1 ) = 0;
(9)
1
2
from equations (6) and (7) we get that h0 (OCH (1)) = 4, and, by (4), we obtain that
h1 (OCH (1)) = 1.
(10)
Finally, from (10), (8) and (9), we get h1 (OF1 (1)), so we have that, if the surface
F1 of case (1b) of Theorem 1 is smooth, it is a speciality one rational surface of P4
(see [1]).
Now, the speciality one rational surfaces of P4 can have degree 8, 9, 10 or 11,
and are known and classified only the surfaces in degrees 8, 9 and 10: see [2]. From
the list of these surfaces, we have that a speciality one surface of degree m1 , with
8 ≤ m1 ≤ 10 contains (at least) a line. 114
P. De Poi
5.2. The case C ⊂ F1
We start with the following
Lemma 2. If C is not a plane curve, then F1 is a plane.
Proof. We will denote by u the number of lines passing through a general point P
of P4 which meet the zero-dimensional scheme F1 ∩ C.
The two cones χF1 ,P and χC,P of the lines passing through P and meeting,
respectively, F1 and C have dimensions 3 and 2 and degrees m1 and m2 . Therefore
they meet in m1 m2 lines and only one of these belongs to , since we have a
first order congruence. So u = m1 m2 − 1, as the lines through F1 ∩ C cannot be
computed as lines of the congruence; in fact, if Q ∈ F1 ∩ C, the lines through Q
form a star of lines, which is a first order congruence.
On the other hand, since C is not a plane curve, given a general point Q ∈ C,
the cone χQ,C (which has degree m2 − 1) cannot be contained in the cone χQ,F1 ,
otherwise all the secant lines of C would meet F1 and so they would be focal lines.
Then, through Q there pass m1 (m2 − 1) secant lines of C meeting again F1 , that
must be the u lines passing through Q and that meet F1 ∩ C also. This is due to
the fact that if one of the m1 (m2 − 1) lines met F2 outside C, this would be a focal
line, since it would contain (at least) four focal points. So, would have a focal
hypersurface.
Therefore, we have
u = m 1 m2 − 1
= m1 (m2 − 1)
and so m1 = 1.
From this, we obtain
Proof of Theorem 1, case (2). We start considering the case in which C is not a
plane curve, showing that we will be in case (2a) of the theorem. Let P1F1 be the
pencil of 3-planes containing F1 . If H is a general element of P1F1 , |H is a first
order congruence of H ∼
= P3 ; besides, length(H ∩ C) = m2 , i.e. a finite number of
points. Therefore, |H must be a star of lines with centre PH ∈ C; so c = m2 − 1
and C is rational.
Then, we consider the case in which C is a plane curve; we will see that the
possibilities are cases (2b), (2c) and (2d) of the theorem. Let η be a plane containing
C and P1η the pencil of 3-planes containing η. If H is a general element of P1η , |H is
a first order congruence of H ∼
= P3 ; in particular we are in the case of Theorem 4, 2.
Besides, since C is a plane curve, it can either be a focal line or a fundamental curve
for |H and with m2 > 1.
1. If C is a fundamental line of Theorem 4, then H ∩ F1 is a rational curve CH of
degree m1 such that length(CH ∩C) = m1 −1. Therefore, by Proposition 5, F1 is
the projection either of the Veronese surface of P5 or of a rational normal scroll,
and C is an (m1 − 1)-secant line of F1 . To see that all the cases are effective, we
On first order congruences of lines of P4 with a fundamental curve
115
recall that if we choose m1 − 1 points on either the Veronese surface (in which
case m1 = 4) or a rational normal scroll of degree m1 , these are in general
position and so they generate an (m1 − 2)-plane, $. If we project the surface
from an (m1 − 3)-plane contained in $ which does not intersect the surface, we
obtain a surface with rational sections of P4 whose (m1 − 1)-secant line is the
image of $ under this projection.
2. If C is a rational curve but not a line, then H ∩ F1 must be a line " out the plane
η of C and length(" ∩ C) = m2 − 1, by Theorem 4, so F1 must be a cone with
vertex on a point P ∈ C of multiplicity m2 − 1 for C and F1 ∩ η is given by
m1 − 1 lines.
Vice versa, as we did at the end of the proof of Theorem 2, case (1), it is easy
to prove that these cases are effective, since in the general hyperplane containing
either F1 in case (2a) or the plane of the curve C in cases (2b), (2c) and (2d), we
have a first order congruence of P3 . By this result, we are able to give the following
Proof of Theorem 2, case (2).
If a surface of Theorem 1, case (2b), is smooth, it can only be the rational normal
cubic scroll S1,2 , in which case C is a secant line of it. 5.3. Final remarks on the congruences with a fundamental curve
Let us consider a congruence B as a subvariety of dimension three in G(1, 4);
from the article [3] we can deduce which are the smooth congruences B of P4
with a fundamental curve. We recall that a result of this article is that the smooth
congruences with a fundamental 1-locus are classified, and it is explained how to
construct them.
Therefore, we get
Proof of Theorem 3. It is straightforward to see that the only possible cases of first
order congruence of P4 from [3] are that C can be either a line or a conic. The case
in which we have a line is immediate from [3]. Concerning the case of the conic,
we see that we can exclude case (1c) of Theorem 1 since in this case B is not a
scroll, while in case (2d) of Theorem 1, the only way to get a conic with B a scroll
is with m1 = 1 and m2 = 2. We could conclude the case of the conic also directly,
by quoting the explicit construction given in this case in [3]. Acknowledgements. This article is a part of my Ph. D. Thesis, written under the supervision
of Prof. Emilia Mezzetti, and discussed on 22nd October, 1999; first of all, I want to thank
her and Prof. Enrique Arrondo for their help in the elaboration of this paper.
I would also like to thank Prof. Fyodor L. Zak for interesting comments and some
references, especially [18], and A. A. Oblomkov for the useful correspondence on the subject.
Finally, I want to thank the referee for useful comments and remarks and for pointing us an
error in the first version of the article.
116
P. De Poi
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