Related papers: Toric varieties from cyclic matrix semigroups
We study the Zariski closure of the monodromy group $\mathbf{Mon}$ of Lauricella's hypergeometric function $F_C$. If the identity component $\mathbf{Mon}^0$ acts irreducibly, then $\overline{\mathbf{Mon}} \cap \mathbf{SL}_{2^n}(\mathbb{C})$…
A toric variety is a normal complex variety which is completely described by combinatorial data, namely by a fan of strongly convex rational (with respect to a lattice) cones. Due to this rationality condition, toric varieties are…
Consider an integer associated with every subset of the set of columns of an $n\times k$ matrix. The collection of those matrices for which the rank of a union of columns is the predescribed integer for every subset, will be denoted by…
The article demonstrates the procedure how to compute the Zariski closure of an orbit by an algebraic action of finitely generated group on the affine plane. First half of the algorithm is about deciding whether given finitely generated…
In this paper, we describe a way to construct cycles which represent the Todd class of a toric variety. Given a lattice with an inner product we assign a rational number m(s) to each rational polyhedral cone s in the lattice, such that for…
Let $G$ be a simple algebraic group over an algebraically closed field and let $X$ be an irreducible subvariety of $G^r$ with $r \geqslant 2$. In this paper, we consider the general problem of determining if there exists a tuple $(x_1,…
In a previous paper (joint with Hausel and Rodriguez-Villegas) we gave a conjectural formula for the mixed Hodge polynomials of character varieties with generic semisimple conjugacy classes at punctures and we prove a formula for the…
Let $X$ be an affine scheme of $k \times \mathbb{N}$-matrices and $Y$ be an affine scheme of $\mathbb{N} \times \cdots \times \mathbb{N}$-dimensional tensors. The group Sym$(\mathbb{N})$ acts naturally on both $X$ and $Y$ and on their…
Let $n$ be an even natural number. We compute the periods of any $\frac{n}{2}$-dimensional complete intersection algebraic cycle inside an $n$-dimensional non-degenerated intersection of a projective simplicial toric variety. Using this…
A description of transitive actions of a semisimple algebraic group G on toric varieties is obtained. Every toric variety admitting such an action lies between a product of punctured affine spaces and a product of projective spaces. The…
We study holomorphic foliations of aribitrary codimension in smooth complete toric varieties. We show that split foliations are stable if some good behaviour of their singular set is provided. As an application of these results, we exhibit…
We explore systems of polynomial equations where we seek complex solutions with absolute value 1. Geometrically, this amounts to understanding intersections of algebraic varieties with tori -- Cartesian powers of the unit circle. We study…
In this paper, we prove that: For any given finitely many distinct points $P_1,...,P_r$ and a closed subvariety $S$ of codimension $\geq 2$ in a complete toric variety over a uncountable (characteristic 0) algebraically closed field, there…
We consider the set of forms of a toric variety over an arbitrary field: those varieties which become isomorphic to a toric variety after base field extension. In contrast to most previous work, we also consider arbitrary isomorphisms…
We investigate Gauss maps of (not necessarily normal) projective toric varieties over an algebraically closed field of arbitrary characteristic. The main results are as follows: (1) The structure of the Gauss map of a toric variety is…
Empirical properties of generating systems for complex reflection groups and their braid groups have been observed by Orlik-Solomon and Brou\'e-Malle-Rouquier, using Shephard-Todd classification. We give a general existence result for…
We study the divisorial Zariski decomposition on varieties whose first Chern class is zero. We first prove that any exceptional divisor is contractible (up to a birational map that is an isomorphism in codimension one). We then characterize…
We study, for plane complex branches of genus one, the topological type of its generic polar curve, as a function of the semigroup of values and the Zariski invariant of the branch. We improve some results given by Casas-Alvero in 2023,…
We classify the Rauzy-Veech groups of all connected components of all strata of the moduli space of translation surfaces in absolute homology, showing, in particular, that they are commensurable to arithmetic lattices of symplectic groups.…
A framework is developed to describe the Zariski topologies on the prime and primitive spectra of a quantum algebra $A$ in terms of the (known) topologies on strata of these spaces and maps between the collections of closed sets of…