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Work in progress - Ph.D. thesis Olaf Teschke
Since September 1997 I work on my Ph.D. thesis "Classification of vector bundles on double covers of P3". My supervisor is Prof. H. Kurke.
Program:
The main purpose of the work is the construction of moduli of semistable vektorbundles on double covers p: Z -> P3 ("double solids"). Such a bundle E determines data (p*E, y: p*E -> p*E(m)) with y2=f * IdE, where V(f) is the branch locus of p.
In the case of hyperelliptic Curves C (i.e. double covers of P1), R.-O. Buchweitz studied moduli of vector bundles on C in terms of data on P1 by applying methods developed in [1]. My aim is the generalization of these methods to the three-dimensional case.
The first idea is to examine non-splitting bundles with vanishing intermediate cohomology H1,2(E(l)). For these bundles, the criterium of Horrocks gives us in analogy to the one-dimensional case, that p*E is a direct sum of line bundles. If the branch locus has degree 2m, then the existence of such bundles is guaranteed for m>1. For instance, if m=2 the stable bundles of rank r=2 with vanishing H1,2(E(l)) and odd first Chern class are (modulo twist with line bundles) exactly the stable bundles with c1(E)=-1, c2(E)=2 (under the identification Pic(Z)=< p*O(1) > = Z). The moduli of these bundles is a 126 :1 - cover of the P3; the bundles correspond via Serre-correspondence to the 63 one-dimensional families of lines on Z, classified by I. Hadan in [2]. In the case of even first Chern class, there are semistable bundles with c1(E)=0, c2(E)=1 corresponding to the bitangents of the branch quartic, and stable bundles with c1(E)=0, c2(E)=2 corresponding to elliptic curves of degree 4 in the P3. The next aim is the description of the moduli for general m and r.
In the general case p*E splits not into the sum of line bundles. But there is a representation of p*E as a cohomology of a monad of sums of line bundles possible. To construct the moduli via geometric invariant theory, one has to examine how the induced structure on p*E lifts to the monad.
As an application, I try to construct framed vector bundles on compactifications of special double covers of the Cn to decide the (non-)isomorphy of these spaces to the Cn, which may give answers to the cancellation and the C* problem (see [3] for construction of these spaces).
[1] R.-O. Buchweitz, F.-O.Schreyer: Intersection of two quadrics, hyperelliptic curves and their Clifford algebras; not published
[2] I.Hadan: Tangent Conics at Quartic Surfaces and Conics in Quartic Double Solids. Ph.D. thesis, Humboldt-University 1997
[3] Sh. Kaliman, M. Zaidenberg: Affine Modifications and affine hypersurfaces with a very transitive automorphism group; e-print, not yet published
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Hinweise und Kommentare bitte an:
teschke@mathematik.hu-berlin.de
Letzte Änderung: 17.07.1998