Cox rings are significant global invariants of algebraic varieties, naturally generalizing
homogeneous coordinate rings of projective spaces. This book provides a largely self …

Let X be any Q Q-Fano variety and Aut (X) _0 Aut (X) 0 be the identity component of the
automorphism group of X. Let GG be a connected reductive subgroup of Aut (X) _0 Aut (X) 0 …

We investigate the Cox ring of a normal complete variety X with algebraic torus action. Our
first results relate the Cox ring of X to that of a maximal geometric quotient of X. As a …

The absolute regularity of a Fano variety, denoted by reg^(X), is the largest dimension of the
dual complex of a log Calabi–Yau structure on X. The absolute coregularity is defined to be …

We state a number of conjectures that together allow one to classify a broad class of del
Pezzo surfaces with cyclic quotient singularities using mirror symmetry. We prove our …

We prove that for any $\mathbb {Q} $-Fano variety $ X $, the special $\mathbb {R} $-test
configuration that minimizes the $ H $-functional is unique and has a K-semistable $\mathbb …

We prove that any finite energy geodesic ray with a finite Mabuchi slope is maximal in the
sense of Berman-Boucksom-Jonsson, and reduce the proof of the uniform Yau-Tian …

We consider Fano manifolds admitting an algebraic torus action with general orbit of
codimension 1. Using a recent result of Datar and Székelyhidi, we effectively determine the …

The geometry of T-varieties Page 31 The geometry of T-varieties Klaus Altmann, Nathan
Owen Ilten, Lars Petersen, Hendrik Süß, and Robert Vollmert Contents 1 Introduction.. 2 …

Using the language of Altmann, Hausen and Süß, we describe invariant divisors on normal
varieties X which admit an effective codimension one torus action. In this picture, X is given …