Mateusz Michałek

Geometry of phylogenetic group-based models

In this paper we complete the results of Sullivant and Sturmfels [SS] proving that many of the algebraic group-based models for Markov processes on trees are pseudo-toric. We also show in which cases these varieties are normal. This is done by the generalization of the discrete Fourier transform approach. In the next step, following Sullivant and Sturmfels, we describe a fast algorithm finding a polytope associated to these algebraic models. However in our case we apply the notions of sockets and networks extending the work of Buczyńska and Wísniewski [BW] who introduced it for the binary case of the group Z2.

On the toric ideal of a matroid

Abstract Describing minimal generating set of a toric ideal is a well-studied and difficult problem. In 1980 White conjectured that the toric ideal associated to a matroid is equal to the ideal generated by quadratic binomials corresponding to symmetric exchanges. We prove Whites conjecture up to saturation, that is that the saturations of both ideals are equal. In the language of algebraic geometry this means that both ideals define the same projective scheme. Additionally we prove the full conjecture for strongly base orderable matroids.

Constructive degree bounds for group-based models☆

Abstract Group-based models arise in algebraic statistics while studying evolution processes. They are represented by embedded toric algebraic varieties. Both from the theoretical and applied point of view one is interested in determining the ideals defining the varieties. Conjectural bounds on the degree in which these ideals are generated were given by Sturmfels and Sullivant (2005) [25, Conjectures 29, 30] . We prove that for the 3-Kimura model, corresponding to the group G = Z 2 × Z 2 , the projective scheme can be defined by an ideal generated in degree 4. In particular, it is enough to consider degree 4 phylogenetic invariants to test if a given point belongs to the variety. We also investigate G-models, a generalization of abelian group-based models. For any G-model, we prove that there exists a constant d, such that for any tree, the associated projective scheme can be defined by an ideal generated in degree at most d.

On the full, strongly exceptional collections on toric varieties with Picard number three

We investigate full strongly exceptional collections on smooth, complete toric varieties. We obtain explicit results for a large family of varieties with Picard number three, containing many of the families already known. We also describe the relations between the collections and the split of the push forward of the trivial line bundle by the toric Frobenius morphism.

Secant Cumulants and Toric Geometry

We study the secant line variety of the Segre product of projective spaces using special cumulant coordinates adapted for secant varieties. We show that the secant variety is covered by open normal toric varieties. We prove that in cumulant coordinates its ideal is generated by binomial quadrics. We present new results on the local structure of the secant variety. In particular, we show that it has rational singularities and we give a description of the singular locus. We also classify all secant varieties that are Gorenstein. Moreover, generalizing [SZ12], we obtain analogous results for the tangential variety.

Secants of minuscule and cominuscule minimal orbits

We study the geometry of the secant and tangential variety of a cominuscule and minuscule variety, e.g. a Grassmannian or a spinor variety. Using methods inspired by statistics we provide an explicit local isomorphism with a product of an affine space with a variety which is the Zariski closure of the image of a map defined by generalized determinants. In particular, equations of the secant or tangential variety correspond to relations among generalized determinants. We also provide a representation theoretic decomposition of cubics in the ideal of the secant variety of any Grassmannian.

Toric geometry of the 3-Kimura model for any tree

In this paper we present geometric features of group based models. We focus on the 3-Kimura model. We present a precise geometric description of the variety associated to any tree on a Zariski open set. In particular this set contains all biologically meaningful points. Our motivation is a conjecture of Sturmfels and Sullivant on the degree in which the ideal associated to 3-Kimura model is generated.

A short proof of Combinatorial Nullstellensatz

Abstract In this note we give a short, direct proof of the combinatorial Nullstellensatz.

On the graph labellings arising from phylogenetics

We study semigroups of labellings associated to a graph. These generalise the Jukes-Cantor model and phylogenetic toric varieties defined in [Buczynska W., Phylogenetic toric varieties on graphs, J. Algebraic Combin., 2012, 35(3), 421–460]. Our main theorem bounds the degree of the generators of the semigroup by g + 1 when the graph has first Betti number g. Also, we provide a series of examples where the bound is sharp.

Family of counterexamples to King's conjecture

Abstract In this short Note we present an infinite family of arbitrary high dimensional counterexamples to the Kings conjecture.