arXiv subjects

Lauren Cranton Heller

Publications and source records attributed to Lauren Cranton Heller.

8 recordsLinked to original sources

arXiv2026

Characterizing Multigraded Regularity and Virtual Resolutions on Products of Projective Spaces

We explore the relationship between multigraded Castelnuovo--Mumford regularity, truncations, Betti numbers, and virtual resolutions on a product of projective spaces $X$. After proving a uniqueness theorem for certain virtual resolutions, we show that the multigraded regularity region of a module $M$ is determined by the minimal graded free resolutions of the truncations $M_{\geq\mathbf d}$ for $\mathbf d\in\operatorname{Pic} X$. Further, by relating the minimal graded free resolutions of $M$ and $M_{\geq\mathbf d}$ we provide a new bound on multigraded regularity of $M$ in terms of its Betti numbers. Using this characterization of regularity and this bound we also compute the multigraded Castelnuovo--Mumford regularity for a wide class of complete intersections in products of projective spaces.

math.AC↗

arXiv2025

Cellular free resolutions for normalizations of toric ideals

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 cellular free resolutions. As applications, we unify several constructions for a resolution of the diagonal embedding of a toric variety, and compare the locally free resolutions for toric subvarieties introduced by Hanlon--Hicks--Lazarev and Brown--Erman.

math.AC↗

arXiv2025

King's Conjecture and the Cox category

We state and prove a realization of King's Conjecture for a category glued from the derived categories of all of the toric varieties arising from a given Cox ring. Our perspective extends ideas of Beilinson and Bondal to all semiprojective toric varieties.

math.AG↗

arXiv2025

Seshadri Regions and the Asymptotic Shape of Multigraded Regularity

We introduce the Seshadri region of a subvariety, a convex region packaging the classical Seshadri constants with respect to every line bundle simultaneously. We develop the theory of Seshadri regions as a measure of positivity along subvarieties and apply it to determine asymptotic Castelnuovo-Mumford regularity for ideal powers and symmetric powers on smooth projective toric varieties.

math.AG↗

arXiv2025

Explicit constructions of short virtual resolutions of truncations

We propose a concept of truncation for arbitrary smooth projective toric varieties and construct explicit cellular resolutions for nef truncations of their total coordinate rings. We show that these resolutions agree with the short resolutions of Hanlon, Hicks, and Lazarev, which were motivated by symplectic geometry, and we use our definition to exhibit nontrivial homology in the commutative algebraic analogue of their construction.

math.AG↗

arXiv2025

Bounds on Multigraded Regularity

Multigraded Castelnuovo--Mumford regularity of a module $M$ over the total coordinate ring $S$ of a smooth projective toric variety $X$ is a region $\operatorname{reg} M \subset \operatorname{Pic} X$ invariant under translation by the nef cone $\operatorname{Nef} X$. We prove that the multigraded regularity of a finitely generated faithful module is contained in a translate of $\operatorname{Nef} X$ determined by the degrees of the generators of $M$, and thus contains only finitely many minimal elements. We show that this condition can fail even for cyclic modules if $M$ has torsion and the rank of the Picard group is at least two. As an application, we exhibit asymptotic bounds for the multigraded regularity of powers of ideals. For $I$ an ideal in $S$, we bound $\operatorname{reg}(I^n)$ by proving that it contains a translate of $\operatorname{reg} S$ and is contained in a translate of $\operatorname{Nef} X$, where each bound translates by a fixed vector as $n$ increases.

math.AC↗

arXiv2021

On an equivalence of divisors on $\bar{M}_{0,n}$ from Gromov-Witten theory and conformal blocks

We consider a conjecture that identifies two types of base point free divisors on $\bar{M}_{0,n}$. The first arises from Gromov-Witten theory of a Grassmannian. The second comes from first Chern classes of vector bundles associated to simple Lie algebras in type A. Here we reduce this conjecture on $\bar{M}_{0,n}$ to the same statement for $n=4$. A reinterpretation leads to a proof of the conjecture on $\bar{M}_{0,n}$ for a large class, and we give sufficient conditions for the non-vanishing of these divisors.

math.AG↗

arXiv2021

Linear Truncations Package for Macaulay2

We introduce the Macaulay2 package $\mathtt{LinearTruncations}$ for finding and studying the truncations of a multigraded module over a standard multigraded ring that have linear resolutions.

math.AC↗