On Fano Arrangement Varieties

Abstract

This thesis contributes to the study of projective varieties with torus action. At first, we present an explicit approach to algebraic varieties with torus action by constructing them suitably embedded inside toric varieties. This extends existing constructions of rational varieties with a torus action of complexity one and delivers i.a. all Mori dream spaces with torus action. Our major example class are the (general/special) arrangement varieties, for which the torus action gives rise to a specific rational quotient to a projective space having as critical values a hyperplane arrangement in (general/special) position. This class comprises i.a. all toric varieties and all rational varieties with a torus action of complexity one. We present an explicit description of their Cox rings, which we use to obtain access to the geometry of these varieties, and give classification results in the smooth case. Then, we turn to singular Fano varieties. For these varieties the anticanonical complex has been introduced as a natural generalization of the toric Fano polytope and so far has been successfully used for the study of varieties with a torus action of complexity one. Using our explicit approach, we enlarge the area of application of the anticanonical complex to varieties with a torus action of higher complexity, for example, the arrangement varieties. As an application, we obtain several classification results. At first we consider intrinsinc quadrics, which are easily verified to be general arrangement varieties. We classify all Q-factorial Fano intrinsic quadrics of dimension three and Picard number one with at most canonical singularities. Moreover, considering the case of complexity two, we obtain the full classification of all Q-factorial canonical Fano intrinsic quadrics of dimension three without restrictions on the Picard number. Finally, we give classification results in the three-dimensional canonical case, where the maximal orbit quotient is the projective plane having a line arrangement of five lines in special position as its critical values.