On K3 surfaces with large complex structure

We discuss a notion of large complex structure for elliptic K3 surfaces with section inspired by the eight-dimensional F-theory/heterotic duality in string theory. This concept is naturally associated with the Type II Mumford partial compactification of the moduli space of periods for these structures. The paper provides an explicit Hodge-theoretic condition for the complex structure of an elliptic K3 surface with section to be large. We also establish certain geometric consequences of this large complex structure condition in terms of the Kodaira types of the singular fibers of the elliptic fibration.     

Chern numbers of certain Lefschetz fibrations

We address the geography problem of relatively minimal Lefschetz fibrations over surfaces of nonzero genus and prove that if the fiber-genus of the fibration is positive, then (equivalently, ) holds for those symplectic 4-manifolds. A useful characterization of minimality of such symplectic 4-manifolds is also proved.

     

DERIVED LENGTHS OF SOLVABLE GROUPS SATISFYING THE ONE-PRIME HYPOTHESIS. III

Abstract

The purpose of this paper is to extend the result in Park (Park, J. (2001). Birational maps of del Pezzo fibrations. J. Reine Angew. Math. 538:213–221) in the case of del Pezzo fibrations of degree 1. To this end we investigate the anticanonical linear systems of del Pezzo surfaces of degree 1. We then classify all possible effective anticanonical divisors on Gorenstein del Pezzo surfaces of degree 1 with canonical singularities.