Minimal sections of conic bundles

Let the threefold X be a general smooth conic bundle over the projective plane P(2), and let (J(X), Theta) be the intermediate jacobian of X. In this paper we prove the existence of two natural families C(+) and C(-) of curves on X, such that the Abel-Jacobi map F sends one of these families onto a copy of the theta divisor (Theta), and the other -- onto the jacobian J(X). The general curve C of any of these two families is a section of the conic bundle projection, and our approach relates such C to a maximal subbundle of a rank 2 vector bundle E(C) on C, or -- to a minimal section of the ruled surface P(E(C)). The families C(+) and C(-) correspond to the two possible types of versal deformations of ruled surfaces over curves of fixed genus g(C). As an application, we find parameterizations of J(X) and (Theta) for certain classes of Fano threefolds, and study the sets Sing(Theta) of the singularities of (Theta).Comment: Duke preprint, 29 pages. LaTex 2.0

On Four-Dimensional Compactifications of F-Theory

Branches of moduli space of F-theory in four dimensions are investigated. The transition between two branches is described as a 3-brane-instanton transition on a 7-brane. A dual heterotic picture of the transition is presented and the F-theory - heterotic theory map is given. The F-theory data - complex structure of the Calabi-Yau fourfold and the instanton bundle on the 7-brane is mapped to the heterotic bundle on the elliptic Calabi-Yau threefold CY_3. The full moduli space has a web structure which is also found in the moduli space of semi-stable bundles on $CY_3$. Matter content of the four-dimensional theory is discussed in both F-theory and heterotic theory descriptions.Comment: 40 pages, TeX, references added, minor changes in Section

F-theory and linear sigma models

We present an explicit method for translating between the linear sigma model and the spectral cover description of SU(r) stable bundles over an elliptically fibered Calabi-Yau manifold. We use this to investigate the 4-dimensional duality between (0,2) heterotic and F-theory compactifications. We indirectly find that much interesting heterotic information must be contained in the `spectral bundle' and in its dual description as a gauge theory on multiple F-theory 7-branes. A by-product of these efforts is a method for analyzing semistability and the splitting type of vector bundles over an elliptic curve given as the sheaf cohomology of a monad.Comment: 40 pages, no figures; minor cosmetic reorganization of section 4; reference [6] update

Lines on the Dwork Pencil of Quintic Threefolds

We present an explicit parametrization of the families of lines of the Dwork pencil of quintic threefolds. This gives rise to isomorphic curves which parametrize the lines. These curves are 125:1 covers of certain genus six curves. These genus six curves are first presented as curves in P^1*P^1 that have three nodes. It is natural to blow up P^1*P^1 in the three points corresponding to the nodes in order to produce smooth curves. The result of blowing up P^1*P^1 in three points is the quintic del Pezzo surface dP_5, whose automorphism group is the permutation group S_5, which is also a symmetry of the pair of genus six curves. The subgroup A_5, of even permutations, is an automorphism of each curve, while the odd permutations interchange the two curves. The ten exceptional curves of dP_5 each intersect each of the genus six curves in two points corresponding to van Geemen lines. We find, in this way, what should have anticipated from the outset, that the genus six curves are the curves of the Wiman pencil. We consider the family of lines also for the cases that the manifolds of the Dwork pencil become singular. For the conifold the genus six curves develop six nodes and may be resolved to a P^1. The group A_5 acts on this P^1 and we describe this action.Comment: 48 pages, 2 figure