By our investigations of conics in double solids we have achieved a quite good understanding of these objects. We could characterise those conics which are candidates for twistor lines as the elements of an irreducible component in the space of conicsXC in Z. Beside the characterisatione given in [Kr3] this component can be characterised in several ways:
• It is the “largest” component.
• Among the components ofXCthat have more than just one connected component in a general fibre over ˇIP3 it is the only component that can have real points over a generalH∈IPˇ3.
• In the spaceX parametrising tangent conics there is exactly one component whose preimage under XC−→X (induced byZ −→IP3) is irreducible.
• It is the preimage of the unique component ofX whose one-parameter families in a general plane do not contain reducible elements one or both bitangents of which are contained in one of the tropes.
Another description of the corresponding component inX could be that it is the component with the elements that are most difficultly to get a handle to. To explain this – note that the only bitangents at our quartic B which we have a priori are those in the tropes. Our distinguished component in X is just that one “which has nothing to do” with these bitangents. As another incarnation of that “inaccessability” consider the cubic threefoldK as constructed in Section 2.2.
Fix any line`inK (different from the six that are contracted under the projection toIP3). Then every plane in IP4 containing` intersects K in a reducible curve consisting of ` and a conic. A general conic obtained in this way is mapped (via the projectionK−−→IP3) to a tangent conic.
By constructingKfor each of the four trope planes ofB, one obtains in this way all tangent conics atB – except those in our distinguished component.
4.4. CONCLUDING REMARKS AND OPEN PROBLEMS 53
In spite of the results achieved in this paper, most of the problems stated in [Kr3] still remain open. Even one of the first questions – which of the conics in Ze have the correct normal bundle O(1)⊕ O(1) – could not be solved (in spite of considerable efforts). In [Kr3] Kreußler gave a criterion when the two conics inZ over a tangent conic have the correct normal bundle:
Proposition 4.13 Let F be the equation of a quartic surface B, L the equation of a plane that intersects B transversally and Q the equation of a quadric surface such that L =Q = 0 defines a smooth tangent conic C at B. Let C1 and C2 be the two conics over C in the double solid Z branched alongB. C1andC2have the same normal bundle and this normal bundle is different from O(1)⊕O(1)if and only if there are quadratic formsP1,P2 andP3such thatF =P1L2+P2Q+P32. It is not clear by now, whether this Proposition and the construction of Section 4.3 are sufficient to answer the question for the normal bundle. From the Proposition it is obvious that the property to have the correct normal bundle only depends on the one-parameter family in which the corre-sponding tangent conic is contained. Moreover, the normal bundle of the elements in the family F as defined by Corollary 4.11 does not depend on the chosen small resolutionZe−→Z – at least for those S ∈
ω−e1/2
Z
that do not contain an exceptional curve. This follows from the fact that the one-parameter family of tangent conics corresponding toF is determined independently of the chosen resolution.
Indeed – letH be a real plane (transversal toB) and letH1be a real plane through the real point of B that does not contain any other singular points ofB. B∩H1 then is a quartic curve with exactly one double point. (AsB has only one real point all further singularities ofB∩H1 would come in pairs of conjugate singular points. A line connecting such two conjugate singular points of B ∩H1 would be a real double tangent at B outside the tropes.) Let γ: [0,1] −→ IPˇ3IR be a path which connects H =γ(0) with H1 =γ(1) such that the planeγ(t) is transversal to B for t < 1. Let Xγ be the restriction ofX −→ IPˇ3 to the path γ and let (YF)γ be the restriction of YF (see (1.4)) to γ. (YF)γ is ramified over 1 ∈ [0,1] where six pairs of branches of (YF)γ meet (cf. Proposition 2.1). Let (`11(t), `12(t)), . . . ,(`61(t), `62(t)) (t∈ [0,1]) be the pairs of bitangents in the fibre ofYF overt that “meet” when tapproaches 1. ri(t) :=`i1∪`i2(t) then are reducible tangent conics for each t <1 (i= 1, . . . ,6) and ri(1) are double lines through the singular point of B. These reducible conics ri(t) are, of course, contained in Xγ. We claim that they are all contained in the same one-parameter family in the fibre of X over γ(t)∈IPˇ3. In Section 1.2 we have determined the tangent conics at a plane quartic with one ordinary double point. There we have seen that all tangent conics which are different from double lines are parametrised by one-parameter families and that non of these families contains double lines. Therefore, the “limit”
of the families containing the ri(t) for t → 1 cannot be among the 46 one-parameter families found in that section. On the other hand, there is at most one one-parameter family in the plane γ(t) (t < 1) without “limit” among these 46 families: Each of the 16 families with base point (cf. Proposition 1.3) must be the limit of two one-parameter families over γ(t). All other one-parameter families are the limit of exactly one one-one-parameter family. (Look at the reducible conics and compare with the bitangents!) Therefore, there is a unique one-parameter family left that must contain the six reducible conics ri(t). Fortapproaching 1 this family becomes the family of double lines in the planeγ(t) through the real singular point ofB. (This “limit family” consisting of double lines corresponds to the “exceptional” linear system found on page 4.) This way, γ determines a one-parameter family in the fibre ofX overH =γ(0) – namely the family containing the six reducible conicsri(0). Now, letFH1 be the element of Pic(ZeH1) withFH1·C= 2 (C – the real exceptional curve inZ) ande FH1·D±i = 1 (cf. page 48) and letFγ(t)be the distinguished class obtained fromFH1 by monodromy (cf. Corollary 4.11). Then it is obvious that the one-parameter family in the planeγ(t) which contains the six reducible conicsri(t) must be the image of the linear
systemFγ(t) (via the the map XC −→X induced from Z −→ IP3). But FH =Fγ(0) does not depend on the chosen pathγ and, hence, our distinguished one-parameter family does not depend onγeither. Conversely, the distinguished one-parameter family (containingri(1)) does not depend on the choice of a small resolutionZe−→Z. Finally, by Proposition 4.13 the normal bundle of the elements of |FH| depends only on the corresponding tangent conic and so these normal bundles are independent of the chosen resolution.
List of notations
∆ ∆⊂IPˇ3 the subvariety of planes in IP3 not transversal to B (i.e. having singular intersection withB)
B B⊂IP3 quartic surface with at most ordinary nodes as singularities Ci (as a (−1)-curve in a Del Pezzo surface which is the blow-up ofIP2 in 6
points) the strict transform of the conic in IP2 through all but the i-th point (cf. page 25)
Cij (as a (−1)-curve in a Del Pezzo surface which is the blow-up ofIP2 in 7 points) the strict transform of the conic inIP2through all but thei-th and thej-th point (cf. page 25)
“double six” pair of sextuples of lines in a cubic surface where the lines of each sextuple are mutually skew and each line in one sextuple is skew to exactly one line of the other sextuple, e.g. [(E1, . . . , E6),(C1, . . . , C6)] (cf. page 28) Ei (as a (−1)-curve in a Del Pezzo surface which is the blow-up of IP2 in r
points) denotes the exceptional divisor over thei-th of the blown-up points (cf. page 25)
Fano(V) Fano variety of lines in the hyper surfaceV ⊂IPN
Gij (as a (−1)-curve in a Del Pezzo surface which is the blow-up of IP2 in r points) the strict transform of the line inIP2through thei-th and thej-th point (cf. page 25)
Ki (as a (−1)-curve in a Del Pezzo surface which is the blow-up ofIP2 in 7 points) the strict transform of the cubic in IP2 through all seven points having a node in thei-th point (cf. page 26)
Rr the root system
L ∈Pic(V)| L2=−2,L ·ω= 0 ⊂Pic(V)⊗IR associ-ated to a Del Pezzo surfaceV of degree 9−r(cf. Theorem 3.1 on page 26)
“trope” plane that intersects a quartic surface along a conic and is tangent to that quartic in every intersection point (cf. page 15)
X the variety parametrising the tangent conics at the quarticB
XC the variety (as defined on page 10) parametrising conics in the double solid branched overB
Y0 Y0⊂Grass(2,4) the variety of bitangents at the quarticB 55
YF YF ⊂ Y0×IPˇ3 the variety of pairs (`, H) satisfying ` ⊂ H (see (1.4) on page 11)
Z the double cover ofIP3(double solid) branched over B
ZH the restriction of the double solid branched overB to the planeH ⊂IP3 Ze a small resolution of the double points of the double solidZ branched over
a quarticB with ordinary double points ωV canonical sheaf/class of the varietyV
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