### Tangent Conics at Quartic Surfaces and Conics in Quartic Double Solids

### Dissertation

zur Erlangung des akademischen Grades Doctor rerum naturalium

im Fach Mathematik

eingereicht an der

Mathematisch-Naturwissenschaftlichen Fakult¨at II der Humboldt-Universit¨at zu Berlin

von Ingo Hadan

geb. 31. Mai 1968 in Berlin

Pr¨asident der Humboldt-Universit¨at zu Berlin Prof. Dr. Dr. Hans Meyer

Dekan der Mathematisch-Naturwissenschaftlichen Fakult¨at II Prof. Dr. Wolfgang Reisig

Gutachter/Gutachterin

1. . . . 2. . . . 3. . . .

Tag der m¨undlichen Pr¨ufung: . . . .

## Introduction

The present paper is concerned with a problem of a very classical nature – namely with the
description of the space of conics which are tangent to a given quartic surface *B* *⊂* *IP*^{3}. This
question is naturally raised once the space of bitangents at *B* is understood. The latter space
repeatedly was the subject of interest until recently. First investigations of bitangents at quartic
curves date back into the middle of the nineteenth century. Pl¨ucker [P] and Jacobi [Ja] seem to be
the first to have shown that there are 28 bitangents at a smooth plane quartic curve. Afterwards,
Steiner [St] Hesse [Hes] and Clebsch [Cleb] (to mention only a few) intensively studied the symmetry
of the 28 bitangents at a plane quartic. In Frames paper [Fra] the properties of the corresponding
symmetry group are briefly summerised. Finally Harris’ work on bitangents must not be forgotten
when talking about bitangents at a plane quartic. In [Har1] he realises the symmetry group of the
bitangents as the image of a certain monodromy action.^{1}) whereas in [Har2] he is concerned with
bitangents at singular plane quartics obtaining partially classical results with modern methods.

The bitangents at a quartic surface are still of use to contemporary mathematics. Especially in the theory of threefolds the family of bitangents at a quartic surface and the corresponding Abel-Jacobi map were used to study the intermediate Jacobian of quartic double solids (cf. [Clem]).

However, nearly nothing could be found in the literature on tangent conics at a quartic surface.

This is quite astonishing as this subject – just like the bitangents – contains a lot of beautiful
geometry and combinatorics. Certainly some (if not many) of the ideas presented throughout
this paper are already (at least implicitly) contained elsewhere. E.g., a considerable part of the
work needed to describe the tangent conics at a plane quartic is already contained in Salmon’s
book [Sa]. A part of these ideas also appear in C.T.C. Wall’s paper [Wa]. There we also found
the reference [Co]. Coble determines (in Chapter I,*§*14) the 63 one-parameter families of tangent
conics at a smooth plane quartic in a way similar to the method used in Section 1.2. But apparently
by now nobody worked on the question raised above.

Ceresa and Verra [CV] study tangent conics at sextic surfaces (to get information on the Abel- Jacobi map associated to the family of “conics” in a sextic double solid). Unfortunately, their methods seemingly do not apply to quartic surfaces. Generally, quartics seem to be exceptional with regard to bitangents and tangent conics. E.g., in [Har1] (page 687) it is shown that quartics are the only curves whose set of bitangents has an inherent symmetry.

Closely connected with tangent conics at quartic surfaces are what we will call “conics” in double
solids: If*Z−→IP*^{3}is the double cover branched along a quartic surface (= double solid), and if*H*
denotes the pull-back of*O**IP*^{3}(1) to*Z* then a conic in*Z*is a smooth rational curve*F*in*Z*satisfying
*F·H* = 2. As in the case of tangent conics and bitangents at a quartic surface, the conics in double
solids are natural successors of lines in these threefolds. (The latter being defined to be smooth
rational curves *L* such that *L·H* = 1.) While *Z* *−→IP*^{3} maps lines in*Z* to bitangents at the
branch locus – conics are mapped to tangent conics. Again, there are already results concerning

1However, the monodromy action that will appear in the present paper is associated to a family of quartics with a much smaller base space. Hence the smaller monodromy groups!

i

The only reference we could find where conics in double solids play a role is [SW]. They use
conics in*Z* to classify stable rank-2 vector bundles on Fano threefolds (of which double solids are
examples). They state that the space of all conics in a double solid with smooth branch locus is
smooth and irreducible. But in the reference [I] they give the proof of the claim could not be found;

Iskovskikh uses a quite different notion of conics which is related to the canonical embedding of*Z*.
Moreover, most of the results mentioned above hold only for non-singular double covers of *IP*^{3}.
Donagi and Smith [DS] consider conic bundle structures on double solids with nodal branch locus.

The conics occurring as fibres of these bundles are conics in our sense but the base of these fibrations is (of course) only two-dimensional and thus represents only a codimension-2 subset in the space of all conics.

A further (and in fact the initial) motivation for the present subject arises from the attempt to construct twistor spaces over the connected sum of three complex projective planes. In [KK] it is shown that these twistor spaces generically are small resolutions of double solids branched along singular quartics. For the construction of these twistor fibrations it would be helpful to know the twistor fibres (or at least candidates for twistor fibres). These fibres must be conics in the double solid (cf. Lemma 4.5).

### Organisation of the paper and results

Throughout this paper we work over the fieldCof complex numbers. The starting point for the in-
vestigation of tangent conics resp. conics in double solids is the observation that the corresponding
parameter spaces*X* and*X** _{C}* are fibred over the projective three-space. After removing a critical
set ∆ from

*IP*

^{3}the restricted fibration is smooth with disconnected fibres. Those irreducible com- ponents of these spaces which dominate

*IP*

^{3}are then the closure of (arc-)connected components of the restricted fibration. Thus if

*X*

*−→X*

^{0}*−→IP*

^{3}resp.

*X*

_{C}*−→X*

_{C}

^{0}*−→IP*

^{3}denote the Stein factorisations then the irreducible components of

*X*resp.

*X*

*(which dominate*

_{C}*IP*

^{3}) correspond to the monodromy orbits of the finite covers

*X|*

*IP*

^{ˇ}

^{3}

*\*∆

*−→IP*ˇ

^{3}

*\*∆ resp.

*X*

*C*

*|*

*IP*

^{ˇ}

^{3}

*\*∆

*−→IP*ˇ

^{3}

*\*∆.

The determination of this monodromy action is essentially based on two ingredients:

*•* a structure on the fibres which is left invariant by the monodromy action

*•* information on the size of the monodromy-orbits.

The Picard group of certain Del Pezzo surfaces and, in particular, the intersection pairing therein provide the first ingredient. The parameter space of bitangents at the quartic surface – especially the number and the size of its irreducible components – yield the desired information on the size of the monodromy orbits.

Chapter 2 is devoted to the study of this space of bitangents at a quartic surface with double points.

As a by-product we obtain a partial classification of nodal quartic surfaces (cf. Theorem 2.12).

Of course, this result is not at all new. One will find it, e.g., in Rohn’s paper [R] (though not explicitly).

Unfortunately, the above ingredients are not sufficient for our purposes (see the discussion in Chapter 3 on page 29). Therefore we have to restrict ourselves to the case of quartic surfaces with

“trope”-planes (cf. page 15). Quartics of this kind provide a further invariant of the monodromy
action: a (*−*1)-curve in the above Del Pezzo surfaces which is fixed under the monodromy. For

ii

this restricted class of branch loci we succeed in giving a quite complete picture of the monodromy action. These results are obtained in Chapter 3.

As another by-product we obtain the fact that the dual surfaces to the quartics we consider have a complement with “highly” non-abelian fundamental group. This is immediately clear from the fact that the groups generated by the monodromy action are non-abelian.

Though not explicitly stated there, Chapter 3 also contains results on lines (curves *L* satisfying
*L·H* = 1) in double solids with singular branch locus. The irreducible components of the parameter
space of lines can be easily determined using the methods and results of this chapter.

The restricted class of quartic surfaces is still large enough to cover the branch loci occurring in connection with twistor spaces. Using the results of the first chapters, we re-obtain results found by Kreußler in [Kr3]. These results now appear in a broader context which, hopefully, deepens the understanding of the matter. Moreover, the results permit a more explicit (at least a different) characterisation of the candidates for twistor fibres. (See Section 4.3.)

Acknowledgements: I would like to thank my advisor, Prof. Kurke, for pointing me to this beautyful subject and for his steady support, as well as Bernd Kreußler for many fruitful dis- cussions. Furthermore, I would like to thank the “Graduiertenkolleg ‘Geometrie und nichtlineare Analysis’ ” for financial support during my work on this thesis.

iii

## Contents

Introduction i

Contents v

1 Tangent conics at a quartic surface 1

1.1 Parameter space of conics . . . 1

1.2 Tangent conics at a plane quartic . . . 2

1.3 Tangent conics and double covers . . . 6

1.4 Conics in Double Solids . . . 9

1.5 Next steps . . . 11

2 The space of bitangents at a quartic surface 13 2.1 General assertions . . . 13

2.2 Bitangents at a quartic with tropes . . . 15

2.3 Lines and planes in a nodal cubic threefold . . . 18

2.4 The components of the space of bitangents . . . 22

3 The parameter space of tangent conics 25 3.1 The monodromy . . . 25

3.2 Tangent conics at a quartic with tropes . . . 29

4 Applications to Twistor spaces 43 4.1 Generalities . . . 43

4.2 Lines and Twistor fibres . . . 44

4.3 An explicit description . . . 51

4.4 Concluding remarks and open problems . . . 52

List of notations 55

Bibliography 57

v

### Chapter 1

## Tangent conics at a quartic surface

### 1.1 Parameter space of conics

In this section we will construct the parameter space of all conics in*IP*^{3}. The space *X* of tangent
conics will be contained within this parameter space. Let ˇ*IP*^{3}be the space of planes in *IP*^{3} and*S*
the universal subbundle over ˇ*IP*^{3} = Grass(3,4). Then *P* :=*IP*(Sym^{2}*S** ^{∨}*) is the parameter space
of all conics in

*IP*

^{3}. (Every conic is determined by a plane and a symmetric form of degree two in this plane. On the other hand, every conic determines a unique plane which it sits in and in that plane it determines a symmetric 2-form which is unique up to multiplication by scalars. Even for a double line there is a unique plane in which it is contained. It is determined by the non-reduced subscheme structure of the double line.) The projection

*p:P*

*−→*

*IP*ˇ

^{3}assigns to each conic the unique plane which it is contained in.

There is a universal family over*P*, constructed as follows: Let *H* :=*IP*(p^{∗}*S*) be the pull-back of
the universal plane over ˇ*IP*^{3},*τ:H* *→P* the projection, and*O**H*(1) the relative tautological bundle
of*H* over*P. Then there is a distinguished section in (τ*^{∗}*O**P**|**IP*ˇ^{3}(*−*1))^{∨}*⊗ O** ^{H}*(2), for we have

*τ*^{∗}*O**P**|**IP*ˇ^{3}(*−*1)
*∨*

*⊗ O** ^{H}*(2) = Hom

*τ*^{∗}*O**P**|**IP*ˇ^{3}(*−*1)*,* *O** ^{H}*(

*−*1)

^{∨}*⊗*2

= Hom

*τ*^{∗}*O*_{P}_{|}*IP*ˇ^{3}(*−*1)*, Sym*^{2}(*O**H*(*−*1))* ^{∨}*
and there are canonical injections of vector bundles over

*H:*

*O**H*(*−*1)*,→τ*^{∗}*p*^{∗}*S.*

*τ*^{∗}*O*_{P}_{|}*IP*ˇ^{3}(*−*1)*,→τ*^{∗}*p** ^{∗}*(Sym

^{2}

*S*

*) =*

^{∨}*τ*

^{∗}*Sym*

^{2}(p

^{∗}*S*)

^{∨}*.*The distinguished section is given by the composition

*τ*^{∗}*O**P**|**IP*ˇ^{3}(*−*1)*,→τ*^{∗}*Sym*^{2}*p*^{∗}*S*^{∨}*−→Sym*^{2}*O** ^{H}*(

*−*1)

^{∨}*.*The universal family over

*P*is the zero locus of this section.

Restricting the projection *P* *−→* *IP*ˇ^{3} to the parameter space *X* *⊂* *P* of *tangent* conics we get a
map*X* *−→IP*ˇ^{3}assigning to each conic the plane which it sits in. In Section 1.2 the general fibres
of this projection are studied.

1

### 1.2 Tangent conics at a plane quartic

Throughout this section, let*B* be a plane quartic curve with at most one ordinary node (P_{0}) as
singularity. A tangent conic at *B* is an effective Cartier divisor *C* of degree 2 on *IP*^{2} such that
*B* *·C* = 2D where *D* is a (Weil) divisor on *B. Since* *H*^{0}(IP^{2}*,O**IP*ˇ^{2}(2)) *∼*= *H*^{0}(B,*O**B*(2)) *C* is
uniquely determined by its restriction to*B. Therefore, finding all conicsC* that are tangent to*B*
is the same as finding all divisors*D*of degree 4 on*B*such that 2D is Cartier and*O*(2D)*∼*=*O** ^{B}*(2).

Suppose first that*B*is*smooth. ThenD*is a Cartier divisor and we seek for line bundles*L*:=*O*(D)
satisfying*L*^{⊗}^{2}*∼*=*O** ^{B}*(2) and

*L 6∼*=

*O*

*(1). (The latter condition is to exclude the double lines from our search.) All line bundles*

^{B}*L*with these properties are of the form

*O*

*(1)*

^{B}*⊗ L*

^{0}with

*L*

^{0}

*∈J*2(B) and

*L*0

*6∼*=

*O*

*B*(where

*J*

_{2}(B) denotes the subgroup of elements of order two in Pic(B)). As Pic

^{0}(B) is isomorphic to the torus C

^{g}Z^{2g} with *g* = genus(B) = 3 there are 2^{6}*−*1 = 63 line bundles
*L 6∼*=*O**B* satisfying*L*^{⊗}^{2}*∼*=*O**B*(2).

To get a complete overview over the variety of tangent conics we next have to determine the number
of sections of the line bundles *L*. For this purpose first observe that the canonical sheaf*ω**B* of *B*
is isomorphic to*O** ^{B}*(1) (by adjunction formula). Now by Riemann-Roch

*h*^{0}(*L*) =*h*^{0}(*L*^{−}^{1}*⊗ω** _{B}*) + deg

*L*+ 1

*−g*=

*h*

^{0}(

*L*

^{−}^{1}

*⊗ω*

*) + 4*

_{B}*−*2

*≥*2.

If*h*^{0}(*L*) were greater than two then*L*would be special and of degree 4 = deg(ω*B*) which is only
possible if*L ∼*=*ω**B* *∼*=*O** ^{B}*(1). Hence, for each of the

*L*with the above properties the complete linear system

*|L|*has dimension one and thus each

*L*determines a one-parameter family of tangent conics.

Let*{s, t} ⊂H*^{0}(*L*) be a basis. Then the linear system*|L|*is the set of divisors*D*_{(λ:µ)}=*Z(λs*+*µt)*
((λ:µ)*∈IP*^{1}). For each (λ:µ)*∈IP*^{1} the corresponding tangent conic is given by the section

(λs+*µt)*^{2}*∈H*^{0}(B,*L*^{⊗}^{2}) =*H*^{0}(B,*O**B*(2))*∼*=*H*^{0}(IP^{2}*,O**IP*^{2}(2)).

This one-parameter family *λ*^{2}*s*^{2}+ 2λµ st+*µ*^{2}*t*^{2} is embedded in *IP*(H^{0}(IP^{2}*,O*^{IP}^{2}(2)))*∼*=*IP*^{5} as a
smooth conic. Otherwise the three sections *s*^{2}, *t*^{2}, and *st* would be linearly dependent and the
mapping

*IP*^{1}*∼*=*|L| −→* *IP H*^{0}(B,*L*^{⊗}^{2})*∼*=*IP H*^{0}(IP^{2}*,O**IP*^{2}(2))
(λ:*µ)* *7−→* *λ*^{2}*s*^{2}+ 2λµ st+*µ*^{2}*t*^{2}

would be a double cover of the line spanned by *s*^{2} and *t*^{2} in *IP H*^{0}(IP^{2}*,O*^{IP}^{2}(2)). But then each
of the conics (except two of them) parametrised by that line would be the image of two *different*
divisors of the linear system *|L|*. This is impossible as for each conic its intersection with *B* is
*uniquely*determined.

Let, again,*L* be one of the 63 bundles, *{s, t} ⊂H*^{0}(*L*) be a basis, and*s*^{2}*, st, t*^{2}*∈H*^{0}(B,*L*^{⊗}^{2})*∼*=
*H*^{0}(IP^{2}*,O**IP*^{2}(2)). After a choice of a basis in*IP*^{2}, the three sections correspond to 3*×*3-matrices
*A** _{s}*2,

*A*

*st*, and

*A*

*2 and the one-parameter family of tangent conics is given by*

_{t}*λ*^{2}*A** _{s}*2+ 2λµ A

*+*

_{st}*µ*

^{2}

*A*

*2*

_{t}The determinant of the above matrix is a homogeneous polynomial of degree six in (λ: *µ) and,*
hence, there are at most six singular conics in each one-parameter family. These singular conics
cannot be double lines as the double lines correspond to the line bundle *L* =*O** ^{B}*(1). Therefore,
each one-parameter family contains up to six reducible conics which must consist of two double
tangents. Now, as the families are necessarily disjoint and as there are

^{28}

_{2}

= 63*·*6 pairs of double
tangents^{1}) and consequently each of the 63 families must contain exactly six reducible elements.

1Any smooth plane quartic curve has exactly 28 double tangents (or lines with fourth order contact^{2})) cf. [GH]

Section 4.4

2Those lines of fourth order contact will always be considered to be bitangents. So if we talk about bitangents then lines of fourth order contact are always meant to be included.

*1.2. TANGENT CONICS AT A PLANE QUARTIC* 3

Summerising, we have proven the

Proposition 1.1 *For any smooth plane quartic the variety of tangent conics is the disjoint union*
*of 63 one-parameter families which are embedded inIP*^{5}*∼*=*IP*(H^{0}(IP^{2}*,O**IP*^{2}(2)))*as smooth conics.*

*Each of the families contains exactly six reducible conics each of which is the union of two (different)*

*double tangents.*

Next, we want to treat quartics with an ordinary double point in a similar manner. For this purpose we need the following (technical) lemma.

Lemma 1.2 *Let* *B* *be a quartic inIP*^{2} *with exactly one ordinary nodeP*0 *and letσ:B*e*→B* *be the*
*normalisation. Then:*

*(i) There are exactly two reduced points* *P*1 *andP*2 *in* *B*e *overP*0*.*

*(ii) Let* m*P*0 *be the ideal sheaf of the point* *P*_{0} *∈* *B* *(i.e.* m*P*0 *is the sheaf whose stalk in* *P*_{0}
*is the maximal ideal of* *O**B,P*0 *and which is isomorphic to* *O**B* *outside* *P*_{0}*). Then* m_{P}_{0} =
*σ*_{∗}*O**B*e(*−P*_{1}*−P*_{2}) *and*m^{2}_{P}

0=*σ*_{∗}*O**B*e(*−*2P_{1}*−*2P_{2}).

*(iii)* *ω*_{B}_{e}=*σ*^{∗}*ω**B*(*−P*1*−P*2)*is the canonical sheaf of* *B.*e

*(iv) There is a Cartier divisorE* *with associated Weil divisor*2P0 *and*m^{2}_{P}

0*O*(E) =m*P*_{0}*.*
To determine the variety of tangent conics at a plane quartic with an ordinary double point we
first try to proceed as in the non-singular case. Let *D* be a (Weil) divisor on *B* such that 2D is
Cartier and*O*(2D)*∼*=*O**B*(2). Suppose that*D*itself is Cartier (i.e.,*D*is the Weil divisor associated
to a Cartier divisor). Then the method used for the case of a smooth quartic works with slight
modifications.

For counting the elements of order two in the Picard group of*B* we consider the exact sequence
1*−→ O**B*^{∗}*−→* *σ*_{∗}*O*^{∗}_{B}_{e} *−→* C^{∗}*−→*1.

*f* *7−→* ^{f(P}*f(P*^{1}_{2}^{)})

From the corresponding cohomology sequence we get

1*−→*C^{∗}*−→*Pic(B)*−→*Pic(*B)*e *−→*1 (1.1)

and by applying Hom(IF_{2}*,−*) we get

1*−→ {±*1*} −→* ^{2}Pic(B)*−→* ^{2}Pic(*B)*e *−→*Ext^{1}(IF2*,*C* ^{∗}*) = 0 (1.2)
(where2

*A*denotes the elements of order two in the group

*A).*

*J*2(

*B) consists of 16 elements and*e consequently there are 31 line bundles

*L*different from

*O*

*(1) whose square equals*

^{B}*O*

*(2). As above the dimension of the corresponding complete linear systems*

^{B}*|L|*is determined to be one.

Choosing a basis *{s, t} ∈H*^{0}(B,*L*) we – again – get a one-parameter family

*λ s*^{2}+ 2λµ st+*µ t*^{2} (λ:*µ)∈IP*^{1} *⊂H*^{0}(B,*L*^{⊗}^{2})*∼*=*H*^{0}(IP^{2}*,O*^{IP}^{2}(2))

of tangent conics. By the same argument as above this one-parameter family is embedded as
smooth conic in *IP*^{5}*∼*=*IP*(H^{0}(IP^{2}*,O**IP*^{2}(2))).

In the present (singular) case we have one “exceptional” linear system: From the exact se-
quence (1.2) it is clear that there is one nontrivial line bundle *L*^{0} in the kernel of 2Pic(B) *−→*^{σ}^{∗}

2Pic(*B*e). If*L*:=*L*^{0}*⊗ O** ^{B}*(1) is the corresponding “square root” of

*O*

*(2) then*

^{B}*P*0 is a base point of the linear system

*|L|*. This follows from:

*h*^{0}(m*P*_{0}*L*) = *h*^{0}(σ* _{∗}*(σ

^{∗}*L*(

*−P*1

*−P*2))) =

*h*

^{0}(σ

^{∗}*L*(

*−P*1

*−P*2))

= *h*^{0}(ω_{B}_{e}*⊗σ*^{∗}*L*^{−}^{1}(P1+*P*2)) + deg*L −*2 + (1*−*e*g)*

= 1 +*h*^{0}(σ^{∗}*ω**B**⊗σ*^{∗}*L*^{−}^{1})

= 1 +*h*^{0}(σ^{∗}*L*0).

(1.3)

(notation as in Lemma 1.2). (In particular, this special*L*is the only one among the 31 line bundles
found above that has*P*_{0} as base point.) Then, again using Lemma 1.2, we have

m*P*_{0}*⊗ L* = *σ*_{∗}*O**B*e(*−P*1*−P*2)*⊗ω**B**⊗ L*^{0}

= *σ*_{∗}*σ*^{∗}*ω**B**⊗ O**B*e(*−P*1*−P*2)

= *ω**B**⊗σ*_{∗}*O**B*e(*−P*1*−P*2)

= *ω*_{B}*⊗*m_{P}_{0}

= m*P*_{0}*⊗ O** ^{B}*(1)
and, consequently,

*H*^{0}(*L*) =*H*^{0}(m_{P}_{0}*L*) =*H*^{0}(m_{P}_{0}*O**B*(1)).

Thus the elements of*|L|*correspond to the*double lines*through*P*0. (Though we have excluded the
case*L*=*O**B*(1) we still can obtain Weil divisors corresponding to Cartier divisors with line bundle
*O**B*(1), for on singular varieties a Weil divisor may be represented by different Cartier divisors.)
For divisors with support in the regular locus of*B* the correspondence between Weil divisors and
Cartier divisors is bijective. In particular, the one-parameter families arising from the other 30
line bundles are all different from each other and different from divisors corresponding to double
lines: None of the corresponding linear systems has base points, hence, each of them contains an
element with support in the regular locus of*B* and the one-parameter families must be different.

(This does not mean that these one-parameter families are disjoint. In fact they come in 15 pairs
of families that intersect in one element.) To show the 30 line bundles being base point free, recall
that *P*0 is not a base point of any of these bundles (which followed from (1.3)). But a smooth
point*P* *∈B*cannot be a base point of any of the 31 line bundles. Suppose*P* were a base point of

*|L|*. Then

*h*^{0}(*L*(*−P*)) =*h*^{0}(*L*) = 2
and by Riemann-Roch we would have

*h*^{0}(*L*(*−P*))*−h*^{0}(*L*^{−}^{1}(P)*⊗ω**B*) =*h*^{0}(*L*(*−P))−h*^{0}(*L*^{0}(P)) = 1

where *L*0*∈J*_{2}(B) is the bundle*L ⊗ O**B*(*−*1) (i.e. *L*=*ω*_{B}*⊗ L*0). Thus *h*^{0}(*L*0(P)) = 1 and from
the corresponding section we would get the sequence

0*−→ L** ^{−}*0

^{1}(

*−P*)

*−→ O*

^{B}*−→ F −→*0

with a torsion sheaf*F*of length 1. Consequently,*L** ^{−}*0

^{1}(

*−P*) would be the ideal sheaf of a point

*Q∈*

*B, henceL*0(P)

*∼*=

*O*

*B*(Q). Finally

*O*

*B*

*∼*=

*L*

*0*

^{⊗}^{2}

*∼*=

*O*

*B*(2Q

*−*2P) and

*B*would be hyperelliptic – in contradiction to the ampleness of

*ω*

_{B}*∼*=

*O*

*B*(1). Therefore,

*|L|*cannot have base points.

It remains to consider the case where the divisor *D* (satisfying *O**B*(2D)*∼*=*O**B*(2)) is *not*Cartier.

Then*D*contains the singular point*P*_{0}of*B*with an odd coefficient and*D+P*_{0}is Cartier. Then*L*:=

*1.2. TANGENT CONICS AT A PLANE QUARTIC* 5

*O** ^{B}*(D+

*P*0) must satisfy

*L*

^{⊗}^{2}

*∼*=

*O*

*(2)(2P0). First, we again determine the set of bundles*

^{B}*L*with

*L*

^{⊗}^{2}

*∼*=

*O*

*(2)(2P0). For this purpose consider the sequence (1.1). The pull-back*

^{B}*σ*

^{∗}*O*

*(2)(2P0) has a square root in Pic(*

^{B}*B) (since it is of even degree and Pic(*e

*B) is an extension of the divisible*e group Pic

^{0}(

*B*e) withZ). So

*O*

*B*(2)(2P

_{0}) has a square root

*L*1 in Pic(B) since C

*is divisible. All square roots of*

^{∗}*O*

*B*(2)(2P

_{0}) are obtained by multiplying

*L*1 by the 32 elements of

_{2}Pic(B).

To get a complete overview over the corresponding tangent conics, we again consider the linear
systems associated to the *L* (with*L*^{⊗}^{2} *∼*=*O**B*(2)(2P_{0})). This time we don’t have to consider the
complete linear system*|L|* but only the space of sections vanishing in *P*_{0}, i.e. *H*^{0}(m_{P}_{0}*L*). From
Lemma 1.2 we easily deduce m*P*_{0}*L ∼*=*σ*_{∗}*σ*^{∗}*L*(*−P*1*−P*2) and hence

*h*^{0}(m*P*_{0}*L*) = *h*^{0}(σ^{∗}*L*(*−P*1*−P*2))

= *h*^{1}(σ^{∗}*L*(*−P*_{1}*−P*_{2})) + deg*σ*^{∗}*L*(*−P*_{1}*−P*_{2}) + 1*−*e*g*

= 2

as*h*^{1}(σ^{∗}*L*(*−P*1*−P*2)) = 0 since deg*σ*^{∗}*L*(*−P*1*−P*2) = 3*>*2e*g−*2. If*s∈H*^{0}(m*P*_{0}*L*) is any section
then*s*^{2} is contained in

*H*^{0}(σ^{∗}*L*^{⊗}^{2}(*−*2P_{1}*−*2P_{2})) = *H*^{0}(σ* ^{∗}*(

*O*

*B*(2)(2P

_{0}))(

*−*2P

_{1}

*−*2P

_{2}))

= *H*^{0}(m^{2}_{P}_{0}*O*(2P0)*O** ^{B}*(2))

= *H*^{0}(m*P*_{0}*O** ^{B}*(2)).

This shows that we obtain one-parameter families of tangent conics in this case as well. By
the same argument as in the smooth case these one-parameter families are embedded in *IP*^{5} *∼*=
*IP*(H^{0}(IP^{2}*,O*^{IP}^{2}(2))).

Now, from the equality of m*P*_{0}*L* and*σ*_{∗}*σ*^{∗}*L*(*−P*1*−P*2) follows that two line bundles which differ
by the non-trivial element in the kernel of 2Pic(B) *−→*^{σ}^{∗}^{2}Pic(*B) yield the same one-parameter*e
family. Consequently, we obtain 16 further families of tangent conics giving a total of 46 one-
parameter families of non-degenerated tangent conics. The following proposition summarises the
above discussion.

Proposition 1.3 *Let* *B* *be a plane quartic with exactly one ordinary double point as its only*
*singularity. Then the variety of those tangent conics at* *B* *that are not double lines consists of*
*46 conics in* *IP*^{5} *∼*= *IP*(H^{0}(IP^{2}*,O**IP*^{2}(2))). There are 16 of these one-parameter families whose
*corresponding tangent conics all contain the singular point.*

We next want to show that none of these 46 one-parameter families contains double lines. For that purpose we need the following lemma.

Lemma 1.4 *Let* *B* *be any (plane) curve with an ordinary node* *P*0 *as its only singularity. LetD*
*be an effective Weil divisor which contains* *P*0 *with coefficient 2 or 4. Then there are exactly two*
*Cartier divisors* *D*1 *andD*2 *such thatD* *is the Weil divisor associated to* *D*1 *as well as toD*2 *and*
2D1= 2D2*.*

Proof: We only have to consider the local equations in *P*0for the divisors in question. The local
ring*O** ^{B,P}*0of

*B*in

*P*0is isomorphic toC[X, Y]

_{(X,Y}

_{)}

(Y^{2}*−εX*^{2}) (where*ε*is a unit inC[X, Y]_{(X,Y}_{)}).

An*f* in that ring defining the divisor 2P0 must be of the form*f* =*e*1*X* +*e*2*Y* where*e**i* is either
zero or a unit of the local ring (i= 1,2). To find the local equations of*D*1and*D*2 in*P*0we have
to find two elements *f*1 and *f*2 of that form such that their quotient is not a unit and that their
squares only differ by a unit. A trivial calculation shows that (up to units) the only solution of
this problem is*f*_{1}=*X* and*f*_{2}=*Y*.

For the case of 4P_{0} the local equations must be of the form *f* = *e*_{1}*X*^{2}+*e*_{2}*XY* and the same

argument works.

Proposition 1.5 *Let* *B* *be a plane quartic curve with exactly one ordinary double point as its*
*only singularity. Then none of the 46 one-parameter families of tangent conics constructed above*
*contains double lines.*

Proof: First observe that a double line in the families in question must contain the singular point
*P*0 of*B: A line inIP*^{2} not through*P*0 intersects*B* in a Weil divisor with support in the regular
locus of*B. For those divisors there is exactly one Cartier divisor with the same associated cycle.*

Now, consider the 30 one-parameter families which arose from Cartier divisors*D* with*O**B*(2D)*∼*=
*O**B*(2). The corresponding linear system was one dimensional and base point free. Hence, each
of these one-parameter families contains exactly one conic which meets *P*_{0}. By Lemma 1.4 this
conic is contained in two of the 30 families. (And this conic is the only element that the two
one-parameter families have in common.) Now, for a plane quartic with one node there are six
lines through the node that are tangent at the quartic at another point (resp. which intersect *B*
only in *P*0). (This fact can already be found in the book of George Salmon [Sa].) From these six
lines we get fifteen reducible conics each of which intersects*B*in “twice a Cartier divisor”. Hence
the only element of each of the 30 families which meets*P*0is a singular conic which is*not*a double
line.

The tangent conics of the remaining 16 one-parameter families all contain the singular point*P*0

of *B. The reducible conics consisting of a bitangent through* *P*0 (of which there are six) and a
bitangent which does not meet *P*0 must be contained in these families. There are 16 bitangents
at *B* which do not pass through *P*0. (This follows from the Pl¨ucker formulas (cf. [Sa]) or from
the theory of theta-characteristics on singular curves – cf. [Har2].) Thus, there are 6*·*16 reducible
conics which must be contained in these 16 one-parameter families. Now, recall that these families
are embedded in *IP*^{5} *∼*= *IP*(H^{0}(IP^{2}*,O*^{IP}^{2}(2))) as smooth conics and, hence, intersect the variety
of singular conics (which is of degree three) in at most six points: None of the families can be
entirely contained in the set of singular conics since then the family would contain only double
lines (as there are only finitely many reducible tangent conics). But there*are*reducible elements
in the families. Hence, there are at most 16*·*6 singular conics in these 16 families which must
be just the 96 reducible conics consisting of two bitangents one of which contains the node of *B.*

Consequently, also these one-parameter families cannot contain double lines.

Remark: Lemma 1.4 equally applies to the family of double lines: The double lines through*P*0

occur in exactly two linear series – namely in the series of elements in *|O** ^{B}*(1)

*|*which contain

*P*0

and in*|O** ^{B}*(1)

*⊗ L*

^{0}

*|*where

*L*

^{0}is the non-trivial element in the kernel of2Pic(B)

*−→*

^{σ}

^{∗}^{2}Pic(

*B).*e

### 1.3 Tangent conics and double covers

In this section another description of tangent conics is to be given. Let*B*be a smooth quartic curve.

Denote by*Z**B* the double cover of*IP*^{2} branched along *B* and by*π:Z**B**−→IP*^{2} the corresponding
morphism. By [GH] Chapter 4.4,*Z**B* is isomorphic to the blow-up of*IP*^{2}in seven points and *π*is
induced by the anticanonical linear systemω^{−}_{Z}^{1}

*B*

. *π*maps the 56 (*−*1)-curves^{3}) in*Z** _{B}* onto the 28
double tangents of

*B*so that the preimage under

*π*of a double tangent consists of two (

*−*1)-curves.

Those pairs*C,C** ^{0}* of (

*−*1)-curves lying over the same double tangent are just the pairs satisfying

*C·C*

*= 2 and (equivalently) [C+*

^{0}*C*

*] = [ω*

^{0}

_{Z}

^{−}^{1}

*B*].

Now, let *C*1 and*C*2 be two (*−*1)-curves that have different images under*π*(i.e. *π(C*1)*6*=*π(C*2))
which means that they are (*−*1)-curves over different double tangents of*B. This is equivalent to*

3By a (*−*1)-curve we always mean a smooth rational curve*C**⊂**Z**B* (resp. the corresponding class in Pic(Z*B*))
such that*C*^{2}=*C**·**ω*_{Z}* _{B}*=

*−*1.

*1.3. TANGENT CONICS AND DOUBLE COVERS* 7

[C1+*C*2]*6*= [ω^{−}_{Z}^{1}

*B*]. Let*C*_{1}* ^{0}* and

*C*

_{2}

*be the (*

^{0}*−*1)-curves defined by [C

*i*+

*C*

_{i}*] = [ω*

^{0}

^{−}

_{Z}^{1}

*B*] (i.e. *C**i*and*C*_{i}* ^{0}*
form a pair of (

*−*1)-curves over the same double tangent). Then these curves intersect as follows:

*C*1*·C*2=*C*_{1}^{0}*·C*_{2}* ^{0}* = 1

*−C*1

*·C*

_{2}

*= 1*

^{0}*−C*

_{1}

^{0}*·C*2

*.*This follows from

1 =*C*1*·ω*^{−}_{Z}^{1}

*B*=*C*1*·*(C2+*C*_{2}* ^{0}*) =

*C*1

*·C*2+

*C*1

*·C*

_{2}

^{0}and analogous identities. Therefore, by eventually exchanging *C*1 and *C*_{1}* ^{0}*, one can achieve that

*C*1

*·C*2= 1 (leaving the corresponding double tangents unchanged).

Proposition 1.6 *If* *C*1 *and* *C*2 *are chosen as above with* *C*1 *·C*2 = 1 *then the linear system*

*|C*_{1}+*C*_{2}*|is one-dimensional. Its generic element is a smooth rational curve that by the projection*
*π:Z*_{B}*−→IP*^{2} *is mapped to a tangent conic.*

Proof: For any of the 56 (*−*1)-curves*C* of*Z**B* consider the exact sequence
0*−→ O*^{Z}*B**−→ O*^{Z}*B*(C)*−→ O** ^{C}*(C)

*−→*0.

Since*C*is a smooth rational curve with *O**C*(C) =*O**C*(C*·C) =O**C*(*−*1) and since*Z** _{B}* is a smooth
rational surface so that 0 =

*h*

^{1}(

*O*

*Z*

*B*) =

*h*

^{1}(

*O*

*Z*

*B*(C)) we get

*h*

^{0}(

*O*

*Z*

*B*) =

*h*

^{0}(

*O*

*Z*

*B*(C)) = 1. Now, the exact sequence

0*−→ O**Z**B*(C_{2})*−→ O**Z**B*(C_{1}+*C*_{2})*−→ O**C*1(C_{1}+*C*_{2})*−→*0
and the fact that*O** ^{C}*1(C1+

*C*2) =

*O*

*1 (since*

^{C}*C*1

*·*(C1+

*C*2) = 0) yield

*h*^{0}(*O*^{Z}*B*(C1+*C*2)) =*h*^{0}(*O*^{Z}*B*(C2)) +*h*^{0}(*O** ^{C}*1) = 2
and, hence, dim

*|C*1+

*C*2

*|*= 1.

*|C*1+*C*2*|* cannot have a fixed component. Since*C*1 and*C*2 are irreducible this fixed component
would have to be one of these two curves and then the linear system would only contain the divisor
*C*1+*C*2in contradiction to the dimension of the system being one. Hence, as (C1+*C*2)^{2}= 0, the
system cannot have base points at all.

By Bertini’s Theorem the generic element of*|C*1+*C*2*|*is smooth away from the base locus. As the
base locus is empty the generic element of the system is smooth everywhere. Let *C* *∈ |C*1+*C*2*|*
be a general and in particular smooth element. We claim that*C* has to be connected and, hence,
irreducible. Consider the morphism *Z**B*

*−→**ψ* *IP*^{1} induced by the linear system *|C*1+*C*2*|*. If the
general element of the linear system were not connected then for general*y∈IP*^{1}

*h*^{0}(y,*O*^{Z}*B*) := dim_{k(y)}*H*^{0}(C*y**,*(*O*^{Z}*B*)*y*) = dim_{k(y)}*H*^{0}(C*y**,O*^{C}*y*)*>*1

where *C** _{y}* is the fibre of

*ψ*over

*y*

*∈IP*

^{1}. But the element

*C*

_{1}

*∪C*

_{2}

*∈ |C*

_{1}+

*C*

_{2}

*|*is connected and, therefore,

*h*

^{0}(y

_{0}

*,O*

*Z*

*B*) = 1 for the corresponding

*y*

_{0}

*∈IP*

^{1}– which is impossible by semicontinuouity ([Hart] Theorem III.12.8). (This argument shows that

*every*element of the linear system

*|C*

_{1}+C

_{2}

*|*must be connected.)

From Adjunction formula we get for generic*C∈ |C*1+*C*2*|*
genus(C) =*ω**Z*_{B}*·C*+*C*^{2}

2 + 1 = 0

so that the generic element of the linear system is a smooth rational curve.

Now, let*R⊂Z**B*be the ramification divisor of the map*Z**B**−→IP*^{2}. *O*^{Z}*B*(R) =*π** ^{∗}*(

*O*

*IP*

^{2}(2)) =

*ω*

_{Z}

^{−}^{2}

*B*

([BPV] Section I.17). Hence

(C_{1}+*C*_{2})*·*[R] = 2 (C_{1}+*C*_{2})*·ω*_{Z}^{−}^{1}

*B* = 4
and by projection formula

4 [pt] =*π** _{∗}*((C

_{1}+

*C*

_{2})

*·*[R]) =

*π*

*((C*

_{∗}_{1}+

*C*

_{2})

*·π*

*(2 [*

^{∗}*l*])) =

*π*

*(C*

_{∗}_{1}+

*C*

_{2})

*·*(2 [

*l*])

where [pt] denotes the class of a point and [*l*] the class of a line in*IP*^{2}. Therefore*π** _{∗}*(C1+

*C*2)

*∈*

*|*2 [*l*]*|*. Let *C* *∈ |C*1+*C*2*|* be a general element. If *π(C) were a (double) line then* *C* would be
contained in the preimage of a line. As [C1+*C*2]*6*= [ω_{Z}^{−}^{1}

*B*] there would exist an effective *C** ^{0}* such
that [C] + [C

*] = [ω*

^{0}

^{−}

_{Z}^{1}

*B*]. But then*C*^{0}*·ω*_{Z}^{−}^{1}

*B* = (ω_{Z}^{−}^{1}

*B*)^{2}*−C·ω*^{−}_{Z}^{1}

*B* = 0 which is impossible as*ω*^{−}_{Z}^{1}

*B* is
ample. So the image of*C* under *π*must be a smooth conic and*π|**C* is of degree one onto*π(C).*

Consider now the preimage*π*^{−}^{1}(π(C)) in*Z**B*. As*π*is a double cover and*π|** ^{C}*is only of degree one,

*π*

^{−}^{1}(π(C)) must either contain other components than

*C*or

*π(C) must be contained in*

*B. The*latter is not possible since

*B*was supposed to be a smooth quartic curve. Since

*π(C) is an element*of

*|O*

^{IP}^{2}(2)

*|*and since

*π:Z*

*B*

*−→IP*

^{2}is induced by the anticanonical linear system

*π*

^{−}^{1}(π(C)) is an element of

*|ω*

_{Z}

^{−}^{2}

*B**|*. Therefore, the sum of the other components of *π*^{−}^{1}(π(C)) must be an element
of *| −*2 [ω_{Z}* _{B}*]

*−*[C

_{1}+

*C*

_{2}]

*|*=

*|C*

_{1}

*+*

^{0}*C*

_{2}

^{0}*|*. (C

_{i}*was defined to be the (*

^{0}*−*1)-curve in

*Z*

*such that [C*

_{B}*+*

_{i}*C*

_{i}*] = [ω*

^{0}

^{−}

_{Z}^{1}

*B*].)

Let*C*^{0}*∈ |C*_{1}* ^{0}*+C

_{2}

^{0}*|*be the divisor which is complementary to

*C*in

*π*

^{−}^{1}(π(C)). Note that (C

_{1}

*+C*

^{0}_{2}

*), as well as (C*

^{0}_{1}+

*C*

_{2}), is the sum of two (

*−*1)-curves with intersection

*C*

_{1}

^{0}*·C*

_{2}

*= 1. Therefore, a general element of*

^{0}*|C*

_{1}

*+*

^{0}*C*

_{2}

^{0}*|*is also a smooth rational curve which by

*π*is mapped onto a smooth conic in

*IP*

^{2}. So if

*C*

*∈ |C*

_{1}+

*C*

_{2}

*|*is sufficiently general then

*C*

*is a smooth rational curve that is mapped onto a smooth conic. Therefore,*

^{0}*π*

^{−}^{1}(π(C)) splits into two components each of which is a smooth rational curve.

Now, to prove that*π(C) is a tangent conic atB* let Spec*A*=*U* *⊂π(C) be an open subset ofπ(C),*
*f* *∈A*the equation of*B* restricted to *U, and* *A**Z* :=*A[T*]

(T^{2}*−f*). Then Spec*A**Z* *∼*=*π*^{−}^{1}(U)*⊂*
*π*^{−}^{1}(π(C)) and Spec*A**Z* is reducible if and only if (T^{2}*−f*) is reducible in*A[T], i.e., if and only if*
there is a*g∈A*with*f* =*g*^{2}. Hence*π(C) is a tangent conic and the proposition is proved.*

Corollary 1.7 *π:Z**B* *−→IP*^{2} *maps any of the linear systems* *|C*1+*C*2*|* *(C*1 *andC*2 *as above) to*
*a one-parameter family of tangent conics. For each one-parameter family there are two of these*
*linear systems which are mapped to that family. Each of the linear systems contains exactly six*
*reducible elements which are the union of two*(*−*1)-curves that intersect each other.

Proof: By Proposition 1.6 *π* induces a morphism *|C*1+*C*2*| ∼*= *IP*^{1} *−→* *IP*^{5} (where *IP*^{5} is the
parameter space of conics in*IP*^{2}). This morphism is necessarily injective as for*C∈ |C*1+*C*2*|*the
preimage of *π(C) only consists of* *C* and an element of the linear system *|*[ω_{Z}^{−}^{1}

*B*]*−*[C]*|* which is
different from*|C*_{1}+C_{2}*|*. There is an open subset in*|C*_{1}+C_{2}*|*which is mapped into the closed subset
of tangent conics in*IP*^{5}. Therefore each element of*|C*_{1}+*C*_{2}*|* is mapped onto a (maybe singular)
tangent conic. Note that none of the elements in *|C*_{1}+*C*_{2}*|* can be mapped to a (double) line.

Furthermore, a reducible tangent conic is the union of two bitangents and an element of*|C*_{1}+*C*_{2}*|*
is mapped to a reducible conic if and only if it is the sum of two (*−*1)-curves. By Proposition 1.1
each one-parameter family of tangent conics contains six reducible elements and so there must be
six reducible elements in each linear system.

Conversely, for each one-parameter family there exist exactly two of these linear systems that are
mapped to this family: Let *C*_{1}, *C*_{1}* ^{0}* and

*C*

_{2},

*C*

_{2}

*be the four (*

^{0}*−*1)-curves over the two lines of a reducible conic of that family. If

*C*

_{1}

*·C*

_{2}= 1 then the two linear systems are just

*|C*

_{1}+

*C*

_{2}

*|*and

*|C*_{1}* ^{0}* +

*C*

_{2}

^{0}*|*(otherwise

*|C*

_{1}

*+*

^{0}*C*

_{2}

*|*and

*|C*

_{1}+

*C*

_{2}

^{0}*|*).

*1.4. CONICS IN DOUBLE SOLIDS* 9

Remark: Let*|L|*=*|C*1+*C*2*|*be a linear system (as above) which by Corollary 1.7 corresponds
to a one-parameter family of tangent conics. Then the other linear system which corresponds to
the same one-parameter family of tangent conics is*| −*2 [ω*Z** _{B}*]

*−*[

*L*]

*|*.

Next, we want to characterise the above linear systems on *Z**B* in a different way. Thereby it will
turn out, that they are related to the root systemE7.

Lemma 1.8 *The divisors* *Lof the form* *L*=*C*1+*C*2 *with* (*−*1)-curves *C**i* *satisfying* *C*1*·C*2= 1
*are characterised by the properties* *L*^{2}= 0*andL ·ω**Z** _{B}* =

*−*2.

Proof: Obviously the divisors*L*=*C*1+*C*2 satisfy*L*^{2}= 0 and*L ·ω**Z** _{B}* =

*−*2. On the other hand, there is a bijection

*L ∈*Pic(Z* _{B}*)

*| L*

^{2}= 0,

*L ·ω*

_{Z}*=*

_{B}*−*2

*←→*

*L*^{0}*∈*Pic(Z* _{B}*)

*| L*

^{0}^{2}=

*−*2,

*L*

^{0}*·ω*

_{Z}*= 0*

_{B}*L* *7−→* *L** ^{0}*:=

*ω*

_{Z}*⊗ L*

In [M] Chapter. IV.3. it is shown (Proposition 3.8 and Corollary 3.9) that there are exactly 126
line bundles *L*^{0}*∈* Pic(Z* _{B}*) with

*L*

^{0}^{2}=

*−*2 and

*L*

^{0}*·ω*

_{Z}*= 0. (They define a root system in (ω*

_{B}*Z*

*)*

_{B}

^{⊥}*⊗*

^{Z}

*IR∼*=

*IR*

^{7}(cf. [M]) which turns out to be of typeE7 – cf. Proposition 3.1 of the present paper.) By Corollary 1.7 and Proposition 1.1 there are 126 divisors of the form

*C*1+

*C*2 (two for each one-parameter family of tangent conics). This proves the lemma.

Remark: Consider again the map *Z**B* *−→* *IP*^{1} induced by the linear system *|C*1+*C*2*|*. By
comparing the Euler numbers of*Z**B*,*IP*^{1}and the fibre of that map one may determine the number
of singular elements in the linear system *|C*1+*C*2*|* in a way different from that used in the proof
of Lemma 1.7. Then, one may count the elements of the form*C*1+*C*2directly by determining the
number of pairs of intersecting (*−*1)-curves. This way, using arguments of the preceeding proofs,
one obtains an alternative proof of Proposition 1.1.

### 1.4 Conics in Double Solids

Using the results of the previous section, we now construct the parameter space*X** _{C}*of conics in a
double solid. (We will soon become more precise about what

*X*

*is meant to be.) Let*

_{C}*B⊂IP*

^{3}be a quartic surface with only isolated singularities. Denote by

*Z*

*−→*

^{π}*IP*

^{3}the double cover branched along

*B. The following facts onZ*may be found in [Kr2].

Lemma 1.9 *The canonical line bundle ofZ* *is the pull-back (via* *π) ofO**IP*^{3}(*−*2). In particular, it
*is locally free, its dual has a square rootO** ^{Z}*(1) :=

*ω*

^{−}1 2

*Z* *and the covering mapZ−→*^{π}*IP*^{3} *is induced*

*by the three-dimensional linear system−*^{1}2[ω*Z*].

Definition: *A “conic” in the double solid* *Z* *is a smooth rational curve* *C* *⊂* *Z* *such that*
*C· O** ^{Z}*(1)= 2.

Lemma 1.10 *Any conicC* *⊂Z* *is contained in an element* *S* *of the linear system|O**Z*(1)*|. If* *S*
*is smooth thenC* *satisfiesC*^{2}= 0 *andC·ω** _{S}* =

*−*2

*inS.*

Proof: The linear system *|O**Z*(1)*|* is three-dimensional, hence, through any three points in *Z*
there is an element of*|O**Z*(1)*|*containing these points. In particular, for a conic*C*in*Z* there is an