arXiv subjects
Publications and source records attributed to Christopher Manon.
We give an unconditional proof of the Coba conjecture for wonderful compactifications of adjoint type for semisimple Lie groups of type $A_n$. We also give a proof of a slightly weaker conjecture for wonderful compactifications of adjoint type for arbitrary Lie groups.
We introduce the notion of a polyptych lattice, which encodes a collection of lattices related by piecewise linear bijections. We initiate a study of the new theory of convex geometry and polytopes associated to polyptych lattices. In certain situations, such a polytope associated to a polyptych lattice encodes a compactification of an affine variety whose coordinate ring can be equipped with a valuation into a certain semialgebra associated to the polyptych lattice. We show that aspects of the geometry of the compactification can be understood combinatorially; for instance, under some hypotheses, the resulting compactifications are arithmetically Cohen-Macaulay, and have finitely generated class group and finitely generated Cox rings.
The notion of polyptych lattices, introduced by Escobar, Harada, and Manon, wraps the data of a collection of lattices related by piecewise-linear bijections together into a single semi-algebraic object, equipped with its own notions of convexity and polyhedra. The main purpose of this manuscript is to construct an explicit family of polyptych lattices, and to illustrate via explicit computations the abstract theory introduced by Escobar-Harada-Manon. Specifically, we first construct a family of rank-$2$ polyptych lattices $\mathcal{M}_s$ with $2$ charts, compute their space of points, and prove that they are full and self-dual. We then give a concrete sample computation of a point-convex hull in $\mathcal{M}_s \otimes \mathbb{R}$ to illustrate that convex geometry in the polyptych lattice setting can exhibit phenomena not seen in the classical situation. We also give multiple examples of $2$-dimensional ``chart-Gorenstein-Fano'' polytopes, which give rise to pairs of mutation-related $2$-dimensional (classical) Gorenstein-Fano polytopes. Finally, we produce detropicalizations $(\mathcal{A}_s, \mathfrak{v}_s)$ of $\mathcal{M}_s$, and in the case $s=1$ where the detropicalization is a UFD, and with respect to a certain choice of PL polytope $\mathcal{P}$, we give an explicit generators-and-relations presentation of the (finitely generated) Cox ring of the compactification $X_{\mathcal{A}_s}(\mathcal{P})$ of $\mathrm{Spec}(\mathcal{A}_s)$ with respect to $\mathcal{P}$.
We give a combinatorial classification of torus equivariant vector bundles on a (normal) projective T-variety of complexity-one. This extends the classification of equivariant line bundles on complexity-one T-varieties by Petersen-S\"uss on one hand, and Klyachko's classification of equivariant vector bundles on toric varieties on the other hand. A main ingredient in our classification is the classification of torus equivariant vector bundles on toric schemes over a DVR in terms of piecewise affine maps to the (extended) Bruhat-Tits building of the general linear group.
We introduce a notion of tropical vector bundle on a tropical toric variety which is a tropical analogue of a torus equivariant vector bundle on a toric variety. Alternatively it can be called a toric matroid bundle. We define equivariant $K$-theory and characteristic classes of these bundles. As a particular case, we show that any matroid comes with tautological tropical toric vector bundles over the permutahedral toric variety and the corresponding equivariant $K$-classes and Chern classes recover the tautological classes of matroids constructed in the recent work of Berger-Eur-Spink-Tseng. In analogy with toric vector bundles, we define sheaf of sections and Euler characteristic as well as positivity notions such as global generation, ampleness and nefness for tropical toric vector bundles. Moreover, we prove a vanishing of higher cohomologies result. Finally, we study the splitting of our tropical toric vector bundles and, in particular, an analogue of Grothendieck's theorem on splitting of vector bundles on projective line.
We define equivariant Chern classes of a toric vector bundle over a proper toric scheme over a DVR. We provide a combinatorial description of them in terms of piecewise polynomial functions on the polyhedral complex associated to the toric scheme, which factorize through to an extended Bruhat--Tits building. We further motivate this definition from an arithmetic perspective, connecting to the non-Archimedean Arakelov theory of toric varieties.
We use presentations of the Cox rings of projectivized toric vector bundles and elements of matroid theory to compute Newton-Okounkov bodies, effective cones, and nef cones of these spaces. As an application we analyze the Fano property and establish Fujita's freeness and ampleness conjectures for several classes of projectivized toric vector bundles.
We construct a presentation for the Cox ring of the projectivization $\mathbb{P}\mathcal{E}$ of any rank $n$ irreducible toric vector bundle on $\mathbb{P}^n$. We use this presentation to show that $\mathbb{P}\mathcal{E}$ always satisfies Fujita's freeness and ampleness conjectures.
A toric vector bundle $\mathcal{E}$ is a torus equivariant vector bundle on a toric variety. We give a valuation theoretic and tropical point of view on toric vector bundles. We present three (equivalent) classifications of toric vector bundles, which should be regarded as repackagings of the Klyachko data of compatible $\mathbb{Z}$-filtrations of a toric vector bundle: (1) as piecewise linear maps to space of $\mathbb{Z}$-valued valuations, (2) as valuations with values in the semifield of piecewise linear functions, and (3) as points in tropical linear ideals over the semifield of piecewise linear functions. Moreover, we interpret the known criteria for ampleness and global generation of $\mathcal{E}$ as convexity conditions on its piecewise linear map in (1). Finally, using (2) we associate to $\mathcal{E}$ a collection of polytopes indexed by elements of a certain (representable) matroid encoding the dimensions of weight spaces of global sections of $\mathcal{E}$. This recovers and extends the Di Rocco-Jabbusch-Smith matriod and parliament of polytopes of $\mathcal{E}$. This is a follow up paper to arXiv:1806.05613.
We show that a certain compactification $\mathfrak{X}_g$ of the $\mathrm{SL}_2(\mathbb{C})$ free group character variety $\mathcal{X}(F_g, \mathrm{SL}_2(\mathbb{C}))$ is Fano. This compactification has been studied previously by the second author, and separately by Biswas, Lawton, and Ramras. Part of the proof of this result involves the construction of a large family of integral reflexive polytopes.
We give a classification of rank $r$ torus equivariant vector bundles $\mathcal{E}$ on a toric scheme $\mathfrak{X}$ over a discrete valuation ring $\mathcal{O}$, in terms of graded piecewise linear maps $\Phi$ from the fan of $\mathfrak{X}$ to the (extended) building of $GL(r)$. This is an extension of Klyachko's classification of torus equivariant vector bundles on toric varieties over a field on one hand, and Mumford's classification of equivariant line bundles on toric schemes over $\mathcal{O}$ on the other hand. We also give a simple criterion for equivariant splitting of $\mathcal{E}$ into a sum of toric line bundles in terms of its piecewise linear map. Among other things, this work lays the foundations for study of arithmetic geometry of toric vector bundles.
Work of Gonz\'alez, Hering, Payne, and S\"uss shows that it is possible to find both examples and non-examples of Mori dream spaces among projectivized toric vector bundles. This result, and the combinatorial nature of the data of projectivized toric vector bundles make them an ideal test class for the question: what makes a variety a Mori dream space? In the present paper we consider this question with respect to natural algebraic operations on vector bundles. Suppose $\mathcal{E}$ is a toric vector bundle such that the projectivization $\mathbb{P}\mathcal{E}$ is a Mori dream space, then when are the direct sum bundles $\mathbb{P}(\mathcal{E} \oplus \mathcal{E})$, $\mathbb{P}(\mathcal{E} \oplus \mathcal{E} \oplus \mathcal{E})\ldots$ also Mori dream spaces? We give an answer to this question utilizing a relationship with the associated full flag bundle $\mathcal{FL}(\mathcal{E})$. We describe several classes of examples, and we compute a presentation for the Cox ring of the full flag bundle for the tangent bundle of projective space.
Phylogenetic networks can model more complicated evolutionary phenomena that trees fail to capture such as horizontal gene transfer and hybridization. The same Markov models that are used to model evolution on trees can also be extended to networks and similar questions, such as the identifiability of the network parameter or the invariants of the model, can be asked. In this paper we focus on finding the invariants of the Cavendar-Farris-Neyman (CFN) model on level-1 phylogenetic networks. We do this by reducing the problem to finding invariants of sunlet networks, which are level-1 networks consisting of a single cycle with leaves at each vertex. We then determine all quadratic invariants in the sunlet network ideal which we conjecture generate the full ideal.
An embedded variety is said to be well-poised when the associated initial ideal degenerations coming from points of the tropical variety are reduced and irreducible. Varieties with a well-poised embedding admit a large collection of explicitly constructible Newton-Okounkov bodies. This paper aims to study the well-poised property under torus quotients. Our first result states that GIT quotients of normal well-poised varieties by quasi-tori also have well-poised embeddings. As an application, we show that several Hassett spaces, $\overline{M}_{0,\beta}$, are well-poised under Alexeev's embedding. Conversely, given an affine $T$-variety $X$ with polyhedral divisor $\mathfrak{D}$ on a well-poised base $Y$, we construct an embedding of $X \subseteq \mathbb{A}^N$ and provide conditions on $Y$ and $\mathfrak{D}$ which if met, imply $X$ is well-poised under this embedding. Then we show that any affine arrangement variety meets the specified criteria, generalizing results of Ilten and the second author for rational complexity 1 varieties. Using this result, we explicitly compute many Newton-Okounkov cones of $X$ and provide a criterion for the associated toric degenerations to be normal. Our final application combines these two results to show that hypertoric varieties have well-poised embeddings.
We study the generic tropical initial ideals of a positively graded Cohen-Macaulay algebra $R$ over an algebraically closed field $\mathbf{k}$. Building on work of R\"omer and Schmitz, we give a formula for each initial ideal, and we express the associated quasivaluations in terms of certain $I$-adic filtrations. As a corollary, we show that in the case that $R$ is a domain, every initial ideal coming from the codimension-$1$ skeleton of the tropical variety is prime, so "generic presentations of Cohen-Macaulay domains are well-poised in codimension-$1$."
An ideal $I$ is said to be "well-poised" if all of the initial ideals obtained from points in the tropical variety $Trop(I)$ are prime. This condition was first defined by Nathan Ilten and the third author. We classify all well-poised hypersurfaces over an algebraically closed field. We also study the tropical varieties and associated Newton-Okounkov bodies of these hypersurfaces.
Using the notion of a valuation into the semifield of piecewise linear functions, we give a classification of torus equivariant flat families of finite type over a toric variety base, by certain piecewise linear maps between fans. As a consequence we derive a classification of toric vector bundles phrased in terms of tropicalized linear spaces. We use these tools to give a characterization of the Mori dream space property for a projectivized toric vector bundle.
We construct a family of compactifications of the affine cone of the Grassmannian variety of 2-planes. We show that both the tropical variety of the Pl\"ucker ideal and familiar valuations associated to the construction of Newton-Okounkov bodies for the Grassmannian variety can be recovered from these compactifications. In this way, we unite various perspectives for constructing toric degenerations of flag varieties.