Nathan Owen Ilten

Toric Degenerations of Fano Threefolds Giving Weak Landau-Ginzburg Models

Abstract We show that every Picard rank one smooth Fano threefold has a weak Landau–Ginzburg model coming from a toric degeneration. The fibers of these Landau–Ginzburg models can be compactified to K3 surfaces with Picard lattice of rank 19. We also show that any smooth Fano variety of arbitrary dimension which is a complete intersection of Cartier divisors in weighted projective space has a very weak Landau–Ginzburg model coming from a toric degeneration.

THE GEOMETRY OF T-VARIETIES

This survey paper is based on my IMPANGA lectures given in the Banach Center, Warsaw in January 2011. We study the moduli of holomorphic map germs from the complex line into complex compact manifolds with applications in global singularity theory and the theory of hyperbolic algebraic varieties.We give a light introduction to some recent developments in Mori theory, and to our recent direct proof of the finite generation of the canonical ring.We introduce a variety

Deformations of rational -varieties

\hat{G}_2

Polarized complexity-1 T-varieties

parameterizing isotropic five-spaces of a general degenerate four-form in a seven dimensional vector space. It is in a natural way a degeneration of the variety

K-Stability for Fano Manifolds with Torus Action of Complexity 1

G_2

Deformations of smooth toric surfaces

, the adjoint variety of the simple Lie group

Upgrading and downgrading torus actions

\mathbb{G}_2

Versal deformations and local Hilbert schemes

. It occurs that it is also the image of

Degenerations to unobstructed Fano Stanley–Reisner schemes

\mathbb{P}^5

Mutations of Laurent Polynomials and Flat Families with Toric Fibers

by a system of quadrics containing a twisted cubic. Degenerations of this twisted cubic to three lines give rise to degenerations of