principal elliptic fibrations, Math. Res. Lett. 12 (2005), 251-264.

normal functions – a subject of increasing interest due to their recent spectacular use in open string mirror symmetry [ 24 ] – which is further amplified by explicit examples in Section 5 . The second half of the paper takes up the question of how to use the geometry of polarized K 3 $K3$ surfaces with high Picard rank to construct indecomposable cycles (Sections 5 and 6 ). Elliptic fibrations yield an extremely natural source of families of cycles, whose image under the real and transcendental regulator maps have apparently not been previously studied. Our

## 1. Introduction

Let *X* be an algebraic variety defined over a number field *F*. We will say that rational points are *potentially dense* if there exists a finite extension *K/F* such that the set of *K*-rational points *X*(*K*) is Zariski dense in *X*. The main problem is to relate this property to geometric invariants of *X*. Hypothetically, on varieties of general type rational points are not potentially dense. In this paper we are interested in smooth projective varieties such that neither they nor their unramified coverings admit a dominant map onto varieties of general type. For these varieties it seems plausible to expect that rational points are potentially dense (see [2]).

that the support of the (non-trivial) negative part of the Zariski decomposition of every big divisor on X consists of pairwise disjoint curves. Indeed, the condition in question implies that if the intersection matrix of two irreducible negative curves C 1 , C 2 ⊂ X ${C_{1},C_{2}\subset X}$ is negative-definite, then it is diagonal. After proving Theorem 3 in Section 2 , we study the relation between elliptic fibrations and Zariski chambers on Enriques and K3 surfaces in Section 3 . It should be mentioned, that this note was motivated and inspired by the

Adv. Geom. 14 (2014), 735–756 Advances in Geometry DOI 10.1515 / advgeom-2014-0018 c© de Gruyter 2014 On quartics with lines of the second kind Sławomir Rams and Matthias Schütt∗ (Communicated by T. Grundhöfer) Abstract. We study the geometry of quartic surfaces in P3 that contain a line of the second kind over algebraically closed fields of characteristic different from 2, 3. In particular, we correct Segre’s claims made for the complex case in 1943. Key words. Line, quartic, elliptic fibration, K3 surface. 2010 Mathematics Subject Classification. Primary: 14J28

J. reine angew. Math. 600 (2006), 81—116 DOI 10.1515/CRELLE.2006.087 Journal für die reine und angewandte Mathematik ( Walter de Gruyter Berlin New York 2006 Weighted Fano threefold hypersurfaces By Ivan Cheltsov at Edinburgh and Jihun Park at Pohang Abstract. We study birational transformations into elliptic fibrations and birational automorphisms of quasismooth anticanonically embedded weighted Fano 3-fold hypersur- faces with terminal singularities classified by A. R. Iano-Fletcher, J. Johnson, J. Kollár, and M. Reid. 1. Introduction Let S be a smooth cubic

} . (ii) The previous set is equidistributed in S with respect to μ. (iii) If P ∨ / P {P^{\vee}/P} has no non-trivial isotropic subgroup, then the set of points s ∈ S {s\in S} (counted with multiplicity) for which 𝒳 s {\mathcal{X}_{s}} admits an elliptic fibration of norm less than n is equidistributed with respect to μ as n tends to infinity. For the definition of a parabolic line bundle of type ( γ , n ) {(\gamma,n)} and the norm of an elliptic fibration, we refer to Definition

give explicitly the structure of this compactified Simpson Jacobian for the following projective curves: tree-like curves and all reduced and reducible curves that can appear as Kodaira singular fibers of an elliptic fibration, that is, the fibers of types III , IV and IN with N f 2. 1. Introduction The problem of compactifying the (generalized) Jacobian of a singular curve has been studied since Igusa’s work [16] around 1950. He constructed a compactification of the Jacobian of a nodal and irreducible curve X as the limit of the Jacobians of smooth curves

curve of poles is either a rational or an elliptic curve of null self-intersection or it has the combinatorics of a singular fiber of an elliptic fibration. This result is then globalized by proving that, always up to a birational transformation, a semi- complete meromorphic vector field on a compact complex Kähler surface must satisfy at least one of the following conditions: to be globally holomorphic, to possess a non-trivial meromorphic first integral or to preserve a fibration. In particular, this extends the results established by Brunella for complete

## Abstract

We classify six-dimensional F-theory compactifications in terms of simple features of the divisor structure of the base surface of the elliptic fibration. This structure controls the minimal spectrum of the theory. We determine all irreducible configurations of divisors (“clusters”) that are required to carry nonabelian gauge group factors based on the intersections of the divisors with one another and with the canonical class of the base. All 6D F-theory models are built from combinations of these irreducible configurations. Physically, this geometric structure characterizes the gauge algebra and matter that can remain in a 6D theory after maximal Higgsing. These results suggest that all 6D supergravity theories realized in F-theory have a maximally Higgsed phase in which the gauge algebra is built out of summands of the types su(3), so(8), f4, e6, e8, e8, (g2 ⊕ su(2)); and su(2) ⊕ so(7) ⊕ su(2), with minimal matter content charged only under the last three types of summands, corresponding to the non-Higgsable cluster types identified through F-theory geometry. Although we have identified all such geometric clusters, we have not proven that there cannot be an obstruction to Higgsing to the minimal gauge and matter configuration for any possible F-theory model. We also identify bounds on the number of tensor fields allowed in a theory with any fixed gauge algebra; we use this to bound the size of the gauge group (or algebra) in a simple class of F-theory bases.