Another classification of Incidence Scrolls

10 Pages·0.17 MB·English

by  Rosa Cid

Another Classification of Incidence Scrolls
2
0
0
2 Rosa Cid-Mun˜oz Manuel Pedreira

n
a
J
Authors’ address: DepartamentodeAlgebra,UniversidaddeSantiagodeCompostela.
1
2 15782 Santiago de Compostela. Galicia. Spain. Phone: [phone]-ext.13152. Fax: 34-
81597054. e-mail: [email protected]; [email protected]

] Abstract: Theaimofthispaperisthecomputationofthedegreeandgenusofallincidence
G
scrolls in IPn. For this, we fix the dimension of a linear space which must be contained in
A
the base of anincidence scroll. Then wewillfind all the incidence scrollswhichhave abase
. spaceof this fixed dimension. In thisway, wecan obtainall theincidence scrollswithaline
h
t asdirectrixcurve,thosewhosebasecontains aplane,andsoon.
a
MathematicsSubject Classifications: Primary,14J26;secondary, 14H25,14H45.
m
Introduction: Throughoutthis paper, the base field for algebraicvarieties
[
is C. Let IPn be the n-dimensional complex projective space and G(l,n) the

1 Grassmannian of l-planes in IPn. Then Rd ⊂ IPn denotes a scroll of degree d
v g
and genus g. We will follow the notation and terminology of [3].
3
9 It is useful to represent a scroll in IPn by a curve C ⊂ G(1,n) ⊂ IPN. The
1 lines which intersect a given subspace IPr ⊂ IPn are represented by the points
1 of the special Schubert variety Ω(IPr,IPn). Each Ω(IPr,IPn) is the intersection
0 of G(1,n) with a certain subspace of IPN. Since G(1,n) has dimension 2n−2
2
and we search a curve, we must impose 2n−3 linear conditions on G(1,n).
0
/ Consequently, the choice of subspaces is not arbitrary. Any set of subspaces of
h IPn which imposes 2n−3 linear conditions on G(1,n) is the base of a certain
t
a incidence scroll. The backgroundabout Schubert varieties can be found in [4].
m
The aim of this paper is to obtain another classification of the scrolls in
: IPn which are defined by a one-dimensional family of lines meeting a certain
v
i set of subspaces of IPn, a first classification is given in paper [1]. These ruled
X
surfaces are called incidence scrolls, and such an indicated set is a base of the
r incidence scroll. Unless otherwise stated, we can assume that the base spaces
a
are in general position. We will fix the dimension of a linear space which must
be contained in the base of an incidence scroll. In this way, we can obtain all
the incidence scrolls which have a base space of this fixed dimension, i.e., all
the incidence scrolls with a line as directrix curve, those whose base contains a
plane, and so on.
In section 1 we explain our notation and collect background material. Our
first step will be to summarize some properties of ruled surfaces and, in par-
ticular, some general properties of incidence scrolls. For more details we refer
the reader to [3] and [1] respectively. Having revised the notion of incidence
scroll, we have compiled some basic properties of such a scroll. This section
1
2 R. Cid-Mun˜oz-M. Pedreira
containsa briefsummaryandthe detailedproofswill appearin[1]. The degree
of the scroll given by a general base is provided by Giambelli’s formula which
appears in [2]. Moreover, the study of deformations of a given incidence scroll
is a powerful tool in this paper in order to simplify our proofs. If the incidence
scroll Rd ⊂ IPn breaks up into Rd1 ⊂ IPr and Rd2 ⊂ IPs with δ generators in
g g1 g2
common, then d=d +d and g =g +g +δ−1.
1 2 1 2
Section 2 contains our main results about incidence scrolls. We identify all
the incidence scrolls. In this way, we find all incidence scrolls which contain a
directrix line, those which contain a base plane and finally those which contain
a base space of dimension 3. A more complete theory may be obtained by a
combinationofthese processes. Accordingly,allthe other incidence scrollsmay
be obtained. Moreover,wewillseethatthe fixedbasespaceimposesconditions
on the genus of the incidence scroll.
TheresultsonthispaperbelongtothePh.D.thesisofthefirstauthorwhose
advisor is the second one.
1 Incidence Scrolls
A ruled surface is a surfaceX together with a surjectivemorphismπ: X −→C
to a smooth curve C such that the fibre X is isomorphic to IP1 for every point
y
y ∈ C, and such that π admits a section. There exists a locally free sheaf E
of rank 2 on C such that X ∼= IP(E) over C. Conversely, every such IP(E) is a
ruled surface over C.
A scroll is a ruled surface embedded in IPn in such a way that the fibres f
b
have degree 1. If we take a very ample divisor on X, D ∼ aC + f, then the
o
embedding Φ : X −→ Rd ⊂ IPn = IP(H0(O (D))) determines a scroll when
D g X
a=1. AscrollRd ⊂IPn issaidtobean incidence scrollifitisgeneratedbythe
g
lines which meet a certain set B of linear spaces in IPn, or equivalently, if the
correspondent curve in G(1,n) is an intersection of special Schubert varieties
Ω(IPr,IPn), 0≤r <n−1. Sucha setis calledabase ofthe incidence scrolland
such a base will be denoted by:
B ={IPn1,IPn2,··· ,IPnr}.
WewillwriteitsimplyBwhennoconfusioncanarise,wheren ≤n ≤···≤n ,
1 2 r
i.e., Cd = r (Ω(IPni,IPn))⊂G(1,n).
g Ti=1
Therefore, unless otherwise stated, we will work with linear spaces in gen-
eral position. By general position we will mean that (IPn1,··· ,IPnr) ∈ W =
G(n ,n)×···×G(n ,n) is contained in a nonempty open subset of W. For
1 r
simplicity of notation, we abbreviate it to base in general position.
Proposition 1.1 The intersection C = r Ω(IPni,IPn) of special Schubert
Ti=1
varieties associated to linear spaces IPni, i =1,··· ,r, in general position is an
irreducible curve of G(1,n) if and only if it verifies the following equality
rn−(n +n +···+n )−r =2n−3 (IS)
1 2 r
Another Classification of Incidence Scrolls 3
Proof. See [1], Proposition 2.4.
Moreover, the incidence scroll generated by a base B have degree d if and
only if we obtain the following equality of Schubert cycles:
Ω(n ,n)···Ω(n ,n)=dΩ(0,2).
1 r
Furthermore, we present one of the three main theorems of the symbolic
formalism, known as Schubert calculus, for solving enumerative problems.
Theorem 1.2 (Pieri’s formula) For all sequences of integers 0 ≤ a < ··· <
0
a ≤n and for h =0,··· ,n−l, the following formula holds in the cohomology
l
ring H⋆(G(l,n);Z):
Ω(a0,··· ,al)Ω(h,n)=XΩ(b0,··· ,bl)
where the sum ranges over all sequences of integers b < ··· < b satisfying
0 l
l l
0≤b ≤a <b ≤a <···<b ≤a and b = a −(n−l−h).
0 0 1 1 l l Pi=0 i Pi=0 i
Proof. See [4], p. 1073.
Finally,letusmentionanimportantpropertyofdegenerationofthesescrolls.
Proposition 1.3 Let Rd ⊂ IPn be an incidence scroll with base B in general
g
position. Suppose that IPni and IPnj lie in a hyperplane IPn−1 and have in
common IPm, m=n +n −n+1. Then the scroll breaks up into:
i j
- Rgd11 ⊂ IPn with base B˙ = {IPm,IPn1,··· ,IdPni,··· ,IPdnj,··· ,IPnr} (which
is possibly degenerate);
- Rd2 ⊂IPn−1 withbaseB¨={··· ,IPni−1−1,IPni,IPni+1−1,··· ,IPnj−1−1,IPnj,
IPgn2j+1−1,···}
which have κ≥1 generators in common. Then, d=d +d and g =g +g +
1 2 1 2
κ−1.
Moreover, if m=0, then the incidence scroll breaks up into a plane and an
incidence scroll Rd−1 ⊂IPn−1 with base B¨ in general position.
g
Proof. See [1], Proposition 3.1.
If m = 0, then shall refer to this particular degeneration as join IPni and
IPnj (i.e., ni +nj = n−1) and to the inverse as separate IPni and IPnj (i.e.,
n +n =n).
i j
2 Classification of Incidence Scrolls
In [1] we have obtained a classification of incidence scrolls of genus 0 and 1.
We canstudy a new point ofview to giveanotherclassificationofthe incidence
scrolls. For this we will fix the dimension of a base space. In this way, we will
obtain all the incidence scrolls with a line as directrix curve, those whose base
containsaplane,andfinally,thosewhosebasecontainsa3-plane. Alltheothers
may be obtained by combinations of these processes.
4 R. Cid-Mun˜oz-M. Pedreira
2.1 Incidence Scrolls with a Directrix Line
The following theorem gives all linearly normal incidence scrolls with a line as
directrix. For the proof we refer the reader to [1], Proposition 4.3.
Theorem 2.1 In IPn, n≥3, the scroll given by
B ={IP1,(n−1)IPn−2}
n
in general position is the rational normal scroll of degree n−1 with a line as
minimum directrix.
We see at once that the previous theorem gives all rational normal scrolls
with a line as directrix. Then these are projective models of rational ruled
surfaces X ,e ≥ 0. For each X , the unisecant complete linear system which
e e
gives the immersion is defined by the very ample divisor H ∼C +(e+1)f.
o
TABLE 1. INCIDENCE SCROLLSWITH A DIRECTRIX LINE
ni,i=1,···,7 b
Scroll 1 2 3 4 5 6 7 Normalized Min. Dir.(⋆) deg( )
R20⊂IP3 3 - - - - - - OIP1⊕OIP1 IP1(∞1) 1
R30⊂IP4 1 3 - - - - - OIP1⊕OIP1(−1) IP1(1) 2
R40⊂IP5 1 - 4 - - - - OIP1⊕OIP1(−2) IP1(1) 3
R50⊂IP6 1 - - 5 - - - OIP1⊕OIP1(−3) IP1(1) 4
R60⊂IP7 1 - - - 6 - - OIP1⊕OIP1(−4) IP1(1) 5
R70⊂IP8 1 - - - - 7 - OIP1⊕OIP1(−5) IP1(1) 6
R80⊂IP9 1 - - - - - 8 OIP1⊕OIP1(−6) IP1(1) 7
⋆
Numberofminimumdirectrixcurves
2.2 Incidence Scrolls with IP2 ∈ B
Theorem 2.2 For every n ≥ 4 and 0 ≤ i ≤ n, there is an incidence scroll of
2
degree dn =(n−i)+i−1 and genus gn =(n−i−2) given by
i 2 i 2
Bn ={IP2,iIPn−3,(n−2i)IPn−2}
i
in general position. The directrix curve in the plane has degree (n − i− 1).
Moreover, these are all incidence scrolls with base IP2.
Proof. If a nondegenerate incidence scroll R ⊂ IPn has a plane as base space,
thenthe otherbasespaceshavedimensionn−3orn−2. FixIP2 ∈B. Thenwe
can vary the number of IPn−3’s, written i, between 0 and n (respectively n−1)
2 2
if n is even (respectively if n is odd). Fix IP2 and iIPn−3. Then the number of
IPn−2’s is determined by (IS).
Theproofisbyinductiononi. Ifi=0,wewillproofthattheincidencescroll
Rdn0 ⊂IPn with base Bn ={IP2,nIPn−2} in generalposition has degree (n)−1
gn 0 2
0
Another Classification of Incidence Scrolls 5
and genus (n−2), and the plane directrix curve has degree (n−1). To do this,
2
we suppose that IP2∨IPn−2 = IPn−1 ⊂ IPn, for IP2 ∈ Bn and any IPn−2 ∈ Bn.
0 0
Then the incidence scroll Rdn0 ⊂ IPn degenerates into: Rdn0−1 ⊂ IPn−1 and the
g0n g0n−1
rational normal scroll of degree n−1 with n−2 generators in common. By
Proposition 1.3, it is obvious that dn = dn−1+n−1 and gn = gn−1+n−3.
0 0 0 0
Since the incidence scroll R5 ⊂IP4 is given by B4 ={5IP2} (apply Proposition
1 0
1.3 to 2IP2 ∈B4),
0
d4 =5⇒ d5 =9⇒ d6 =14⇒ dn =(n)−1;
0 0 0 0 2
g4 =1⇒ g5 =3⇒ g6 =6⇒ gn =(n−2).
0 0 0 0 2
Moreover, the degree of the plane directrix curve is (n−1) because it is the
number of lines in IPn which meet a IP1 and nIPn−2 (by Giambelli’s formula).
According to Pieri’s formula, we have:
Ω(1,n)Ω(n−2,n)n =
=Ω(0,n)Ω(n−2,n)n−1+Ω(1,n−1)Ω(n−2,n)n−1 =
=1+Ω(0,n−1)Ω(n−2,n)n−2+Ω(1,n−2)Ω(n−2,n)n−2 =
=···=(n−1)Ω(0,1)=n−1.
Assume the theorem holds for i−1≥ 0; we will obtain the incidence scroll
Rdni−1 ⊂ IPn with base Bn = {IP2,(i − 1)IPn−3,(n − 2i + 2)IPn−2} and a
gin−1 i−1
directrix curve in IP2 of degree (n−i).
Separate plane and a IPn−2 in Bn . Then, we obtain an incidence scroll in
i−1
IPn+1ofdegree(n−i+1)+i−1andgenus(n−i−1)withbaseB′ ={IP2,iIPn−2,(n−
2 2
2i+1)IPn−1},andaplanedirectrixcurveofdegree(n−i). Hence,writing(n−1)
instead of n, we conclude the proof.
Remark 2.3 Since we will work under the hypothesis n ≥ 4, this theorem
gives all the incidence scrolls with a plane directrix curve. For n = 3, the
incidence scroll R2 ⊂IP3 with a plane directrix curve also appears in Theorem
0
2.1. Moreover,it is important to note that:
1. Ingeneral,ifRdni ⊂IPn(n≥4)hasaplanedirectrixcurve,thentheplane
gn
i
which contains such a directrix is a base space (see [1], Proposition 2.5).
This is not true for n=4 and i=2. In this case,we obtain the incidence
scroll R2 ⊂ IP3 with base B4 = {3IP1} where IP2 ∈/ B4 because a plane
0 2 2
directrixcurveisahyperplanesectionandthenanyplanedoesnotimpose
conditions on G(1,3).
2. Theplanedirectrixcurveisnotnecessarilythedirectrixcurveofminimum
degree. For example, for i = 1 and n = 4, we obtain R3 ⊂ IP4 which is
0
defined by B4 = {IP1,3IP2} with a line as minimum directrix curve (see
1
Theorem 2.1).
3. Forn=4,theplanedirectrixcurveisnotnecessarilyunique. Forexample,
if i=0 andn=4,then we obtainR5 ⊂IP4 with base B4 ={5IP2}where
1 0
each plane directrix curve is of type C3 ⊂IP2.
1
6 R. Cid-Mun˜oz-M. Pedreira
4. If we take a plane between the base spaces, then we impose conditions
on the genus of the incidence scroll (the same is true for IP1 because the
incidence scroll is always rational). In the notation of Theorem 2.2 the
genus of the scroll (written g) is subject to the condition
2g =(n−i−2)(n−i−3).
For example, for g = 2, there exists no incidence scrolls with a plane
directrix curve.
5. Therearescrollswithaplanedirectrixcurvewhicharenotincidence. For
example,wecantakeR6 ⊂IP7 withadirectrixconic,athree-dimensional
0
familyofdirectrixquarticsandafive-dimensionalfamilyofdirectrixquin-
tics. It is not defined by incidences (see [1], Example 4.8).
TABLE2. INCIDENCESCROLLSWITHABASEPLANE
ni,i=1,···,7
Scroll 1 2 3 4 5 6 7 Min. Dir.(⋆⋆)
Genus 0 (n=i+3)⇒(n≤6)
R2⊂IP3 3 - - - - - - IP1(∞1)
0
R3⊂IP4 1 3 - - - - - IP1(1)
0
R4⊂IP5 - 3 1 - - - - C2⊂IP2(∞1)
0 0
R5⊂IP6 - 1 3 - - - - C2⊂IP2(1)
0 0
Genus 1 (n=i+4)⇒(n≤8)
R5⊂IP4 - 5 - - - - - C3⊂IP2(∞1)
1 1
R6⊂IP5 - 2 3 - - - - C3⊂IP2(2)
1 1
R7⊂IP6 - 1 2 2 - - - C3⊂IP2(1)
1 1
R8⊂IP7 - 1 - 3 1 - - C3⊂IP2(1)
1 1
R9⊂IP8 - 1 - - 4 - - C3⊂IP2(1)
1 1
Genus 3 (n=i+5)⇒(n≤10)
R9⊂IP5⋆ - 1 5 - - - - C4⊂IP2(1)
3 3
R10⊂IP6⋆ - 1 1 4 - - - C4⊂IP2(1)
3 3
R11⊂IP7⋆ - 1 - 2 3 - - C4⊂IP2(1)
3 3
R12⊂IP8⋆ - 1 - - 3 2 - C4⊂IP2(1)
3 3
R13⊂IP9⋆ - 1 - - - 4 1 C4⊂IP2(1)
3 3
R14⊂IP10⋆ - 1 - - - - 5 C4⊂IP2(1)
3 3
⋆ Incidence special scroll
(⋆⋆) Number of minimum directrix curves
In Table 2 we see some examples of incidence scrolls which have a plane
directrix curve. Moreover, the table contains all the incidence scrolls with a
plane directrix curve of genus 0,1 and 3.
Another Classification of Incidence Scrolls 7
2.3 Incidence Scrolls with IPr ∈ B,r ≥ 3
Wecancontinuewithasimilarmethodforobtainanewtheoremwhichprovides
allthe incidencescrollswhichhavea IP3 betweenthe basespaces. Without loss
of generality we can assume n≥5 because the other cases appear in Theorems
2.1and2.2. Forexample,R3 ⊂IP4 withIP1 asminimumdirectrixcurveisgiven
0
by Theorem 2.1.
Fix IP3. The other base spaces have dimension ≥n−4 because the scrollis
non-degenerate. Then the other base spaces are necessarily jIPn−4’s, iIPn−3’s
and n+1−3j−2iIPn−2’s. In fact, we see at once that
1. IP3 imposes (n−4) conditions but we need (2n−3) conditions;
2. 0≤j ≤ 1(n+1);
3
3. fixing j between the above limits, we obtain 0≤i≤ 1(n+1−3j).
2
Lemma 2.4 In IPn(n ≥ 5), the incidence scroll given by Bn = {IP3,(n +
0,0
1)IPn−2} in general position has degree dn = (n+1)−n−1 and genus gn =
0,0 3 0,0
(n)+(n−1)−2n+4. The directrix curve in IP3 has degree (n)−1.
3 3 2
Proof. We proceed by induction on n. If n = 5, then we will prove that B5
0,0
definesthescrollR14 ⊂IP5 andthedirectrixcurveinIP3 hasdegree9. Suppose
8
that2IP3’shavebyintersectionaplaneinsteadofaline. Sincethescrollbreaks
upintoR9 ⊂IP5 andR5 ⊂IP4 (seeTheorem2.2)with5generatorsincommon,
3 1
Proposition1.3 shows that the scrollhas degree 14 and genus 8. Moreover,the
directrix curve in each IP3 has degree equal to the number of lines in IP5 which
meet a IP2 and 6IP3’s. From Pieri’s formula we have:
Ω(2,5)Ω(3,5)6 =
=Ω(1,5)Ω(3,5)5+Ω(2,4)Ω(3,5)5 =
=Ω(0,5)Ω(3,5)4+2Ω(1,4)Ω(3,5)4+Ω(2,3)Ω(3,5)4 =
=···=9Ω(0,2)Ω(3,5)=9Ω(0,1)=9.
Suppose that the incidence scroll with base Bn in general position has
0,0
degree dn and genus gn and each directrix curve in IP3 has degree (n)−1.
0,0 0,0 2
Then we will proof the lemma for n+1.
Let Rd ⊂IPn+1 be the incidence scroll defined by Bn+1 in general position.
g 0,0
We will calculate its degree and genus. If IP3 ∨IPn−1 = IPn, then the scroll
breaks up into:
- R(n2+1)−1 ⊂IPn+1 with base Bn+1 (see Theorem 2.2);
(n−1) 0
2
- Rdn0,0 ⊂IPn with base Bn , by hypothesis of induction, and (n+1)−n−1
gn 0,0 2
0,0
generators in common.
8 R. Cid-Mun˜oz-M. Pedreira
An easy computation shows that d = (n+2)−n−2 = dn+1 and g = (n)+
3 0,0 3
(n+1)−2n+2=gn+1. Moreover,the degree ofthe directrix curve in IP3 is the
3 0,0
number of lines in IPn+1 which meet a IP2 and (n+2)IPn−1’s. It is exactly:
Ω(2,n+1)Ω(n−1,n+1)n+2 =
=Ω(1,n+1)Ω(n−1,n+1)n+1+Ω(2,n)Ω(n−1,n+1)n+1 =
=nΩ(0,1)+(n−1)Ω(0,1)+Ω(2,n−1)Ω(n−1,n+1)n =
=···=n+(n−1)+(n−2)+···+2= 1(n−1)(n+2).
2
Lemma 2.5 InIPn(n≥5),theincidencescrollwithbaseBn ={IP3,IPn−3,(n−
0,1
1)IPn−2} in general position has degree dn =(n)−1 and genus gn =(n−2)+
0,1 3 0,1 3
(n−1)−n+3. The directrix curve in IP3 has degree (n−1).
3 2
Proof. Suppose that IP3∨IPn−3 =IPn−1. Then the incidence scroll Rd ⊂IPn
g
with base Bn degenerates into:
0,1
- Rdn0,−01 ⊂IPn−1 with base Bn−1 (see Lemma 2.4);
gn−1 0,0
0,0
- R0n−1 ⊂IPn with base Bn−1 ={IP1,(n−1)IPn−2} (see Theorem 2.1) and
n−2 generators in common.
Whence,weconcludethatd=(n)−1=dn andg =(n−2)+(n−1)−n+3=
3 0,1 3 3
gn .
0,1
The degree of the directrix curve which is contained in IP3 is given by:
Ω(2,n)Ω(n−3,n)Ω(n−2,n)n−1 =
=Ω(0,n)Ω(n−2,n)n−1+Ω(1,n−1)Ω(n−2,n)n−1+
+Ω(2,n−2)Ω(n−2,n)n−1 =···=
=1+(n−2)+(n−3)+···+2= 1(n−1)(n−2).
2
Lemma 2.6 InIPn(n≥5),theincidencescrollwithbaseBn ={IP3,IPn−4,(n−
1,0
2)IPn−2} in general position has degree dn =(n−1) and genus gn =(n−2)+
1,0 3 1,0 3
(n−3)−n+4. The directrix curve in IP3 has degree (n−2).
3 2
Proof. Separate IP3 and IPn−4. Then we conclude from Proposition 1.3 that
our scroll Rd ⊂ IPn with base Bn degenerates into Rd−1 ⊂ IPn−1 with base
g 1,0 g
Bn−1, hence that d=(n−1) and g =(n−2)+(n−3)−n+4, by Lemma 2.5.
0,1 3 3 3
If we work these lemmas, then we can obtain a general theorem which will
give every incidence scroll with a IP3 between the base spaces. Moreover, we
will obtain the degree and the genus of the incidence scroll for any i and j
between suitable limits. The degree of the directrix curve of such an incidence
scroll which is given by the following theorem may easily be found by referring
to each particular case and using Pieri’s formula.
Another Classification of Incidence Scrolls 9
Theorem 2.7 For every n ≥ 5, 0 ≤ j ≤ 1(n+1) and 0 ≤ i ≤ 1(n+1−3j),
3 2
there is an incidence scroll of degree
dn =dn−j +j =(n−i−2j+1)−(n−i−2j)+(i+j)(n−i−2j−1)+j−1
j,i 0,i+j 3
and genus
gn =gn−j =(n−i−2j)+(n−i−2j−1)−2(n−i−2j)+(i+j)(n−i−2j−2)+4
j,i 0,i+j 3 3
withbase Bn ={IP3,jIPn−4,iIPn−3,(n+1−3j−2i)IPn−2}in general position.
j,i
Proof. We proceed by induction on j.
For j = 0, we will proof that the incidence scroll with base Bn , in general
0,i
position, has degree dn = (n−i+1)−(n−i)+i(n−i)−1 and genus gn =
0,i 3 0,i
(n−i)+(n−i−1)−2(n−i)+i(n−i−2)+4. Todothis,weproceedbyinduction
3 3
on i. The theorem is true for i = 0, by Lemma 2.4. Supposing the theorem
true for i−1 ≥ 0, we prove it for i. Suppose that IP3 ∨IPn−3 = IP1, for any
IPn−3 ∈ Bn . Then the incidence scroll Rd ⊂ IPn with base Bn in general
0,i g 0,i
position degenerates into: Rdn0,−i−11 ⊂ IPn−1 with base Bn−1 ( by hypothesis of
g0n,−i−11 0,i−1
induction) and R0n−i ⊂ IPn−i+1 with base Bn−i = {IP1,(n−i)IPn−i−1} (see
Theorem 2.1) and (n−i−1) generators in common.
Whence,wefindthatd=dn−1 +(n−i)=dn andg =gn−1 +(n−i−2)=
0,i−1 0,i 0,i−1
gn .
0,i
Supposing the theorem true for j−1≥0, we prove it for j. Separating IP3
and IPn−3 in Bn we obtain a base B = {IP3,jIPn−3, (i−1) IPn−2,(n+1−
j−1,i
3(j−1)−2i)IPn−1} in general position which defines Rdnj−1,i+1 ⊂IPn+1.
gjn−1,i
Writeninsteadn+1andiinsteadi−1. ThentheincidencescrollRd ⊂IPn
g
with base Bn in general position has degree d = dn−1 +1 and genus g =
j,i j−1,i+1
gn−1 . Therefore, d=dn−j +j and g =gn−j , by hypothesis of induction.
j−1,i+1 0,i+j 0,i+j
Remark 2.8 The determination of the degree of the directrix curve in IP3,
writtendeg(C),ismoresubtlethanintheothercasesbecauseitisΩ(2,n)Ω(n−
4,n)jΩ(n−3,n)iΩ(n−2,n)n+1−3j−2i. From what has already been proved it
may be concluded that for every j,i between suitable limits,
deg(C)=(n−i−2j)+i+j−1.
2
If we now apply this argument again, with IP3 replaced by IPr, r ≥ 4, then
we can obtain a general theorem which will give all incidence scrolls with IPr
between the base spaces. Its formulation is very complicated because we must
work with, at least, three index: i,j,k. But the proof is similar to that of
Theorem 2.7 for all cases. Therefore we can obtain degree and genus of the
incidence scroll with base
Bin1,···,ir−1 ={IPr,i1IPn−r−1,··· ,ir−2IPn−4,ir−1IPn−3,
(n+r−2−ri1−···−3ir−2−2ir−1)IPn−2}
for any i1,··· ,ir−1 between suitable limits.
10 R. Cid-Mun˜oz-M. Pedreira
TABLE3. INCIDENCESCROLLSWITHIP3∈B
ni,i=1,···,4
Scroll 2 3 4 5 DirectrixinIP3
R14⊂IP5⋆ - 7 - - C9⊂IP3
8 8
R5⊂IP6 1 3 - - C3⊂IP3
0 0
R7⊂IP6 1 2 2 - C4⊂IP3
1 1
R10⊂IP6⋆ 1 1 4 - C5⊂IP3
3 3
R9⊂IP6 - 4 1 - C5⊂IP3
2 2
R13⊂IP6⋆ 3 3 - C7⊂IP3
5 5
R19⊂IP6⋆ - 2 5 - C10⊂IP3
11 11
R28⊂IP6⋆ - 1 7 - C14⊂IP3
22 22
R6⊂IP7 - 3 1 - C3⊂IP3
0 0
R8⊂IP7 - 3 - 2 C4⊂IP3
1 1
R10⊂IP7 - 2 2 1 C5⊂IP3
2 2
R14⊂IP7⋆ - 2 1 3 C7⊂IP3
5 5
R20⊂IP7⋆ - 2 - 5 C10⊂IP3
11 11
R12⊂IP7 - 1 4 - C6⊂IP3
3 3
⋆
Incidencespecialscroll
References
[1] CID-MUN˜OZ,R.-PEDREIRA,M.Classification of Incidence Scrolls(I);
preprint math.AG/0005272.
To appear in Manuscripta Mathematica.
[2] GRIFFITHS, P. - HARRIS, J. Principles of algebraic geometry.
Pureandappliedmathematics. AWiley-Intersciencepublication,pp.193-
211 (1978).
[3] HARTSHORNE, R. Algebraic Geometry.
Graduate Texts in Mathematics 52. Springer-Verlag, New York. 1977.
[4] KLEIMAN, S. L. - LAKSOV, D. Schubert calculus.
Amer. Math. Monthly 79, pp. 1061-1082(1972).