Related papers: Explicit constructions of short virtual resolution…
We construct minimal resolutions of pushforwards of structure sheaves of toric substacks of smooth toric stacks by line bundles as strong deformation retracts of cellular resolutions constructed by Hanlon, Hicks and Lazarev. We also provide…
We give a short, new proof of a recent result of Hanlon-Hicks-Lazarev about toric varieties. As in their work, this leads to a proof of a conjecture of Berkesch-Erman-Smith on virtual resolutions and to a resolution of the diagonal in the…
Resolutions of the diagonal of toric varieties has been an active area of study since Beilinson's celebrated resolution of the diagonal for $\PP^n$ and the disproof of King's conjecture. The author generalized a cellular resolution of the…
We introduce the notion of a conical symplectic variety, and show that any symplectic resolution of such a variety is isomorphic to the Springer resolution of a nilpotent orbit in a semisimple Lie algebra, composed with a linear projection.
Hanlon, Hicks and Lazarev constructed resolutions of structure sheaves of toric substacks by certain line bundles on the ambient toric stacks. In this paper, we give a new and substantially simpler proof of their result.
Beilinson first gave a resolution of the diagonal for $\mathbb{P}^n$. Generalizing this, a modification of the cellular resolution of the diagonal given by Bayer-Popescu- Sturmfels gives a (non-minimal, in general) virtual resolution of the…
For any toric ideal $I$ in a polynomial ring $S$, we provide a combinatorial description of a free resolution of the integral closure of the $S$-module $S/I$. These new complexes arise from an extension of Bayer--Sturmfels' theory of…
We consider a finite collection of line bundles $\Phi$ introduced by Bondal on a smooth, projective toric variety $X$. For any coherent sheaf $F$ on $X$, we construct minimal resolutions of $F$ by line bundles in $\Phi$, up to twist, with…
Let $k$ be a perfect field and let $C_0:f=0$ be a smooth curve in the torus $\mathbb{G}_{m,k}^2$. Let $\mathbb{T}_\Delta$ be the toric variety associated to the Newton polygon of $f$. Extending the toric resolution of $C_0$ on…
We present a summary of some results from our article [BZ1] and other recent results on the so-called LVMB manifolds. We emphasize some features by taking a different point of view. We present a simple variant of the Delzant construction,…
Syzygies capture intricate geometric properties of a subvariety in projective space. However, when the ambient space is a product of projective spaces or a more general smooth projective toric variety, minimal free resolutions over the Cox…
This paper has three main goals : (1) To give an axiomatic formulation of the construction of "reduced \v{C}ech complexes", complexes using fewer than the usual number of intersections but still computing cohomology of an appropriate class…
We introduce topological notions of normal crossings symplectic divisor and variety and establish that they are equivalent, in a suitable sense, to the desired geometric notions. Our proposed concept of equivalence of associated topological…
We use cellular resolutions of monomial ideals to prove an analog of Hilbert's syzygy theorem for virtual resolutions of monomial ideals on smooth toric varieties.
In recent work, we introduced topological notions of simple normal crossings symplectic divisor and variety, showed that they are equivalent, in a suitable sense, to the corresponding geometric notions, and established a topological…
Given any toric subvariety $Y$ of a smooth toric variety $X$ of codimension $k$, we construct a length $k$ resolution of $\mathcal O_Y$ by line bundles on $X$. Furthermore, these line bundles can all be chosen to be direct summands of the…
We study smooth projective varieties with small dual variety using methods from symplectic topology. We prove the affine parts of such varieties are subcritical, and that the hyperplane class is invertible in their quantum cohomology. We…
In this thesis we use the Beauville-Bogomolov decomposition to compute the LLV algebra of smooth projective complex varieties admitting a holomorphic symplectic form, generalizing known results from hyperk\"ahler and abelian varieties.…
Given a smooth projective toric variety $X$ of Picard rank 2, we resolve the diagonal sheaf on $X \times X$ by a linear complex of length $\dim{X}$ consisting of finite direct sums of line bundles. As applications, we prove a new case of a…
In the algebraic setting, cluster varieties were reformulated by Gross-Hacking-Keel as log Calabi-Yau varieties admitting a toric model. Building on work of Shende-Treumann-Williams-Zaslow in dimension 2, we describe the mirror to the GHK…