Related papers: Virtual Resolutions of Monomial Ideals on Toric Va…
The central fiber of a Gross-Siebert type toric degeneration is known to satisfy homological mirror symmetry: its category of coherent sheaves is equivalent to the wrapped Fukaya category of a certain exact symplectic manifold. Here we show…
Recently, Marian-Oprea-Pandharipande established (a generalization of) Lehn's conjecture for Segre numbers associated to Hilbert schemes of points on surfaces. Extending work of Johnson, they provided a conjectural correspondence between…
We use multiplication maps to give a characteristic-free approach to vanishing theorems on toric varieties. Our approach is very elementary but is enough powerful to prove vanishing theorems.
We use the Taylor resolution of a monomial ideal to compute the Tor algebra of the Stanley-Reisner ring of a simplicial complement of a simplicial complex.
For four elements of a Noetherian ring we construct complexes of free modules of length three (resp. five) by an explicit description of the homomorphisms of the free modules. We provide exactness criteria for them. As an application we use…
Given multigraded free resolutions of two monomial ideals we construct a multigraded free resolution of the sum of the two ideals.
Tropical polytopes are images of polytopes in an affine space over the Puiseux series field under the degree map. This viewpoint gives rise to a family of cellular resolutions of monomial ideals which generalize the hull complex of Bayer…
Toric quiver varieties (moduli spaces of quiver representations) are studied. Given a quiver and a weight there is an associated quasiprojective toric variety together with a canonical embedding into projective space. It is shown that for a…
Proofs of two results about a monomial ideal -- describing membership in auxiliary ideals associated to the monomial ideal -- are given which do not invoke resolution of singularities. The AM--GM inequality is used as a substitute for…
In this paper we find monomial bases for the integer cohomology rings of compact wonderful models of toric arrangements. In the description of the monomials various combinatorial objects come into play: building sets, nested sets, and the…
In this thesis, we focus on the study of some classes of monomial ideals, namely lexsegment ideals and monomial ideals with linear quotients.
In this paper we study the topology of the moduli spaces of representations of degree $2$ for free monoids. We calculate the virtual Hodge polynomials of the character varieties for several types of $2$-dimensional representations.…
We use the language of multiplier ideals in order to relate the syzygies of an abelian variety in a suitable embedding with the local positivity of the line bundle inducing that embedding. This extends to higher syzygies a result of Hwang…
In this paper we study some algebraic and combinatorial behaviors of expansion functor. We show that on monomial ideals some properties like polymatroidalness, weakly polymatroidalness and having linear quotients are preserved under taking…
Building on work of Briggs, Grifo and Pollitz arXiv:2506.10827, we give an example of two cohomological support varieties of monomial ideals which are not unions of linear subspaces. We provide a procedure for the computation of the…
Infinite hyperplane arrangements whose vertices form a lattice are studied from the point of view of commutative algebra. The quotient of such an arrangement modulo the lattice action represents the minimal free resolution of the associated…
For each squarefree monomial ideal $I\subset S = k[x_{1},\ldots, x_{n}] $, we associate a simple graph $G_I$ by using the first linear syzygies of $I$. In cases, where $G_I$ is a cycle or a tree, we show the following are equivalent: (a) $…
A short proof of the "Rigidity theorem" using the sheaf theoretic model for Hilbert modules over polynomial rings is given. The joint kernel for a large class of submodules is described. The completion $[\mathcal I]$ of a homogeneous…
We survey recent developments in the study of SYZ mirror symmetry for compact toric and toric Calabi-Yau varieties, with a special emphasis on works of the author and his collaborators.
In a toric symplectic manifold, regular fibres of the moment map are Lagrangian tori which are called toric fibres. We discuss the question which two toric fibres are equivalent up to a Hamiltonian diffeomorphism of the ambient space. On…