We consider Fano manifolds admitting an algebraic torus action with general orbit of codimension one. Using a recent result of Datar and Székelyhidi, we effectively determine the existence of Kähler-Ricci solitons for those manifolds via the notion of equivariant K-stability. This allows us to give new examples of Kähler-Einstein Fano threefolds, and Fano threefolds admitting a non-trivial Kähler-Ricci soliton.

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.

This is a survey of the language of polyhedral divisors describing T -varieties. This language is explained in parallel to the well established theory of toric varieties. In addition to basic constructions, subjects touched on include singularities, separatedness and properness, divisors and intersection theory, cohomology, Cox rings, polarizations, and equivariant deformations, among others.

Abstract. Given a finite set of lattice points A, we consider the associated homogeneous binomial ideal IA and projective toric variety XA. We give a concise combinatorial description of all linear subspaces contained in the variety XA, or, equivalently, all solutions in linear forms to the system of binomial equations determined by IA. More precisely, we study the Fano scheme F k (XA) whose closed points correspond to k-dimensional linear spaces contained in XA. We show that the irreducible components of F k (XA) are in bijection to maximal Cayley structures for A of length at least k. We explicitly describe these irreducible components and their intersection behavior, characterize when F k (XA) is connected, and prove that if XA is smooth in dimension k, then every component of F k (XA) is smooth in its reduced structure. Furthermore, in the special case k = dim XA − 1, we describe the nonreduced structure of F k (XA).