arXiv subjects

Sasha Zotine

Publications and source records attributed to Sasha Zotine.

7 recordsLinked to original sources

arXiv2025

Dynamics of Projectivized Toric Vector Bundles

We study surjective endomorphisms of projective bundles over toric varieties, achieving three main results. First, we provide a structural theorem describing endomorphisms of projectivized split bundles over arbitrary base varieties, which we use to classify all surjective endomorphisms of Hirzebruch surfaces and construct novel families of examples. Second, for non-split equivariant bundles over toric varieties, we prove that the dynamical degree of an endomorphism of the projectivization is controlled by the base morphism; as a consequence, we establish the Kawaguchi--Silverman conjecture for such bundles. Third, using an explicit transition function method, we prove that projectivizations of tangent and cotangent bundles of smooth toric varieties admit no non-automorphic surjective endomorphisms commuting with toric morphisms on the base.

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arXiv2025

$C_p$-Mackey functors in Macaulay2

We introduce the CpMackeyFunctors package for Macaulay2, which allows for computations with Mackey functors over a cyclic group of prime order.

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arXiv2025

Computing higher direct images in Macaulay2

This article highlights the ToricHigherDirectImages package in Macaulay2. The central feature is a method for computing (higher) direct images of line bundles under surjective toric morphisms.

math.AG↗

arXiv2025

Reduced \v{C}ech complexes and computing higher direct images under toric maps

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 of sheaves; (2) To give a construction of such a reduced \v{C}ech complex for every semi-proper toric variety $X$, where every open used in the complex is torus stable, and such that the cell complex governing the reduced \v{C}ech complex has dimension the cohomological dimension of $X$; and (3) to give an algorithm to compute the higher direct images of line bundles relative to a toric fibration between smooth proper toric varieties.

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arXiv2025

The $h$-vectors of toric ideals of odd cycle compositions revisited

Let $G$ be a graph consisting of $s$ odd cycles that all share a common vertex. Bhaskara, Higashitani, and Shibu Deepthi recently computed the $h$-polynomial for the quotient ring $R/I_G$, where $I_G$ is the toric ideal of $G$, in terms of the number and sizes of odd cycles in the graph. The purpose of this note is to prove the stronger result that these toric ideals are geometrically vertex decomposable, which allows us to deduce the result of Bhaskara, Higashitani, and Shibu Deepthi about the $h$-polyhomial as a corollary.

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arXiv2024

The multigraded BGG correspondence in Macaulay2

We give an overview of a Macaulay2 package for computing with the multigraded BGG correspondence. This software builds on the package BGG due to Abo-Decker-Eisenbud-Schreyer-Smith-Stillman, which concerns the standard graded BGG correspondence. In addition to implementing the multigraded BGG functors, this package includes an implementation of differential modules and their minimal free resolutions, and it contains a method for computing strongly linear strands of multigraded free resolutions.

math.AC↗

arXiv2023

Dynamics of Endomorphisms for Projective Bundles on Elliptic Curves

We study the dynamics of surjective endomorphisms of projective bundles on elliptic curves and relate their dynamical properties to the geometry of the bundle. As an application we prove the Kawaguchi--Silverman conjecture for projective bundles on elliptic curves, thereby completing the conjecture for all projective bundles on curves. Our approach is to use the transition functions of the bundles. This allows us to further prove the conjecture for projective split bundles on a smooth projective variety with finitely generated Mori cone.

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