It is pretty well-known that toric Fano varieties of dimension k with terminal singularities correspond to convex lattice polytopes P in R^k of positive finite volume, such that intersection of P and Z^k consists of the point 0 and vertices of P. Likewise, Q-factorial terminal toric singularities essentially correspond to lattice simplexes with no lattice points inside or on the boundary (except the vertices). There have been a lot work, especially in the last 20 years or so on classification of these objects. The main goal of this paper is to bring together these and related results, that are currently scattered in the literature. We also want to emphasize the deep similarity between the problems of classification of toric Fano varieties and classification of Q-factorial toric singularities.
Toric geometry provides a bridge between the theory of polytopes and algebraic geometry: one can associate to each lattice polytope a polarized toric variety. In this thesis we explore this correspondence to classify smooth lattice polytopes having small degree, extending a classification provided by Dickenstein, Di Rocco and Piene. Our approach consists in interpreting the degree of a polytope as a geometric invariant of the corresponding polarized variety, and then applying techniques from Adjunction Theory and Mori Theory. In the opposite direction, we use the combinatorics of fans to descr…
Toric geometry provides a bridge between algebraic geometry and combinatorics of fans and polytopes. For each polarized toric variety (X,L) we have associated a polytope P. In this thesis we use this correspondence to study birational geometry for toric varieties. To this end, we address subjects such as Minimal Model Program, Mori fiber spaces, and chamber structures on the cone of effective divisors. We translate some results from these theories to the combinatorics of polytopes and use them to get structure theorems on space of polytopes. In particular, we treat toric varieties known as 2-F…
Let X and Y be horospherical Mori fibre spaces which are birational equivariantly with respect to the group action. Then, there is a horospherical Sarkisov program from X/S to Y /T .
A matroid is a machine capturing linearity of mathematical objects and producing combinatorial structures. Matroid structure arises everywhere since linearity is a ubiquitous concept. One natural way to obtain matroids is by considering hyperplane arrangements, which give rise to convex polytopes called matroid polytopes. Much research has been conducted on these three areas: matroids, matroid polytopes, and hyperplane arrangements. However, substantial gaps in our knowledge remain, and the correspondence diagram between those areas needs to be more extensive. For instance, currently, there is…
The Fine interior $Δ^{\text{FI}}$ of a $d$-dimensional lattice polytope $Δ$ is a rational subpolytope of $Δ$ which is important for constructing minimal birational models of non-degenerate hypersurfaces defined by Laurent polynomials with Newton polytope $Δ$. This paper presents some computational results on the Fine interior of all $674,\!688$ three-dimensional canonical Fano polytopes.
In this paper we study the geometry of GIT configurations of $n$ ordered points on $\mathbb{P}^1$ both from the the birational and the biregular viewpoint. In particular, we prove that any extremal ray of the Mori cone of effective curves of the quotient $(\mathbb{P}^1)^n//PGL(2)$, taken with the symmetric polarization, is generated by a one dimensional boundary stratum of the moduli space. Furthermore, we develop some technical machinery that we use to compute the canonical divisor and the Hilbert polynomial of $(\mathbb{P}^1)^n//PGL(2)$ in its natural embedding, and its group of automorphism…
We construct, for each convex polytope, possibly nonrational and nonsimple, a family of compact spaces that are stratified by quasifolds, i.e. each of these spaces is a collection of quasifolds glued together in an suitable way. A quasifold is a space locally modelled on $\R^k$ modulo the action of a discrete, possibly infinite, group. The way strata are glued to each other also involves the action of an (infinite) discrete group. Each stratified space is endowed with a symplectic structure and a moment mapping having the property that its image gives the original polytope back. These spaces m…
We give new estimates of lengths of extremal rays of birational type for toric varieties. We can see that our new estimates are the best by constructing some examples explicitly. As applications, we discuss the nefness and pseudo-effectivity of adjoint bundles of projective toric varieties. We also treat some generalizations of Fujita's freeness and very ampleness for toric varieties.
Bott-Samelson varieties are a twisted product of $\mathbb{C}\mathbb{P}^1$'s with a map into $G/B$. These varieties are mostly studied in the case in which the map into $G/B$ is birational to the image; however in this paper we study a fiber of this map when it is not birational. We will see that in some cases the general fiber, which we christen a brick manifold, is a toric variety. In order to do so we use the moment map of a Bott-Samelson variety to translate this problem into one in terms of the "subword complexes" of Knutson and Miller. Pilaud and Stump realized certain subword complexes a…
We introduce iterated sequences for Grassmannians, a new class of Fang-Fourier-Littelmanns' birational sequences and explain how they give rise to points in $\text{trop}(\text{Gr}(k,\mathbb C^n))$, Speyer-Sturmfels' tropical Grassmannian. For $\text{Gr}(2,\mathbb C^n)$ we show that the associated valuations induce toric degenerations. We describe recursively the vertices of the corresponding Newton--Okounkov polytopes, which are particular vertices of a hypercube and hence integral. We show further that every toric degeneration of $\text{Gr}(2,\mathbb C^{n})$ constructed using the tropical Gra…
In this Lecture Notes we present, in a sufficiently self contained way, our contributions and interests in the field of Minimal Model Theory. We study Fano-Mori spaces, both from the biregular and the birational point of view. For the former we recall and develop Kawamata's Base Point Free technique and some of Mori's deformation arguments. For the latter we lean on Sarkisov and #-Minimal Model Programs. In writing these notes we want to give our point of view on this area of research. We are not trying to give a treatment of the whole subject. These notes collect some topics we presented in t…
In this article we explore compactifications of cluster varieties of finite type in complex dimension two. Cluster varieties can be viewed as the spec of a ring generated by theta functions and a compactification of such varieties can be given by a grading on that ring, which can be described by positive polytopes [17]. In the examples we exploit, the cluster variety can be interpreted as the complement of certain divisors in del Pezzo surfaces. In the symplectic viewpoint, they can be described via almost toric fibrations over $\R^2$ (after completion). Once identifying them as almost toric m…
This paper considers the planar figure of a combinatorial polytope or tessellation identified by the Coxeter symbol $k_{i,j}$ , inscribed in a conic, satisfying the geometric constraint that each octahedral cell has a centre. This realisation exists, and is movable, on account of some constraints being satisfied as a consequence of the others. A close connection to the birational group found originally by Coble in the different context of invariants for sets of points in projective space, allows to specify precisely a determining subset of vertices that may be freely chosen. This gives a unifi…
These notes are based on three lectures given at the 2013 CIME/CIRM summer school. The purpose of this series of lectures is to introduce the notion of a toric fibration and to give its geometrical and combinatorial characterizations. Polarized toric varieties which are birationally equivalent to projective toric bundles are associated to a class of polytopes called Cayley polytopes. Their geometry and combinatorics have a fruitful interplay leading to fundamental insight in both directions. These notes will illustrate geometrical phenomena, in algebraic geometry and neighboring fields, which …
Université de Poitiers（普瓦捷大学）；Université de Bourgogne Europe, CNRS, IMB UMR 5584（勃艮第欧洲大学）；Korea Institute for Advanced Study（韩国高等研究院）；Saitama University（埼玉大学）；Ewha Womans University（梨花女子大学）
Given a Laurent polynomial f, one can form the period of f: this is a function of one complex variable that plays an important role in mirror symmetry for Fano manifolds. Mutations are a particular class of birational transformations acting on Laurent polynomials in two variables; they preserve the period and are closely connected with cluster algebras. We propose a higher-dimensional analog of mutation acting on Laurent polynomials f in n variables. In particular we give a combinatorial description of mutation acting on the Newton polytope P of f, and use this to establish many basic facts ab…
This paper classifies toric Fano 3-folds with singular locus { 1/k(1,1,1) } for any positive integer k, building on the work of Batyrev and Watanabe-Watanabe. This is achieved by completing an equivalent problem in the language of Fano polytopes. Furthermore we identify birational relationships between entries of the classification. For a fixed value k>4, there are exactly two such Fano 3-folds linked by a blow-up in a torus-invariant line.
These notes provide an overview of various notions of hyperbolicity for varieties of log general type from the viewpoint of both arithmetic and birational geometry. The main results are based on our paper entitled "Hyperbolicity and uniformity of varieties of log general type." They are expanded notes from a minicourse the authors gave as part of the Geometry and arithmetic of orbifolds workshop at UQÁM.
