Research

Research Areas: algebraic geometry, derived categories and homological algebra, symplectic geometry, homological mirror symmetry

Other topics I am interested in: enumerative geometry, quantum cohomology and stable maps, localization arguments in equivariant cohomology

Recently, my work has focused on giving presentations of derived categories of smooth projective toric Fano varieties in the presence of a full strong exceptional collection of line bundles. This is related to both the modified King's conjecture and homological mirror symmetry. My research in graduate school focused on derived categories of smooth toric varieties and toric Deligne-Mumford stacks which are global quotients of smooth toric varieties by finite abelian groups. In my dissertation, I developed a cellular resolution of the diagonal for toric D-M stacks, and my current research project focuses on implications of this resolution of the diagonal for full strong exceptional collections of line bundles on toric D-M stacks.

Materials from the Specialty Exam (QEII) at Kansas State University:

Sample Publications and Pre-Prints

"Rationality of cohomological descendent series for Quot schemes on surfaces with $p_g=0$ ." Here we prove rationality for PT descendent series on smooth projective connected algebraic surfaces S in the remaining case p_g(S)=0, \beta\neq 0, N>1 for rank 0 quotients Q. To do this, we apply the wall-crossing machinery from joint work with Dominic Joyce on Pandharipande-Thomas stable pairs developed for smooth projective complex threefolds. Submitted.
``Planar rank-one sheaves on $\mathbb{P}^3$, obstruction bundles, and divisor-supported Donaldson-Thomas series." Studies the moduli space of Gieseker semistable sheaves with Chern character $\alpha_n = (0,1,-1/2, -n+1/6)$ on $\PP^3$, and by relating this moduli space to $\mathrm{Hilb}^n(\PP^2)$ and $(\PP^3)^*$, gives a closed form for the generating series of the natural point-inserted 2-dimensional Donaldon-Thomas invariants on $\PP^3$ in terms of the $\eta$ function. We then generalize to a study of two-dimensional sheaves on Fano 3-folds supported on a smooth divisor, distinguishing moving and rigid divisors, and giving a similar generating series. We work out the examples of the exceptional divisor in $\operatorname{Bl}_p\PP^3$ and of a rigid section in a Fano $\PP^1$-bundle over $\PP^1\times\PP^1$. https://www.researchgate.net/publication/402197182_Planar_rank-one_sheaves_on_PP3_obstruction_bundles_and_divisor-supported_Donaldson-Thomas_series. 3/15/2026. Submitted.

"The Pandharipande-Thomas rationality conjecture for superpositive curve classes on projective complex 3-manifolds." Joint with Dominic Joyce. Here we prove rationality of Pandharipande-Thomas descendent generating series for superpositive curve classes on smooth projective complex 3-folds using Joyce's theory of wall-crossing for enumerative invariants of \C-linear additive categories. arXiv: https://arxiv.org/abs/2604.05664
"Examples of descendent generating series for Pandharipande-Thomas stable pairs on Fano threefolds via one-dimensional wall-crossing." Here we give examples of computation of the sheaf-theoretic Donaldson-Thomas and Pandharipande-Thomas stable pair invariants for smooth projective Fano 3-folds which appear in the setup of joint work with Dominic Joyce on wall-crossing for Pandharipande-Thomas stable pairs. We work in Gross' superpolynomial algebra describing the rational homology of the piecewise linear rigidified higher stack of objects in D^b\coh(X) for X a smooth projective Fano 3-fold, as well as in the homology of Joyce's Pairs category for writing down our invariants. Examples are computed for X = \PP^3, the blow-up of \PP^3 at a torus-invariant point and line, respectively, a quartic threefold, and the projectivization of a rank 2 vector bundle over \PP^1 \times \PP^1: X \cong \PP_{\PP^1 \times \PP^1}( \OO \oplus \OO(-1,-1) )$. We compare with previously computed invariants in the literature due to Pandharipande for \PP^3 and Moreira for a quartic threefold, respectively, and confirm that the difference of our large $n$ tails and the full PT stable pair descendent series is a Laurent polynomial, as expected. https://www.researchgate.net/publication/400040590_Examples_of_descendent_generating_series_for_Pandharipande-Thomas_stable_pairs_on_Fano_threefolds_via_one-dimensional_wall-crossing.
"Enumerative Geometry and Tree-Level Gromov-Witten Invariants." arXiv: https://arxiv.org/abs/2501.03232. Expository work, with some novel exposition in computing genus 0 Gromov-Witten potential for a genus g Riemann surface, and in comparing Kalashnikov's quiver-theoretic formula for Givental's small J-function for smooth Fano zero loci in quiver flag varieties with the conventions in Cox-Katz for the example of CP^1. Develops background in differential geometry and torus-equivariant cohomology to give chern class and localization arguments for 27 lines on a non-singular cubic surface; 2875 lines on a generic non-singular quintic threefold. Defines Gromov-Witten invariants from an algebro-geometric perspective to construct genus 0 Gromov-Witten potential on genus g Riemann surface and CP^2. Constructs small and big quantum cohomology for CP^r, and recovers Kontsevich's recursive formula for the number of rational degree d plane curves through 3d-1 points. Also available here.
"Exceptional Collections for Toric Fano Fourfolds." arXiv: https://arxiv.org/abs/2410.23290. Joint with CMC undergraduate students Jumari Querimit-Ramirez, Hill Zhang, and Justin Son.

Investigates question "For which smooth projective toric Fano 4-folds does the Hanlon-Hicks-Lazarev resolution of the diagonal yield a full strong exceptional collection of line bundles?" We find that for 72/124 such 4-folds, the Hanlon-Hicks-Lazarev resolution of the diagonal yields such a collection. Furthermore, direct calculation yields that the success of the Hanlon-Hicks-Lazarev resolution of the diagonal to yield a full strong exceptional collection of line bundles coincides with a numerical criterion described by Bondal in the context of toric mirror symmetry.

"Strong Full Exceptional Collections for Unimodular Toric Surfaces." arXiv preprint: https://arxiv.org/abs/2403.09663. Submitted.

Shows that the cellular resolution of the diagonal constructed in my dissertation yields a full strong exceptional collection of line bundles for all 5 smooth projective toric Fano surfaces, and that the Hanlon-Hicks-Lazarev resolution of the diagonal yields such a collection for 16/18 smooth projective toric Fano 3-folds.

A Resolution of the Diagonal for Smooth Toric Varieties. Submitted. arXiv preprint: https://arxiv.org/abs/2403.09653

First summary article from my PhD dissertation, which constructs a cellular resolution of the diagonal for smooth projective toric varieties. This generalized Bayer-Popescu-Sturmfels previous cellular resolution of the diagonal in what they called the "unimodular" case, which is more restrictive than being smooth. In the smooth non-unimodular case, extra vertices in a finite quotient cellular complex supporting the resolution appear, which we deal with by using the floor function to assign Laurent monomial labelings on vertices in a convex manner, and show that this assignment is compatible with the class group-graded algebra appearing in the correspondence between coherent sheaves and graded modules over the homogeneous coordinate ring.

A Resolution of the Diagonal for Toric Deligne-Mumford Stacks. Accepted for publication in the Journal of Commutative Algebra.

Second summary article from my PhD dissertation, which uses the above resolution of the diagonal for smooth projective toric varieties to construct a cellular resolution of the diagonal for toric Deligne-Mumford stacks which are a global quotient of a smooth projective variety by a finite abelian group. Local diagonal object in $A \otimes A^{op}$-bimodules constructed via Morita theory, which is then shown to be isomorphic to generalization of diagonal object constructed in Bayer-Sturmfels and Bayer-Popescu-Sturmfels. Gluing argument then given, using the fact that a smooth toric variety is locally isomorphic to the affine n space.

A Resolution of the Diagonal for Toric Deligne-Mumford Stacks. Selected portions of my PhD dissertation for Kansas State University. https://arxiv.org/abs/2303.17497
``The Hilbert Polynomial as an Embedded Subvariety": 10.13140/RG.2.2.16238.69443. https://www.researchgate.net/publication/363796564_The_Hilbert_Polynomial_as_an_Embedded_Subvariety. 23 September 2022.
Anderson, Reginald C. ``What is an Infinitesimal?" AMS Graduate Student Blog, 28 August 2019. Web. https://blogs.ams.org/mathgradblog/2019/08/28/32749
Anderson, R. and D. Havlick. ``History and Values in Ecological Restoration Workshop." Ecological Restoration 31.1 (2013): 7-10. Web.
Anderson, Reginald C. ``Growing Up Between." The Good Men Project - Ethics and Values. Ed. Wisdom Amouzou. The Good Men Project, 3 June 2016. Web.

A current version of my dissertation .

Daniel Erman mentioned the results of my dissertation on a resolution of the diagonal near 37:50 and 45:15 from his Invited Address to JMM 2024 "From Hilbert to Mirror Symmetry." video link

Sample Solution Sets

Note: I do appreciate feedback, typos, and error corrections on these via email.

Bayer's "A Tour to Stability Conditions on Derived Categories" Sections 1-2.

Mirror Symmetry textbook (Hori-Katz-Klemm-Pandharipande-Thomas-Vafa-Vakil-Zaslow). Notes and Solutions to Chapters 1-4.

Sample Solutions from Tu's An Introduction to Manifolds

Sample solutions from Weibel's Homological Algebra

(Handwritten) Notes on Hartshorne Chapter 1

Notes on Hartshorne's Algebraic Geometry Chapter 2+

Hartshorne's Deformation Theory

Notes from a Study Group on Milnor-Stasheff's Characteristic Classes

A start to McDuff-Salamon's Symplectic Topology

Huybrechts' Fourier-Mukai Transforms in Algebraic Geometry