Tangent Conics at Quartic Surfaces and Conics in Quartic Double Solids

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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: . . . .

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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 ⊂ IP³. 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: IfZ−→IP³is the double cover branched along a quartic surface (= double solid), and ifH denotes the pull-back ofOIP³(1) toZ then a conic inZis a smooth rational curveFinZsatisfying 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 −→IP³ maps lines inZ 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!

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The only reference we could find where conics in double solids play a role is [SW]. They use conics inZ 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 ofZ. Moreover, most of the results mentioned above hold only for non-singular double covers of IP³. 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 spacesX andX_C are fibred over the projective three-space. After removing a critical set ∆ fromIP³the restricted fibration is smooth with disconnected fibres. Those irreducible components of these spaces which dominate IP³ are then the closure of (arc-)connected components of the restricted fibration. Thus if X −→X⁰ −→IP³ resp. X_C−→X_C⁰ −→IP³ denote the Stein factorisations then the irreducible components ofX resp. X_C(which dominateIP³) correspond to the monodromy orbits of the finite coversX|IP^ˇ³\∆−→IPˇ³\∆ resp. XC|IP^ˇ³\∆−→IPˇ³\∆.

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

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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.

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Tangent conics at a quartic surface

1.1 Parameter space of conics

In this section we will construct the parameter space of all conics inIP³. The space X of tangent conics will be contained within this parameter space. Let ˇIP³be the space of planes in IP³ andS the universal subbundle over ˇIP³ = Grass(3,4). Then P :=IP(Sym²S^∨) is the parameter space of all conics inIP³. (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 −→ IPˇ³ assigns to each conic the unique plane which it is contained in.

There is a universal family overP, constructed as follows: Let H :=IP(p^∗S) be the pull-back of the universal plane over ˇIP³,τ:H →P the projection, andOH(1) the relative tautological bundle ofH overP. Then there is a distinguished section in (τ^∗OP|IPˇ³(−1))^∨⊗ O^H(2), for we have

τ^∗OP|IPˇ³(−1) ∨

⊗ O^H(2) = Hom

τ^∗OP|IPˇ³(−1), O^H(−1)^∨⊗2

= Hom

τ^∗O_P_|IPˇ³(−1), Sym²(OH(−1))^∨ and there are canonical injections of vector bundles over H:

OH(−1),→τ^∗p^∗S.

τ^∗O_P_|IPˇ³(−1),→τ^∗p^∗(Sym²S^∨) =τ^∗Sym²(p^∗S)^∨. The distinguished section is given by the composition

τ^∗OP|IPˇ³(−1),→τ^∗Sym²p^∗S^∨−→Sym²O^H(−1)^∨. The universal family overP is the zero locus of this section.

Restricting the projection P −→ IPˇ³ to the parameter space X ⊂ P of tangent conics we get a mapX −→IPˇ³assigning to each conic the plane which it sits in. In Section 1.2 the general fibres of this projection are studied.

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1.2 Tangent conics at a plane quartic

Throughout this section, letB be a plane quartic curve with at most one ordinary node (P₀) as singularity. A tangent conic at B is an effective Cartier divisor C of degree 2 on IP² such that B ·C = 2D where D is a (Weil) divisor on B. Since H⁰(IP²,OIPˇ²(2)) ∼= H⁰(B,OB(2)) C is uniquely determined by its restriction toB. Therefore, finding all conicsC that are tangent toB is the same as finding all divisorsDof degree 4 onBsuch that 2D is Cartier andO(2D)∼=O^B(2).

Suppose first thatBissmooth. ThenDis a Cartier divisor and we seek for line bundlesL:=O(D) satisfyingL^⊗²∼=O^B(2) andL 6∼=O^B(1). (The latter condition is to exclude the double lines from our search.) All line bundlesLwith these properties are of the form O^B(1)⊗ L⁰withL⁰∈J2(B) andL06∼=OB (whereJ₂(B) denotes the subgroup of elements of order two in Pic(B)). As Pic⁰(B) is isomorphic to the torus C^g

Z^2g with g = genus(B) = 3 there are 2⁶−1 = 63 line bundles L 6∼=OB satisfyingL^⊗²∼=OB(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 sheafωB of B is isomorphic toO^B(1) (by adjunction formula). Now by Riemann-Roch

h⁰(L) =h⁰(L⁻¹⊗ω_B) + degL+ 1−g=h⁰(L⁻¹⊗ω_B) + 4−2≥2.

Ifh⁰(L) were greater than two thenLwould be special and of degree 4 = deg(ωB) which is only possible ifL ∼=ωB ∼=O^B(1). Hence, for each of theLwith the above properties the complete linear system|L|has dimension one and thus eachLdetermines a one-parameter family of tangent conics.

Let{s, t} ⊂H⁰(L) be a basis. Then the linear system|L|is the set of divisorsD_(λ:µ)=Z(λs+µt) ((λ:µ)∈IP¹). For each (λ:µ)∈IP¹ the corresponding tangent conic is given by the section

(λs+µt)²∈H⁰(B,L^⊗²) =H⁰(B,OB(2))∼=H⁰(IP²,OIP²(2)).

This one-parameter family λ²s²+ 2λµ st+µ²t² is embedded in IP(H⁰(IP²,O^IP²(2)))∼=IP⁵ as a smooth conic. Otherwise the three sections s², t², and st would be linearly dependent and the mapping

IP¹∼=|L| −→ IP H⁰(B,L^⊗²)∼=IP H⁰(IP²,OIP²(2)) (λ:µ) 7−→ λ²s²+ 2λµ st+µ²t²

would be a double cover of the line spanned by s² and t² in IP H⁰(IP²,O^IP²(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 uniquelydetermined.

Let, again,L be one of the 63 bundles, {s, t} ⊂H⁰(L) be a basis, ands², st, t²∈H⁰(B,L^⊗²)∼= H⁰(IP²,OIP²(2)). After a choice of a basis inIP², the three sections correspond to 3×3-matrices A_s2, Ast, andA_t2 and the one-parameter family of tangent conics is given by

λ²A_s2+ 2λµ A_st+µ²A_t2

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 ²⁸₂

= 63·6 pairs of double tangents¹) 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 contact²)) 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.

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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 inIP⁵∼=IP(H⁰(IP²,OIP²(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 inIP² with exactly one ordinary nodeP0 and letσ:Be→B be the normalisation. Then:

(i) There are exactly two reduced points P1 andP2 in Be overP0.

(ii) Let mP0 be the ideal sheaf of the point P₀ ∈ B (i.e. mP0 is the sheaf whose stalk in P₀ is the maximal ideal of OB,P0 and which is isomorphic to OB outside P₀). Then m_P₀ = σ_∗OBe(−P₁−P₂) andm²_P

0=σ_∗OBe(−2P₁−2P₂).

(iii) ω_B_e=σ^∗ωB(−P1−P2)is the canonical sheaf of B.e

(iv) There is a Cartier divisorE with associated Weil divisor2P0 andm²_P

0O(E) =mP₀. 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 andO(2D)∼=OB(2). Suppose thatDitself is Cartier (i.e.,Dis 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 ofB we consider the exact sequence 1−→ OB^∗ −→ σ_∗O^∗_B_e −→ C^∗ −→1.

f 7−→ ^f(Pf(P¹₂⁾)

From the corresponding cohomology sequence we get

1−→C^∗−→Pic(B)−→Pic(B)e −→1 (1.1)

and by applying Hom(IF₂,−) we get

1−→ {±1} −→ ²Pic(B)−→ ²Pic(B)e −→Ext¹(IF2,C^∗) = 0 (1.2) (where2A denotes the elements of order two in the group A). J2(B) consists of 16 elements ande consequently there are 31 line bundles L different from O^B(1) whose square equals O^B(2). As above the dimension of the corresponding complete linear systems |L|is determined to be one.

Choosing a basis {s, t} ∈H⁰(B,L) we – again – get a one-parameter family

λ s²+ 2λµ st+µ t² (λ:µ)∈IP¹ ⊂H⁰(B,L^⊗²)∼=H⁰(IP²,O^IP²(2))

of tangent conics. By the same argument as above this one-parameter family is embedded as smooth conic in IP⁵∼=IP(H⁰(IP²,OIP²(2))).

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In the present (singular) case we have one “exceptional” linear system: From the exact sequence (1.2) it is clear that there is one nontrivial line bundle L⁰ in the kernel of 2Pic(B) −→^σ^∗

2Pic(Be). IfL:=L⁰⊗ O^B(1) is the corresponding “square root” ofO^B(2) then P0 is a base point of the linear system|L|. This follows from:

h⁰(mP₀L) = h⁰(σ_∗(σ^∗L(−P1−P2))) =h⁰(σ^∗L(−P1−P2))

= h⁰(ω_B_e⊗σ^∗L⁻¹(P1+P2)) + degL −2 + (1−eg)

= 1 +h⁰(σ^∗ωB⊗σ^∗L⁻¹)

= 1 +h⁰(σ^∗L0).

(1.3)

(notation as in Lemma 1.2). (In particular, this specialLis the only one among the 31 line bundles found above that hasP₀ as base point.) Then, again using Lemma 1.2, we have

mP₀⊗ L = σ_∗OBe(−P1−P2)⊗ωB⊗ L⁰

= σ_∗ σ^∗ωB⊗ OBe(−P1−P2)

= ωB⊗σ_∗OBe(−P1−P2)

= ω_B⊗m_P₀

= mP₀⊗ O^B(1) and, consequently,

H⁰(L) =H⁰(m_P₀L) =H⁰(m_P₀OB(1)).

Thus the elements of|L|correspond to thedouble linesthroughP0. (Though we have excluded the caseL=OB(1) we still can obtain Weil divisors corresponding to Cartier divisors with line bundle OB(1), for on singular varieties a Weil divisor may be represented by different Cartier divisors.) For divisors with support in the regular locus ofB 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 ofB 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 P0 is not a base point of any of these bundles (which followed from (1.3)). But a smooth pointP ∈Bcannot be a base point of any of the 31 line bundles. SupposeP were a base point of

|L|. Then

h⁰(L(−P)) =h⁰(L) = 2 and by Riemann-Roch we would have

h⁰(L(−P))−h⁰(L⁻¹(P)⊗ωB) =h⁰(L(−P))−h⁰(L⁰(P)) = 1

where L0∈J₂(B) is the bundleL ⊗ OB(−1) (i.e. L=ω_B⊗ L0). Thus h⁰(L0(P)) = 1 and from the corresponding section we would get the sequence

0−→ L⁻0¹(−P)−→ O^B −→ F −→0

with a torsion sheafFof length 1. Consequently,L⁻0¹(−P) would be the ideal sheaf of a pointQ∈ B, henceL0(P)∼=OB(Q). FinallyOB ∼=L^⊗0²∼=OB(2Q−2P) andB would be hyperelliptic – in contradiction to the ampleness ofω_B∼=OB(1). Therefore,|L|cannot have base points.

It remains to consider the case where the divisor D (satisfying OB(2D)∼=OB(2)) is notCartier.

ThenDcontains the singular pointP₀ofBwith an odd coefficient andD+P₀is Cartier. ThenL:=

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1.2. TANGENT CONICS AT A PLANE QUARTIC 5

O^B(D+P0) must satisfyL^⊗²∼=O^B(2)(2P0). First, we again determine the set of bundlesLwith L^⊗² ∼= O^B(2)(2P0). For this purpose consider the sequence (1.1). The pull-back σ^∗O^B(2)(2P0) has a square root in Pic(B) (since it is of even degree and Pic(e B) is an extension of the divisiblee group Pic⁰(Be) withZ). So OB(2)(2P₀) has a square rootL1 in Pic(B) since C^∗ is divisible. All square roots ofOB(2)(2P₀) are obtained by multiplyingL1 by the 32 elements of₂Pic(B).

To get a complete overview over the corresponding tangent conics, we again consider the linear systems associated to the L (withL^⊗² ∼=OB(2)(2P₀)). This time we don’t have to consider the complete linear system|L| but only the space of sections vanishing in P₀, i.e. H⁰(m_P₀L). From Lemma 1.2 we easily deduce mP₀L ∼=σ_∗σ^∗L(−P1−P2) and hence

h⁰(mP₀L) = h⁰(σ^∗L(−P1−P2))

= h¹(σ^∗L(−P₁−P₂)) + degσ^∗L(−P₁−P₂) + 1−eg

= 2

ash¹(σ^∗L(−P1−P2)) = 0 since degσ^∗L(−P1−P2) = 3>2eg−2. Ifs∈H⁰(mP₀L) is any section thens² is contained in

H⁰(σ^∗L^⊗²(−2P₁−2P₂)) = H⁰(σ^∗(OB(2)(2P₀))(−2P₁−2P₂))

= H⁰(m²_P₀O(2P0)O^B(2))

= H⁰(mP₀O^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 IP⁵ ∼= IP(H⁰(IP²,O^IP²(2))).

Now, from the equality of mP₀L andσ_∗σ^∗L(−P1−P2) follows that two line bundles which differ by the non-trivial element in the kernel of 2Pic(B) −→^σ^∗ ²Pic(B) yield the same one-parametere 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 IP⁵ ∼= IP(H⁰(IP²,OIP²(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 P0 as its only singularity. LetD be an effective Weil divisor which contains P0 with coefficient 2 or 4. Then there are exactly two Cartier divisors D1 andD2 such thatD is the Weil divisor associated to D1 as well as toD2 and 2D1= 2D2.

Proof: We only have to consider the local equations in P0for the divisors in question. The local ringO^B,P0ofBinP0is isomorphic toC[X, Y]_(X,Y₎

(Y²−εX²) (whereεis a unit inC[X, Y]_(X,Y₎).

Anf in that ring defining the divisor 2P0 must be of the formf =e1X +e2Y whereei is either zero or a unit of the local ring (i= 1,2). To find the local equations ofD1andD2 inP0we have to find two elements f1 and f2 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 isf₁=X andf₂=Y.

For the case of 4P₀ the local equations must be of the form f = e₁X²+e₂XY and the same

argument works.

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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 P0 ofB: A line inIP² not throughP0 intersectsB in a Weil divisor with support in the regular locus ofB. 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 divisorsD withOB(2D)∼= OB(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 P₀. 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 P0). (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 intersectsBin “twice a Cartier divisor”. Hence the only element of each of the 30 families which meetsP0is a singular conic which isnota double line.

The tangent conics of the remaining 16 one-parameter families all contain the singular pointP0

of B. The reducible conics consisting of a bitangent through P0 (of which there are six) and a bitangent which does not meet P0 must be contained in these families. There are 16 bitangents at B which do not pass through P0. (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 IP⁵ ∼= IP(H⁰(IP²,O^IP²(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 therearereducible 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 throughP0

occur in exactly two linear series – namely in the series of elements in |O^B(1)| which containP0

and in|O^B(1)⊗ L⁰|where L⁰ is the non-trivial element in the kernel of2Pic(B)−→^σ^∗ ²Pic(B).e

1.3 Tangent conics and double covers

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

Denote byZB the double cover ofIP² branched along B and byπ:ZB−→IP² the corresponding morphism. By [GH] Chapter 4.4,ZB is isomorphic to the blow-up ofIP²in seven points and πis induced by the anticanonical linear systemω⁻_Z¹

B

. πmaps the 56 (−1)-curves³) inZ_B onto the 28 double tangents ofBso that the preimage underπof a double tangent consists of two (−1)-curves.

Those pairsC,C⁰ of (−1)-curves lying over the same double tangent are just the pairs satisfying C·C⁰ = 2 and (equivalently) [C+C⁰] = [ω_Z⁻¹

B].

Now, let C1 andC2 be two (−1)-curves that have different images underπ(i.e. π(C1)6=π(C2)) which means that they are (−1)-curves over different double tangents ofB. This is equivalent to

3By a (−1)-curve we always mean a smooth rational curveC⊂ZB (resp. the corresponding class in Pic(ZB)) such thatC²=C·ω_Z_B=−1.

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1.3. TANGENT CONICS AND DOUBLE COVERS 7

[C1+C2]6= [ω⁻_Z¹

B]. LetC₁⁰ andC₂⁰ be the (−1)-curves defined by [Ci+C_i⁰] = [ω⁻_Z¹

B] (i.e. CiandC_i⁰ form a pair of (−1)-curves over the same double tangent). Then these curves intersect as follows:

C1·C2=C₁⁰ ·C₂⁰ = 1−C1·C₂⁰ = 1−C₁⁰ ·C2. This follows from

1 =C1·ω⁻_Z¹

B=C1·(C2+C₂⁰) =C1·C2+C1·C₂⁰

and analogous identities. Therefore, by eventually exchanging C1 and C₁⁰, one can achieve that C1·C2= 1 (leaving the corresponding double tangents unchanged).

Proposition 1.6 If C1 and C2 are chosen as above with C1 ·C2 = 1 then the linear system

|C₁+C₂|is one-dimensional. Its generic element is a smooth rational curve that by the projection π:Z_B−→IP² is mapped to a tangent conic.

Proof: For any of the 56 (−1)-curvesC ofZB consider the exact sequence 0−→ O^ZB−→ O^ZB(C)−→ O^C(C)−→0.

SinceCis a smooth rational curve with OC(C) =OC(C·C) =OC(−1) and sinceZ_B is a smooth rational surface so that 0 = h¹(OZB) =h¹(OZB(C)) we get h⁰(OZB) =h⁰(OZB(C)) = 1. Now, the exact sequence

0−→ OZB(C₂)−→ OZB(C₁+C₂)−→ OC1(C₁+C₂)−→0 and the fact thatO^C1(C1+C2) =O^C1 (since C1·(C1+C2) = 0) yield

h⁰(O^ZB(C1+C2)) =h⁰(O^ZB(C2)) +h⁰(O^C1) = 2 and, hence, dim|C1+C2|= 1.

|C1+C2| cannot have a fixed component. SinceC1 andC2 are irreducible this fixed component would have to be one of these two curves and then the linear system would only contain the divisor C1+C2in contradiction to the dimension of the system being one. Hence, as (C1+C2)²= 0, the system cannot have base points at all.

By Bertini’s Theorem the generic element of|C1+C2|is smooth away from the base locus. As the base locus is empty the generic element of the system is smooth everywhere. Let C ∈ |C1+C2| be a general and in particular smooth element. We claim thatC has to be connected and, hence, irreducible. Consider the morphism ZB

−→ψ IP¹ induced by the linear system |C1+C2|. If the general element of the linear system were not connected then for generaly∈IP¹

h⁰(y,O^ZB) := dim_k(y)H⁰(Cy,(O^ZB)y) = dim_k(y)H⁰(Cy,O^Cy)>1

where C_y is the fibre ofψ over y ∈IP¹. But the element C₁∪C₂ ∈ |C₁+C₂| is connected and, therefore,h⁰(y₀,OZB) = 1 for the correspondingy₀∈IP¹– which is impossible by semicontinuouity ([Hart] Theorem III.12.8). (This argument shows thateveryelement of the linear system|C₁+C₂| must be connected.)

From Adjunction formula we get for genericC∈ |C1+C2| genus(C) =ωZ_B·C+C²

2 + 1 = 0

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

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Now, letR⊂ZBbe the ramification divisor of the mapZB−→IP². O^ZB(R) =π^∗(OIP²(2)) =ω_Z⁻²

B

([BPV] Section I.17). Hence

(C₁+C₂)·[R] = 2 (C₁+C₂)·ω_Z⁻¹

B = 4 and by projection formula

4 [pt] =π_∗((C₁+C₂)·[R]) =π_∗((C₁+C₂)·π^∗(2 [l])) =π_∗(C₁+C₂)·(2 [l])

where [pt] denotes the class of a point and [l] the class of a line inIP². Thereforeπ_∗(C1+C2)∈

|2 [l]|. Let C ∈ |C1+C2| be a general element. If π(C) were a (double) line then C would be contained in the preimage of a line. As [C1+C2]6= [ω_Z⁻¹

B] there would exist an effective C⁰ such that [C] + [C⁰] = [ω⁻_Z¹

B]. But thenC⁰·ω_Z⁻¹

B = (ω_Z⁻¹

B)²−C·ω⁻_Z¹

B = 0 which is impossible asω⁻_Z¹

B is ample. So the image ofC under πmust be a smooth conic andπ|C is of degree one ontoπ(C).

Consider now the preimageπ⁻¹(π(C)) inZB. Asπis a double cover andπ|^Cis only of degree one, π⁻¹(π(C)) must either contain other components thanC or π(C) must be contained in B. The latter is not possible sinceBwas supposed to be a smooth quartic curve. Sinceπ(C) is an element of|O^IP²(2)|and sinceπ:ZB−→IP²is induced by the anticanonical linear systemπ⁻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]−[C₁+C₂]| = |C₁⁰ +C₂⁰|. (C_i⁰ was defined to be the (−1)-curve in Z_B such that [C_i+C_i⁰] = [ω⁻_Z¹

B].)

LetC⁰∈ |C₁⁰+C₂⁰|be the divisor which is complementary toCinπ⁻¹(π(C)). Note that (C₁⁰+C₂⁰), as well as (C₁+C₂), is the sum of two (−1)-curves with intersection C₁⁰ ·C₂⁰ = 1. Therefore, a general element of|C₁⁰ +C₂⁰| is also a smooth rational curve which byπis mapped onto a smooth conic inIP². So ifC ∈ |C₁+C₂|is sufficiently general thenC⁰ is a smooth rational curve that is mapped onto a smooth conic. Therefore,π⁻¹(π(C)) splits into two components each of which is a smooth rational curve.

Now, to prove thatπ(C) is a tangent conic atB let SpecA=U ⊂π(C) be an open subset ofπ(C), f ∈Athe equation ofB restricted to U, and AZ :=A[T]

(T²−f). Then SpecAZ ∼=π⁻¹(U)⊂ π⁻¹(π(C)) and SpecAZ is reducible if and only if (T²−f) is reducible inA[T], i.e., if and only if there is ag∈Awithf =g². Henceπ(C) is a tangent conic and the proposition is proved.

Corollary 1.7 π:ZB −→IP² maps any of the linear systems |C1+C2| (C1 andC2 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 |C1+C2| ∼= IP¹ −→ IP⁵ (where IP⁵ is the parameter space of conics inIP²). This morphism is necessarily injective as forC∈ |C1+C2|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|C₁+C₂|. There is an open subset in|C₁+C₂|which is mapped into the closed subset of tangent conics inIP⁵. Therefore each element of|C₁+C₂| is mapped onto a (maybe singular) tangent conic. Note that none of the elements in |C₁+C₂| can be mapped to a (double) line.

Furthermore, a reducible tangent conic is the union of two bitangents and an element of|C₁+C₂| 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 C₁, C₁⁰ and C₂, C₂⁰ be the four (−1)-curves over the two lines of a reducible conic of that family. IfC₁·C₂ = 1 then the two linear systems are just|C₁+C₂| and

|C₁⁰ +C₂⁰|(otherwise |C₁⁰ +C₂|and|C₁+C₂⁰|).

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1.4. CONICS IN DOUBLE SOLIDS 9

Remark: Let|L|=|C1+C2|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 [ωZ_B]−[L]|.

Next, we want to characterise the above linear systems on ZB 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=C1+C2 with (−1)-curves Ci satisfying C1·C2= 1 are characterised by the properties L²= 0andL ·ωZ_B =−2.

Proof: Obviously the divisorsL=C1+C2 satisfyL²= 0 andL ·ωZ_B =−2. On the other hand, there is a bijection

L ∈Pic(Z_B)| L²= 0,L ·ω_Z_B=−2 ←→

L⁰ ∈Pic(Z_B)| L⁰²=−2,L⁰·ω_Z_B = 0

L 7−→ L⁰:=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 L⁰ ∈ Pic(Z_B) with L⁰² = −2 and L⁰ ·ω_Z_B = 0. (They define a root system in (ωZ_B)^⊥⊗^ZIR∼=IR⁷ (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 formC1+C2 (two for each one-parameter family of tangent conics). This proves the lemma.

Remark: Consider again the map ZB −→ IP¹ induced by the linear system |C1+C2|. By comparing the Euler numbers ofZB,IP¹and the fibre of that map one may determine the number of singular elements in the linear system |C1+C2| in a way different from that used in the proof of Lemma 1.7. Then, one may count the elements of the formC1+C2directly 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 spaceX_Cof conics in a double solid. (We will soon become more precise about whatX_Cis meant to be.) Let B⊂IP³ be a quartic surface with only isolated singularities. Denote byZ −→^π IP³ the double cover branched alongB. The following facts onZ may be found in [Kr2].

Lemma 1.9 The canonical line bundle ofZ is the pull-back (via π) ofOIP³(−2). In particular, it is locally free, its dual has a square rootO^Z(1) :=ω⁻

1 2

Z and the covering mapZ−→^π IP³ is induced

by the three-dimensional linear system−¹2[ωZ].

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|OZ(1)|. If S is smooth thenC satisfiesC²= 0 andC·ω_S =−2 inS.

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

Tangent Conics at Quartic Surfaces and Conics in Quartic Double Solids