Constructing algebraic varieties via finite group actions

The aim of this thesis is to investigate certain properties of two constructions of algebraic varieties based on a finite group action. In the first part we investigate Cox rings of minimal resolutions of (complex) surface quotient singularities C^2/G, where G is a finite (small) subgroup of GL(2;C). As a result we provide two descriptions of these rings. The first one is the single relation between its generators or, in other words, an equation for the spectrum of the Cox ring presented as a hypersurface in an affine space. In addition, we obtain an explicit description of the minimal resolution of C^2/G as a divisor in a toric variety. The second way of describing the Cox ring of the minimal resolution of C^2/G ring relies on viewing it as a subring of the coordinate ring of a product of a torus and another surface quotient singularity, C^2/[G;G]. We give a method of finding a set of generators of such an embedding of the Cox ring, which uses only the information on the intersection numbers of components of the exceptional fibre of the considered resolution and on invariants of the induced action of [G;G] on C^2. We expect that this idea can be generalized to selected classes of resolutions of quotient singularities in higher dimensions. The second part of the thesis concerns geometric models of Markov processes on phylogenetic trees. We concentrate on the case of phylogenetic trees with symmetries, understood as invariance with respect to a transitive action of a finite group. First we investigate the setting with added assumption of the isotropy of the model. The main result of this part concerns models with groups of symmetries containing large abelian subgroups. We prove that in this case the assumption of isotropy is unnecessary and we use these results to show that hyperbinary models are the only isotropic models with abelian group of symmetries. Then we change the setting: we give up the assumption of isotropy and consider geometric properties of general group-based models and G-models. We give the first examples of non-normal models in these classes and compute Hilbert-Ehrhart polynomials to investigate the deformation equivalence of models for trees with the same number of leaves. Moreover, we propose a (conjectural) method of generating phylogenetic invariants of group-based models and prove that for the 3-Kimura model it is equivalent to an important conjecture of Sturmfels and Sullivant concerning the degree of generation of phylogenetic invariants.