ofX for which the mapX −→IPˇ3 (which is described in Section 1.1) is dominant. Note that in the case of quartics with tropes there are always components of X which do not dominate ˇIP3 – namely the variety of conics contained in the trope planes.
Lemma 3.5 The only root system contained in R6 whose Weyl group acts transitively on the 27 (−1)-curves of the corresponding Del Pezzo surface isR6 itself.
Proof: LetR⊂R6 be a root system such that the Weyl group W(R)⊂W(R6) acts transitively on the 27 (−1)-curves. As the stabiliser in W(R) of each of the (−1)-curves then is a subgroup of index 27 the order ofW(R) must be a multiple of 27. On the other hand, all roots of the root systemR6have equal length and, hence,Rcan only be of the typeAn,Dn,E6(if it is irreducible) or a product of root systems of typeAn orDn. The Weyl groups of the systems An andDn have the orders (n+ 1)! and 2n−1n! respectively. Since the dimension ofRmust not exceed six Rcan only be of typeE6 orA2×A2×A2. Thus we only have to rule out the latter case.
Let R be of typeA2×A2×A2; by a transformation with an element of W(R6) we can achieve that the first factor ofRconsists of the six roots
±(0; 1,−1,0,0,0,0), ±(0; 1,0,−1,0,0,0), ±(0; 0,1,−1,0,0,0) (3.4) To see this, let C and C0 with C·C0 = 1 be two roots from the first factor. As W(R6) acts transitively onR6 we may assume thatC= (0; 1,−1,0,0,0,0). Now (e.g. using the list (3.1)) one easily checks that by applying reflections of roots orthogonal toCthe (−1)-curveC0 can be moved into one of the roots (3.4) (keepingC unchanged).
The roots of the other factors ofR must be contained in the orthogonal complement to the space spanned by the elements (3.4). The only roots ofR6 in this complement are
±(0; 0,0,0,1,−1,0), ±(0; 0,0,0,1,0,−1), ±(0; 0,0,0,0,1,−1) and
±(1; 1,1,1,0,0,0), ±(1; 0,0,0,1,1,1), ±(2; 1,1,1,1,1,1)
which already form a root system of type A2×A2. The orbit of the (−1)-curve E1 under W(R) now is easily determined to be the set
{E1, E2, E3, C1, C2, C3, G12, G13, G23}
so thatW(R) does not act transitively on the (−1)-curves.
Theorem 3.6 Let B be a quartic surface with exactly one trope plane and up to eleven ordinary nodes as its only singularities. Then the parameter space XC of conics in Z contains exactly three components. The parameter spaceX of tangent conics atB contains exactly two dominating components.
Proof: From the table (2.5) on page 23 we see that the space Y0 of bitangents at B contains two irreducible components – one component containing the lines in the trope and one component containing all remaining double tangents. (B corresponds to row 1 of the table.) The monodromy must, therefore, act transitively on the 27 (−1)-curves inZH0which are skew toD+0. By Lemma 3.5 the subgroupG⊂W(R7) induced by the monodromy must be the group W(R6) embedded as the stabiliser ofD+0.
We now have to determine the orbits in the set (3.2) under the action of G. Without loss of generality, let’s assume that D+0 is the (−1)-curve E7. Then the set (3.2) splits into three G-invariant setsO0, O1,O2 according to the number [E7]· L which can take the values 0, 1, and 2.
3.2. TANGENT CONICS AT A QUARTIC WITH TROPES 31
From the list (3.3) one easily sees that there are exactly 27 elementsLwith [E7]· L= 0 and 27L with [E7]·L= 2. These are the elements of the form [E7+C] resp. [K7+C] wereCis a (−1)-curve satisfyingE7·C= 1 orE7·C= 0 respectively. AsGacts transitively on the (−1)-curves skew to E7 as well as on the (−1)-curves that intersectE7the two sets ofO0 andO2 are orbits underG.
The linear system |E7+C| contains five further reducible elements (cf. Corollary 1.7) each of which is the union of two intersecting (−1)-curves skew toE7. There are (27·10)/2 = 135 of those pairs which, hence, are all contained in the linear systems of the setO0. Analogously, the 135 pairs of intersecting (−1)-curves which meetE7 spread over the linear systems of the setO2.
The remaining (unordered) pairs (C, C0) of intersecting (−1)-curves are those withC·E7= 0 and C0·E7 = 1. By associating to such a pair (C, C0) with C0 ·E7 = 1 the pair (C,−[ωZH0]−C0) these unordered pairs bijectively correspond to ordered pairs of (−1)-curves skew to E7. (This correspondence is compatible with theG-action.) Now,Gacts transitively on this set of ordered pairs: It acts transitively on the (−1)-curves skew toE7and the stabiliser of a (−1)-curveC acts transitively on the (−1)-curves skew to C. Consequently, Gacts transitively on the (unordered) pairs (C, C0) of intersecting (−1)-curves with (C+C0)·E7= 1, hence, on the setO1.
Thus, the threeG-invariant setsO0,O1,O2are in factG-orbits. The orbitsO0andO2correspond to irreducible components ofXC which are mapped to the same “dominating component” ofX. We can give a characterisation of the two components of X: The linear systems of the orbits O0and O2 correspond to those one-parameter families of tangent conics that contain a reducible element with one of the bitangents contained in the trope. For each of these 27 families there is one linear system in each of the two orbits. The orbit O1 covers the remaining 36 one-parameter
families.
Theorem 3.7 LetB be a quartic surface with exactly one trope and twelve ordinary nodes as its only singularities. Then the parameter spaceXC of conics inZ consists of eleven components and the parameter spaceX of tangent conics atB contains exactly six dominating components.
Proof: A quartic with these properties corresponds to row 2 in the table (2.5) on page 23. First, we determine the subgroupG⊂W(R6) that is induced by the monodromy action. Let H ⊂IP3 be a general plane such that B∩H is smooth. Recall the construction of the cubic threefoldK associated to B as described in Section 2.2. Denote by P the point (0:0:0:0:1) ∈ K ⊂ IP4 and byKe the blowup ofK in P. The hyperplaneHP in IP4 spanned by H ⊂IP3 ⊂IP4 and P cuts a smooth cubic surface KH out of K. Let KgH be the strict transform of KH in K. The mape KgH−→H induced fromKe −→IP3is the double cover (ZH) ofH branched inB∩H. Denote the exceptional divisor of the blow-up ofKgH−→KH byEP.
The fundamental groupπ1( ˇIP3\∆, H) acts on the Picard group ofZH ∼=KgH keeping the class of EP fixed. (EP is one of the (−1)-curves that are mapped to the bitangent at B∩H which lies in the trope plane.) The monodromy orbits in the set of (−1)-curves skew toEP correspond to the sets of bitangents atB∩Hbelonging to the same irreducible component ofY0. On the other hand, these (−1)-curves skew to EP correspond (viaKgH −→KH) to lines in the surfaceKH and lines belonging to the same component of Fano(K) correspond to bitangents of the same component of Y0.
Now, let P1 ∈ K be one of the six nodes of K (P1 6∈ HP). Let Q be the intersection of the tangent cone atK inP1with the hyperplane spanned byH andP. S:=Q∩K=Q∩KH is the
“associated sextic” as considered in Section 2.3. From Table 2.1 it is clear thatS must split into two componentsS1 andS2of the types (2,1) and (1,2). By Proposition 2.8, the projection from P1maps lines inK(not throughP1) to lines inHP with twofold intersection withS. In particular, the lines inKH must have intersection number two with S inKH.
To determine the lines inKH belonging to the same component of Fano(K) we now seek to find the classes in Pic(KH) of the two componentsS1 and S2 of S. From the adjunction formula on IP1×IP1∼=Q⊃S one finds that the genus of Si is zero (i= 1,2). Furthermore,
3 = (S1· OQ(1,1))Q= (S1· OIP3(1))IP3 = (S1·ωK−1
H)KH
(analogously forS2). SinceS has exactly five ordinary nodesS1 andS2 have to intersect properly in five points, hence, (S1·S2) = 5. As [S1] + [S2] = [S] =−2[ωKH] in Pic(KH) [S1] and [S2] must be of the form (a;b1, . . . , b6) with 0≤a≤6. From these conditions on each of theSi we infer that the classes of theSi in Pic(KH) must be in one of the classes
(1; 0,0,0,0,0,0) (2; 1,1,1,0,0,0) (3; 2,1,1,1,1,0) (4; 2,2,2,1,1,1) (5; 2,2,2,2,2,2)
or in a class obtained from the above classes by permuting the numbers following the semicolon.
The only pairs of the above classes which have intersection number five are of the form [(1; 0,0,0,0,0,0),(5; 2,2,2,2,2,2)]
[(2; 1,1,1,0,0,0),(4; 1,1,1,2,2,2)]
[(3; 2,1,1,1,1,0),(3; 0,1,1,1,1,2)]
In any of these cases it turns out that the two sets of lines inKH meetingS1resp. S2twice form a
“double six” whereas the remaining 15 lines meet each of the components ofS once. Without loss of generality, let us assume thatS1 andS2are of the classes (1; 0,0,0,0,0,0) and (5; 2,2,2,2,2,2) respectively. Then the lines meetingS1 twice are the (−1)-curvesC1, . . . , C6 and the lines having twofold intersection with S2 are E1, . . . , E6. These two sextuples must be monodromy orbits.
(They are bisecants of one component of S, hence, the corresponding bitangents belong to the same component ofY0.) The only roots inR6 leaving these two sextuples unchanged areEi−Ej (1 ≤i, j ≤6). These roots form the root system A5 whose Weyl group is the symmetric group S6 acting on Pic(KH) by permuting the elementsE1, . . . , E6. The subgroupG⊂W(R6) induced by the monodromy is, hence, contained inW(A5)∼=S6 and is generated by transpositions of S6 (since this is the way the reflections ofA5 act). But the only subgroup ofS6 which is generated by transpositions and which acts transitively on the set{1, . . . ,6}isS6 itself.
KnowingG⊂W(R6)⊂W(R7) it is now easy to list the orbits ofGin the set (3.2):
• (1; 0,0,0,0,0,0,1) (1 element)
• (1; 1,0,0,0,0,0,0), . . . ,(1; 0,0,0,0,0,1,0) (6 elements)
• (2; 1,1,1,1,0,0,0), . . . ,(2; 0,0,1,1,1,1,0) ( 64
= 15 elements)
• (2; 1,1,1,0,0,0,1), . . . ,(2; 0,0,0,1,1,1,1) ( 63
= 20 elements)
• (3; 2,1,1,1,1,1,0), . . . ,(3; 1,1,1,1,1,2,0) (6 elements)
• (3; 2,0,1,1,1,1,1), . . . ,(3; 1,1,1,1,0,2,1) (30 elements)
• (3; 0,1,1,1,1,1,2), . . . ,(3; 1,1,1,1,1,0,2) (6 elements)
• (4; 0,0,0,2,2,2,0), . . . ,(4; 2,2,2,0,0,0,0) ( 63
= 20 elements)
• (4; 0,0,0,0,2,2,2), . . . ,(4; 2,2,0,0,0,0,2) ( 64
= 15 elements)
3.2. TANGENT CONICS AT A QUARTIC WITH TROPES 33
• (5; 1,2,2,2,2,2,2), . . . ,(5,2,2,2,2,2,1,2) (6 elements)
• (5; 2,2,2,2,2,2,1) (1 element)
So, there are eleven orbits and each of them corresponds to a component of XC. As two of the corresponding linear systems yield the same one-parameter family of tangent conics if and only if their sum equals (6; 2,2,2,2,2,2,2) from the above list we deduce that the parameter space X of tangent conics has one component with one one-parameter family in a general plane, two components with six, two components with fifteen, and one component with twenty one-parameter
families in a general plane ofIP3.
We now turn to the case of quartics with two trope planes:
Theorem 3.8 Let B be a quartic surface with exactly two tropes and only ordinary nodes as singularities. Then the parameter spaceXC of conics inZ consists of nine irreducible components and the parameter spaceX of tangent conics atB contains exactly five dominating components.
Proof: Again we start by determining the subgroupG⊂W(R6) that is induced by the monodromy action. The fact that B has two trope planes means that there is a (−1)-curve – say E6 – which is skew to D+0 and invariant under transformations of G. Therefore, G is contained in W(R5)∼=StabW(R6)(E6)⊂W(R6). From the table (2.5) on page 23 we see (row 3) that Ghas only two orbits in the set of (−1)-curves ofZH0 skew toD+0 (see page 29) and different fromE6. On the other hand, the set of (−1)-curves skew toE6and the set of (−1)-curves meetingE6 must beG-invariant asE6 is fixed. Hence, these two sets must be the two orbits.
These two orbits have cardinality 16 and 10 and, hence, the order of Gmust be divisible by 80.
The only root systems of a dimension not exceeding five with roots of equal lengths and whose Weyl group has an order which is divisible by 80 areA5,D5 andA1×A4. But the root systemA5 cannot be contained in a root system of typeD5as 720 =]W(A5) does not divide 1920 =]W(D5).
To rule out the root systemA1×A4we determine the orthogonal complement inR5of an arbitrary root – say (0; 1,−1,0,0,0):
±(0; 0,0,1,−1,0) ±(0; 0,0,1,0,−1) ±(0; 0,0,1,0,−1)
±(1; 1,1,1,0,0) ±(1; 1,1,1,0,0) ±(1; 1,1,1,0,0) and
±(1; 0,0,1,1,1)
These roots form the root systemA1×A3so that the root systemA1×A4cannot be contained inD5. G, hence, must be the whole groupW(R5). W(R5) is generated by the transformations of Pic(ZH0) induced by permuting the (−1)-curvesE1, . . . E5and the reflection of the root (1; 1,1,1,0,0,0,0).
Now, it is easy to list the nine orbits ofGin the set (3.2):
•
(1; 1,0,0,0,0,0,0), . . . ,(1; 0,0,0,0,1,0,0) (2; 1,1,1,1,0,0,0), . . . ,(2; 0,1,1,1,1,0,0)
(10 elements)
•
(1; 0,0,0,0,0,1,0)
(2; 1,1,1,0,0,1,0), . . . ,(2; 0,0,1,1,1,1,0) (3; 2,1,1,1,1,1,0), . . . ,(3; 1,1,1,1,2,1,0)
(16 elements)
•
(1; 0,0,0,0,0,0,1)
(2; 1,1,1,0,0,0,1), . . . ,(2; 0,0,1,1,1,0,1) (3; 2,1,1,1,1,0,1), . . . ,(3; 1,1,1,1,2,0,1)
(16 elements)
• {(3; 1,1,1,1,1,2,0)} (1 element)
•
(2; 1,1,0,0,0,1,1), . . . ,(2; 0,0,0,1,1,1,1) (3; 2,0,1,1,1,1,1), . . . ,(3; 1,1,1,0,2,1,1) (4; 2,2,2,1,1,1,1), . . . ,(4; 1,1,2,2,2,1,1)
(40 elements)
• {(3; 1,1,1,1,1,0,2)} (1 element)
•
(3; 0,1,1,1,1,2,1), . . . ,(3; 1,1,1,1,0,2,1) (4; 1,1,1,2,2,2,1), . . . ,(4; 2,2,1,1,1,2,1)
(5; 2,2,2,2,2,2,1)
(16 elements)
•
(3; 0,1,1,1,1,1,2), . . . ,(3; 1,1,1,1,0,1,2) (4; 1,1,1,2,2,1,2), . . . ,(4; 2,2,1,1,1,1,2)
(5; 2,2,2,2,2,1,2)
(16 elements)
•
(4; 1,1,1,1,2,2,2), . . . ,(4; 2,1,1,1,1,2,2) (5; 1,2,2,2,2,2,2), . . . ,(5; 2,2,2,2,1,2,2)
(10 elements)
Taking into account that two elements define the same one-parameter family of tangent conics if their sum equals (6; 2,2,2,2,2,2,2), from the above list we find one dominating component ofX with only one one-parameter family in every general plane, one component with 10, one with 20, and two dominating components with 16 one-parameter families each in a general plane ofIP3. Theorem 3.9 Let B be a quartic surface with exactly three tropes and only ordinary nodes as singularities. Then the parameter space XC of conics inZ consists of 21 irreducible components and the parameter spaceX of tangent conics atB contains exactly eleven dominating components.
Proof: As in the previous proofs, we consider the (−1)-curves in ZH0 which are skew toD0+ (see page 29) and the action of G ⊂W(R6) on them. As B is supposed to have three tropes there must be exactly two (−1)-curves (skew to D+0) which are fixed by G. (Any further would cause B to have an additional trope.) These two (−1)-curves then must be skew since otherwise there would be a unique (−1)-curve which intersects them both. This third (−1)-curve would be fixed byG, too, yielding a fourth trope atB. (The intersection behavior of the fixed (−1)-curves also could have been determind by methods similar to that in the proof of Theorem 3.7.) Without loss of generality, we may assume that the two fixed (−1)-curves (skew toD+0) areE6 andE5. G, hence, is contained in the groupW(R4)∼=StabStabW(R
6 )(E6)(E5). Now, note that among the (−1)-curves skew to D+0 there are four G-invariant sets – namely the set of (−1)-curves meeting
3.2. TANGENT CONICS AT A QUARTIC WITH TROPES 35
both E5 and E6, the two sets of (−1)-curves meeting exactly one ofE5 and E6, and the set of (−1)-curves meeting neither of them. From the table (2.5) (row 4) we infer that these invariant sets must be orbits as the number of orbits (including the sets{E5}and{E6}) is six. The cardinalities of the four orbits are 10 (for the set of (−1)-curves skew to bothE5 andE6) and 5 (for the three other). ThereforeG must be the whole of W(R4) as there is no other root system of dimension less than or equal to four, with roots of equal lengths, and such that the corresponding Weyl group has order divisible by five. The action of G =W(R4) is generated by the permutations of the E1, . . . , E4together with the reflection corresponding to the root (1; 1,1,1,0,0,0,0). So it is again easy to write down the 21 orbits ofGin the set (3.2):
•
(1; 1,0,0,0,0,0,0), . . . ,(1; 0,0,0,1,0,0,0) (2; 1,1,1,1,0,0,0)
(5 elements)
•
(1; 0,0,0,0,∗,∗,∗)
(2; 1,1,1,0,∗,∗,∗), . . . ,(2; 0,1,1,1,∗,∗,∗)
where “∗,∗,∗” can be “1,0,0”,
“0,1,0” or “0,0,1” – each possibility yielding one orbit with 5 elements.
•
(2; 1,1,0,0,∗,∗,∗), . . . ,(2; 0,0,1,1,∗,∗,∗) (3; 2,1,1,1,∗,∗,∗), . . . ,(3; 1,1,1,2,∗,∗,∗)
where “∗,∗,∗” can be “1,1,0”,
“1,0,1” or “0,1,1” – each possibility yielding one orbit with 10 elements.
•
(2; 1,0,0,0,1,1,1), . . . ,(2; 0,0,0,1,1,1,1) (3; 2,0,1,1,1,1,1), . . . ,(3; 1,1,0,2,1,1,1) (4; 2,2,2,1,1,1,1), . . . ,(4; 1,2,2,2,1,1,1)
(20 elements)
• {(3; 1,1,1,1,∗,∗,∗)}
where “∗,∗,∗” runs through all per-mutations of 2,1,0 – yielding 6 orbits with 1 element.
•
(3; 0,1,1,1,∗,∗,∗), . . . ,(3; 1,1,1,0,∗,∗,∗) (4; 1,1,2,2,∗,∗,∗), . . . ,(4; 2,2,1,1,∗,∗,∗)
where “∗,∗,∗” can be “1,1,2”,
“1,2,1” or “2,1,1” – each possibility yielding one orbit with 10 elements.
•
(4; 1,1,1,2,∗,∗,∗), . . . ,(4; 2,1,1,1,∗,∗,∗) (5; 2,2,2,2,∗,∗,∗)
where “∗,∗,∗” can be “1,2,2”,
“2,1,2” or “2,2,1” – each possibility yielding one orbit with 5 elements.
•
(4; 1,1,1,1,2,2,2)
(5; 1,2,2,2,2,2,2), . . . ,(5; 2,2,2,1,2,2,2)
(5 elements)
By identifying elements yielding the same one-parameter families of tangent conics, from the above list we find four dominating components of X with five one-parameter families, four dominating components with ten, and three components with one family in a general plane ofIP3.
Theorem 3.10 Let B be a quartic surface with exactly four tropes and only ordinary nodes as singularities. Then the parameter space XC of conics inZ consists of 19 irreducible components and the parameter spaceX of tangent conics atB contains exactly ten dominating components.
Proof: To determine the groupG⊂W(R6) we, again, start with finding the intersection behavior of the three (−1)-curves in ZH0 which are skew to D+0 (cf. page 29) and fixed by the action of G. We proceed in the same manner as in the proof of Theorem 3.7: LetK be the cubic threefold as constructed in Section 2.2 andP the point (0 : 0 : 0 : 0 : 1)∈ K ⊂IP4. For a general plane H ⊂IP3(such thatB∩H is smooth) letKH be the cubic surface cut out ofKby the hyperplane inIP4 spanned byH⊂IP3⊂IP4andP.
Further, let P1 ∈ K be one of the nodes of K and Q the intersection of the tangent cone at K in P1 with the hyperplane spanned by H and P. S := Q∩K = Q∩KH is the “associated sextic” as considered in Section 2.3. From the table 2.5 (row 5) we infer thatS splits in irreducible components of types (2,2) + (1,0) + (0,1) or 3×(1,1).
In the first case, S already contains two lines of the cubic surface KH which are fixed by the monodromy action on the lines ofKH. These two lines intersect (as (1,0)·(0,1) = 1 onQ) and, hence, there is a unique line intersecting them both which (by its uniqueness) must be fixed by the monodromy action, too. The remaining lines come in three monodromy invariant subsets: for each of the fixed lines the set of (eight) lines (different from the fixed ones) intersecting it. As there are only six irreducible components in Fano(K) there can only be six orbits among the lines in KH
and, thus the three invariant sets must be orbits.
In the second case (when S splits in three components of type (1,1)) the components of S have self-intersection zero inKH, intersection number two with ω−K1
H (by an analogous argument as in the proof of Theorem 3.7) and each two of them have intersection number two. Therefore, the three components of S, taken as divisors in the Del Pezzo surfaceKH, must be (1; 1,0,0,0,0,0), (2; 1,1,1,1,0,0), (3; 2,1,1,1,1,1) or obtained from these by permuting the numbers following the semicolon. (This immediately follows from the list (3.3) on page 26.) For each component there is a unique (−1)-curve inKH meeting this component twice (thus being a bisecant of that component).
One easily checks that these three bisecants meet each other. As in the first case one finds the three invariant sets of eight lines which are orbits.
Without loss of generality we may assume that the three fixed lines are the three (−1)-curvesC6, E5 andG56 (the Weyl groupR(R6) acts transitively on the triples of mutually intersecting lines).
Then the roots ofR6 orthogonal to these (−1)-curves are
• Ei−Ej (1≤i, j≤4)
• ±(H−Ei−Ej−E6) (1≤i < j ≤4) (3.5)
These form a root system of typeD4– take
(0; 1,−1,0,0,0,0), (0; 0,1,−1,0,0,0), (0; 0,0,1,−1,0,0), (1; 0,0,1,1,0,1)
as a basis of the root system. Consider, now, the eight lines (E1, G16;E2, G26;E3, G36;E4, G46) meeting the lineC6. They come in pairs of intersecting lines and pairs are moved into pairs byG.
As Gis to act transitively on the eight lines it must act transitively on the four pairs (E1, G16), (E2, G26), (E3, G36), (E4, G46) formed of these lines. The reflections of the roots (3.5) act on these eight lines by swapping the lines in two of the four pairs of intersecting lines. A transitive subgroup ofS4which is generated by transpositions must beS4itself. ThereforeGcontains three reflections swapping the pairs {E1, G16} ↔ {E2, G26}, {E2, G26} ↔ {E3, G36} and {E3, G36} ↔ {E4, G46} respectively. The subgroup ofG generated by these three reflections is isomorphic toS4. It does not act transitively on the eight double tangents. Indeed, one can mark one member of each of the four pairs such that this subgroup of G maps marked elements to marked ones. (In other
3.2. TANGENT CONICS AT A QUARTIC WITH TROPES 37
words: If an element of the subgroup maps E1 to E2 then there is no element in this subgroup mapping E1 to G26.) Thus G is the Weyl group of a root system which is contained inD4 and which containsA3 as a proper subsystem. A4 is not contained in D4 (as 120 =]W(A4) does not divide]W(D4) = 192) nor isA3×A1contained inD4(as the orthogonal complement of a root in D4 is the root system A1×A1). SoGis the Weyl group of the whole root system (3.5).
Having found the subgroup in W(R6)⊂ W(R7) generated by monodromy we, again, can easily list the 19 monodromy orbits in the set (3.2). These are:
•
(1; 1,0,0,0,0,0,0), . . . ,(1; 0,0,0,1,0,0,0) (2; 1,1,1,0,0,1,0), . . . ,(2; 0,1,1,1,0,1,0)
=
{(H−Ei)|i= 1, . . .4} ∪ {(H−E1− · · · −E4−E6+Ei)|i= 1, . . .4} (8 elements)
•
(1; 0,0,0,0,1,0,0)
(2; 1,1,0,0,1,1,0), . . . ,(2; 0,0,1,1,1,1,0) (3; 1,1,1,1,1,2,0)
=
{(H−E5),(3H−E1− · · · −E5−2E6)} ∪ {(2H−Ei−Ej−E5−E6|i < j≤4} (8 elements)
• {(1; 0,0,0,0,0,1,0)}
•
(1; 0,0,0,0,0,0,1)
(2; 1,1,0,0,0,1,1), . . . ,(2; 0,0,1,1,0,1,1) (3; 1,1,1,1,0,2,1)
=
{(H−E7),(3H−E1− · · · −E4−2E6−E7)} ∪ {(2H−Ei−Ej−E6−E7|i < j≤4} (8 elements)
• {(2; 1,1,1,1,0,0,0)}
•
(2; 1,1,1,0,1,0,0), . . . ,(2; 0,1,1,1,1,0,0) (3; 2,1,1,1,1,1,0), . . . ,(3; 1,1,1,2,1,1,0)
=
{(2H−E1− · · · −E5+Ei),(3H−E1− · · · −E6−Ei)|i= 1, . . . ,4} (8 elements)
•
(2; 1,1,1,0,0,0,1), . . . ,(2; 0,1,1,1,0,0,1) (3; 2,1,1,1,0,1,1), . . . ,(3; 2,1,1,1,0,1,1)
=
{(2H−E1− · · · −E4−E7+Ei),(3H−E1− · · · −E4−E6−E7−Ei)|i= 1, . . . ,4} (8 elements)
•
(2; 1,1,0,0,1,0,1), . . . ,(2; 0,0,1,1,1,0,1) (3; 2,0,1,1,1,1,1), . . . ,(3; 1,1,0,2,1,1,1) (4; 1,1,2,2,1,2,1), . . . ,(4; 2,2,1,1,1,2,1)
=
{(2H−Ei−Ej−E5−E7),(4H−E1− · · · −E7−E6−Ei−Ej)|i < j≤4} ∪ {3H−E1− · · · −E7−Ei+Ej|i, j≤4, i6=j}
(24 elements)
•
(2; 1,0,0,0,1,1,1), . . . ,(2; 0,0,0,1,1,1,1) (3; 1,1,1,0,1,2,1), . . . ,(3; 0,1,1,1,1,2,1)
=
{(2H−E5−E6−E7−Ei),(3H−E1− · · · −E7−E6+Ei)|i≤4} (8 elements)
• {(3; 1,1,1,1,2,1,0)}
• {(3; 1,1,1,1,0,1,2)}
•
(3; 1,1,1,2,1,0,1), . . . ,(3; 2,1,1,1,1,0,1) (4; 1,2,2,2,1,1,1), . . . ,(4; 2,2,2,1,1,1,1)
=
{(3H−E1− · · · −E7+E6−Ei),(4H−2E1− · · · −2E4−E5−E6−E7+Ei)|i≤4} (8 elements)
•
(3; 0,1,1,1,2,1,1), . . . ,(3; 1,1,1,0,2,1,1) (4; 1,1,1,2,2,2,1), . . . ,(4; 2,1,1,1,2,2,1)
=
{(3H−E1− · · · −E7−E5+Ei),(4H−E1− · · · −E7−E5−E6−Ei)|i≤4} (8 elements)
•
(3; 0,1,1,1,1,1,2), . . . ,(3; 0,1,1,1,1,1,2) (4; 1,1,1,2,1,2,2), . . . ,(4; 2,1,1,1,1,2,2)
=
{(3H−E1− · · · −E6−2E7+Ei),(4H−E1− · · · −E7−E6−E7−Ei)|i≤4} (8 elements)
• {(4; 1,1,1,1,2,2,2)}
3.2. TANGENT CONICS AT A QUARTIC WITH TROPES 39
•
(3; 1,1,1,1,2,0,1)
(4; 1,1,2,2,2,1,1), . . . ,(4; 2,2,1,1,2,1,1) (5; 2,2,2,2,2,2,1)
=
{(3H−E1− · · · −E4−2E5−E7),(5H−2E1− · · · −2E6−E7} ∪ {(4H−E1− · · · −E7−E5−Ei−Ej)|i < j≤4}
(8 elements)
• {(5; 2,2,2,2,2,1,2)}
•
(3; 1,1,1,1,1,0,2)
(4; 1,1,2,2,1,1,2), . . . ,(4; 2,2,1,1,1,1,2) (5; 2,2,2,2,1,2,2)
=
{(3H−E1− · · · −E5−2E7),(5H−2E1− · · · −2E7+E5} ∪ {(4H−E1− · · · −E6−2E7−Ei−Ej)|i < j≤4}
(8 elements)
•
(4; 1,1,1,2,2,1,2), . . . ,(4; 2,1,1,1,2,1,2) (5; 1,2,2,2,2,2,2), . . . ,(5; 2,2,2,1,2,2,2)
=
{(4H−E1− · · · −E7−E5−E7−Ei),(5H−2E1− · · · −2E7+Ei)|i= 1, . . . ,4} (8 elements)
yielding 10 dominating components inX: 3 components with one family, 6 components with eight one-parameter families and one component with twelve one-parameter families in a general plane
ofIP3.
Theorem 3.11 Let B be a quartic surface with exactly six tropes and only ordinary nodes as singularities. Then the parameter space XC of conics in Z consists of 37 irreducible components and the parameter spaceX of tangent conics atB contains exactly 19 dominating components.
Proof: As the proof is similar to the previous ones we will only outline the key points. The five (−1)-curves inZH0 which are skew toD+0 and fixed by monodromy intersect as sketched below.
B B
B B
B BB
B
B B
B B
BB
One may choose them to beG56, E5, C5, E6, C6. Similar as above one shows that the root system RG (cf. Corollary 3.4) is of typeA3 and consists of all roots ofR6 orthogonal to these five (− 1)-curves. Its Weyl group G∼=S4 acts by permuting the four exceptional divisors E1, . . . , E4. The
37 orbits of G in the set (3.2) then are easily determined. In the consequence, one gets four dominating components of X with six one-parameter families, eight components with four and seven components with one one-parameter family in a general plane ofIP3.
Theorem 3.12 Let B be a quartic surface with exactly ten tropes and only ordinary nodes as singularities. Then the parameter space XC of conics inZ consists of 61 irreducible components and the parameter spaceX of tangent conics atB contains exactly 31 dominating components.
Proof: The nine (−1)-curves inZH0 which are skew toD0+ and fixed by monodromy intersect as follows.
HHH HH
HH HH
HH Q
QQ QQ
QQQ
They may be chosen to beE4, E5, E6, C4, C5, C6, G45, G46, G56. Gmust be the whole Weyl group of the root system of typeA2 which consists of the roots orthogonal to all the nine (−1)-curves.
G∼=S3acts by permuting the exceptional divisorsE1, E2, E3. Under this action the set (3.2) splits into 61 orbits which correspond to sixteen dominating components ofX with three one-parameter families and fifteen components with one family in a general plane ofIP3.
Theorem 3.13 LetBbe a quartic surface with exactly sixteen tropes and sixteen ordinary nodes3).
Then the parameter spaceXC of conics inZ consists of 93 irreducible components and the param-eter spaceX of tangent conics atB contains exactly 47 dominating components.
Proof: As B has sixteen tropes there are 32 (−1)-curves inZH0 (corresponding to the sixteen bitangents atB∩H0in the trope planes) which are fixed by the monodromy action. Hence there is at most one (non-trivial) element inG- namely the reflection leaving these 32 (−1)-curves fixed (cf. Proof of Proposition 3.2). From the table (2.5) it is clear that Gmust not be trivial as the spaceY0of bitangents atBhas 22 irreducible components (and not 28 as it would ifGwere trivial).
SoG ∼=S2 is the Weyl group of the root system A1. A reflection of R7 fixes 60 elements of the set (3.2) and swaps the the remaining 66 in pairs (exactly one of the swapped pairs has elements corresponding to one and the same one-parameter family of tangent conics). X, hence, has 16 dominating components with two one-parameter families and 31 components with one family in a
general plane ofIP3.
With the last theorem we have completed our investigations on conics in double solids resp. tangent conics at B for quartics with at least one trope and only ordinary double points. The results of this chapter are summerised in the following table.
3Bis the well known Kummer quartic – cf. [Hu]. The cubic threefold attached toBas constructed in Section 2.2 is the so-called Segre cubic having 10 nodes and containing fifteen planes – cf. [SR] page 169 and [S-C].
3.2. TANGENT CONICS AT A QUARTIC WITH TROPES 41
Number of tropes
Number of nodes ofB
Type of the root system
Number of components
in XC
Number of dominating comp. inX
1 1 6, . . . ,11 E6 3 2
2 1 12 A5 11 6
3 2 10,11,12 D5 9 5
4 3 13 A4 21 11
5 4 12,13 D4 19 10
6 6 14 A3 37 19
7 10 15 A2 61 31
8 16 16 A1 93 47
(3.6)
Remark: When looking at the above table one realises that the root systems which we found in the course of our investigations are exactly the irreducible subsystems of the root systemE6.
Chapter 4
Applications to Twistor spaces
4.1 Generalities
Twistor spaces occur in four dimensional conformal geometry. To any four dimensional oriented conformal manifold M one can construct a twistor space T which is a six dimensional manifold fibred overM and carrying a natural almost complex structure. If the metric onM is self dual then this almost complex structure is integrable and, hence,T a complex three dimensional manifold.
For further details we refer the reader to [AHS], [Fri] and [Ku1]. For our purposes it is sufficient to regard a twistor space as being a complex three dimensional manifold endowed with the following additional structure:
• a proper differentiable submersionπ:T −→M onto a real four dimensional manifoldM such that the fibres are holomorphic curves isomorphic to CP1 and whose normal bundel in T is isomorphic toO(1)⊕ O(1)
• an anti-holomorphic fixed-point-free involutionσwhich preserves the fibres ofπ(i.e. π◦σ=π).
The fibres ofπare called “real twistor lines” andσthe “real structure”.
We are particularly interested in the case whereM is the connected sum of three complex projective planes CP2]3
. Kreußler and Kurke have shown in [KK] that in this case a generic twistor space T over M is a small resolution of a double solid Z (i.e. a double cover of IP3) branched over a quartic surface with thirteen ordinary double points. More precisely, there is a line bundle ω−1/2 onT whose square is the anticanonical sheafωT−1 ofT and whose linear system|ω−1/2|is three-dimensional. This linear system induces a morphism (resp. a rational map) toIP3. Additionally, the bundleω−1/2is real with respect toσ. (A line bundleLonT is calledrealifσ∗L ∼=L.) Hence, the map induced by |ω−1/2|may be chosen to be compatible with the standard real structure of CP3. The following theorem may be found in [KK]:
Theorem 4.1 LetT be a twistor space over the connected sum of three complex projective planes.
• If |ω−1/2|is base point free then T −→IP3 is a small resolution of a double cover Z of IP3 composed with the covering mapZ−→IP3. The ramification locus ofZ overIP3is contained in a (real) six-parameter family of quartic surfaces which have thirteen ordinary double points in general (but may specialise to quartics with double points and other singularities with higher Milnor number). Moreover these quartics are real (with respect to the standard real structure of IP3) and have exactly one real point which is an ordinary double point.
43
• If |ω−1/2| has base points then the extension of the mapT −−→IP3 to the blow-up of T in the base locus of |ω−1/2|is a small resolutionTeof a conic bundleT overIP1×IP1⊂IP3. T is contained in a (real) five-parameter family of conic bundles.
While the “conic bundle case” is well understood (cf. [LeB] and [Ku2]) for the “double solid case”
it is not clear for which quartics and for which small resolutions Ze of the double solid Z twistor spaces occur. One of the problems in this connection is to single out the real twistor lines. As a first step towards the solution of this problem one may try to describe the parameter space of “lines”
inZeorZ where “lines” are meant to be smooth rational curves with normal bundle O(1)⊕ O(1).
Further, one might ask which of these lines are real with respect to the real structure σ. We will give partial answers to these questions for the general case, i.e. for the case of a quartic with thirteen ordinary nodes. For this purpose we give list some basic facts that we will use in the sequel.
We give a characterisation of the quartic surfaces which may occur as branch loci. By Proposi-tion 4.1, these are real with exactly one real point. We will restrict ourselves to the case of quartics with exactly thirteen ordinary nodes. Quartics of this type have been studied by Kreußler in [Kr2]
where the following proposition may be found.
Proposition 4.2 Every real quartic surfaceB with exactly one real point and 13 ordinary double points can (in suitable coordinates) be defined by an equation of the form
F = x23f2+ 2x3L0L1L2+f22−f2 L20+L21+L22
+L20L21+L20L22+L21L22 (4.1)
= 1
4 Q2−E1E2E3E4
whereE1=x3−L0−L1−L2,E2=x3+L0+L1−L2,E3=x3+L0−L1+L2,E4=x3−L0+L1+L2, f2 =x20+x21+x22, Q = 2f2+x23−L20−L21−L22 andLj =P2
i=0aijxi with aij ∈ IR and such thatf2−L2j (j= 0,1,2)defines three smooth conics with 12 different intersection points. The real point ofB is the point(0:0:0:1). It is one of the double points.
Let T be a twistor space and let the morphism ϕ:T −→ IP3 induced by |ω−1/2| be branched along B. Denote by Hi the trope plane Ei = 0 (i = 1, . . . ,4). Then ϕ−1(Hi) splits into two components Si+ and Si−. The surfaces Si± are of degree one which means that their intersection number with any twistor line is one. Moreover, theS±i are real with respect to the real structureσ ofT. (These facts can be found in [KK] Section 4.) From this we immediately get:
Corollary 4.3 If the quartic surfaceB (as above) is the branch locus of T −→ϕ IP3 for a twistor space T then the linear forms Ei (i = 1, . . . ,4) and hence the linear forms Lj (j = 1,2,3) are
real.
By Theorem 4.1 the morphismT −→IP3factors into a small resolutionT −→Z and the covering map Z −→ IP3 whereZ is the double cover of IP3 branched over B. The real structure σ onT induces a real structure onZ such that the morphismsT −→ZandZ−→IP3are real. (This real structure onZ will also be denoted byσ.) This induced real structure onZ has exactly one fixed point (namely the preimage of the real point of the quarticB).