arXiv subjects

Reginald Anderson

Publications and source records attributed to Reginald Anderson.

9 recordsLinked to original sources

arXiv2026

Examples of descendent generating series for Pandharipande--Thomas stable pairs on smooth projective Fano threefolds via one-dimensional wall-crossing

We study descendent generating series for Pandharipande--Thomas stable pairs on smooth projective Fano threefolds. We use the wall-crossing setup developed by the author and Joyce in Joyce's Lie algebra $H_*(\N^{\pl},\Q)$ of the projective-linear pairs stack, and next pass to Gross's polynomial realization $e^{\kappa}\Q[s_{jk\ell}]$. We compute explicit examples of one-dimensional Donaldson--Thomas invariants on Fano 3-folds and, via wall-crossing, Pandharipande--Thomas stable pair invariants and descendent generating series. We compute examples on $\PP^3$, on a smooth cubic threefold, on $\Bl_p\PP^3$, on $\Bl_\ell\PP^3$, and on the projective-bundle threefold $\PP(\OO_X\oplus \OO_X(-1,-1))$ over $X=\PP^1\times\PP^1$. In the $\PP^3$ and cubic threefold examples we compare the intrinsic large-$n$ tails with the formulas of Pandharipande and Moreira and show that, in the cases treated in common, the differences are Laurent polynomials.

math.AG↗

arXiv2026

Rationality of cohomological descendent series for Quot schemes on surfaces with $p_g=0$

For a smooth projective surface $S$, Johnson--Oprea--Pandharipande defined cohomological descendent generating series for Quot schemes of rank-$0$ quotients of $\OO_S^{\oplus N}$. We prove rationality of these series in the remaining cohomological surface case \[ p_g(S)=0,\qquad \beta\neq 0,\qquad N>1. \] The wall-crossing part of the proof starts from Joyce-style generalized Donaldson--Thomas invariant classes of $H$-Gieseker semistable one-dimensional sheaves. We vary a single real parameter in the fixed-source Pairs stability condition and obtain the large-$c$ stable-pair chamber for maps $\OO_S^{\oplus N}\to F$. We then compare this pair chamber with the open pure Quot locus, meaning the locus inside the Quot scheme whose target quotient is pure one-dimensional, and then with the full Quot scheme, where zero-dimensional torsion in the target is allowed. The first comparison records the zero-dimensional cokernel of the image of a pair. After decomposing the pure Quot locus into locally-closed pieces on which the scheme-theoretic support curve is flat over the base, this comparison is identified with relative Quot theory on those support curves. The resulting curve-Quot contributions factor into smooth-normalization contributions and finitely many punctual factors at singular points of the support curve. The second comparison records the maximal zero-dimensional torsion subsheaf of a Quot target; locally it becomes a punctual Quot problem over the completed smooth surface ring $\C[[x,y]]$, and its contribution is the universal punctual smooth-surface factor.

math.AG↗

arXiv2026

The Pandharipande-Thomas rationality conjecture for superpositive curve classes on projective complex 3-manifolds

Let $X$ be a projective complex 3-manifold. An effective curve class $\beta\in H_2(X,\mathbb Z)$ is called positive if $c_1(X)\cdot\beta>0$, and superpositive if all the effective summands of $\beta$ are positive. If $X$ is Fano then all curve classes are superpositive. In arXiv:2111.04694 the second author developed a theory of enumerative invariants in abelian categories and wall-crossing formulae. We use this theory to prove conjectures by Pandharipande and Thomas on the rationality and poles of generating functions of Pandharipande-Thomas invariants of $X$ with descendent insertions, for superpositive curve classes.

math.AG↗

arXiv2025

Exceptional Collections for Toric Fano Fivefolds

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 diagonal given by Bayer-Popescu-Sturmfels to yield a virtual resolution of the diagonal for smooth projective toric varieties, which extends to toric Deligne-Mumford stacks which are a global quotient of a smooth projective variety by a finite abelian group. Moreover, a celebrated result of Hanlon-Hicks-Lazarev gives a symmetric, minimal resolution of the diagonal for smooth projective toric varieties. This work studies when smooth projective toric Fano varieties in dimension 5 yield exceptional collections of line bundles using a resolution of the diagonal. We give the first known count of 300 out of 866 smooth projective toric Fano 5-folds for which the Hanlon-Hicks-Lazarev resolution of the diagonal yields a full strong exceptional collection of line bundles.

math.AG↗

arXiv2024

Enumerative Geometry and Tree-Level Gromov--Witten Invariants

Here we review background in differential topology related to the calculation of an euler characteristic, and background on localization in equivariant cohomology. We then outline Gromov-Witten invariants in algebraic geometry and give examples of the genus 0 Gromov-Witten potential for $\PP^1, \PP^2$, and a genus $g>0$ Riemann surface. Kontsevich-Manin's recursive formula for $N_d$, the number of degree $d$ rational curves through $3d-1$ points in general position on $\PP^2$ is recovered.

math.GM↗

arXiv2024

Exceptional Collections for Toric Fano Fourfolds

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 diagonal for smooth projective toric varieties and toric Deligne-Mumford stacks which are a global quotient of a smooth projective variety by a finite abelian group. In the past year, Hanlon-Hicks-Lazarev gave in particular a symmetric, minimal resolution of the diagonal for smooth projective toric varieties. We give implications for exceptional collections on smooth projective toric Fano varieties in dimension 4. We find that for 72 out of 124 smooth projective toric Fano 4-folds, the Hanlon-Hicks-Lazarev resolution of the diagonal yields a full strong exceptional collection of line bundles, which coincides exactly with satisfying a numerical criterion due to Bondal.

math.AG↗

arXiv2023

Exceptional Collections of Line Bundles for Smooth Toric Fano Surfaces and Threefolds

The cellular resolution of the diagonal given by Bayer-Popescu-Sturmfels for unimodular projective toric varieties yields a full, strong exceptional collection of line bundles on unimodular projective toric surfaces. The Hanlon-Hicks-Lazarev resolution of the diagonal yields a full, strong exceptional collection of line bundles for 16 of the 18 smooth toric Fano threefolds.

math.AG↗

arXiv2023

A Resolution of the Diagonal for Smooth Projective Toric Varieties

The cellular construction of Bayer-Popescu-Sturmfels extends Beilinson's diagonal resolution from projective space to projective toric varieties whose lattice of principal divisors is unimodular. We investigate the smooth projective case in which this lattice condition fails. The periodic arrangement then contains vertices outside the lattice, and the associated finite cellular complex can acquire homology in degree $0$ and in higher degrees. We use a floor-function labeling to assign Laurent monomials to vertices in a way compatible with the periodic hyperplane arrangement. A counterexample shows that, without further hypotheses, the undeformed complex need not resolve the diagonal. We therefore impose a symmetry hypothesis on the fan (central symmetry across the origin) under which the construction yields a locally free resolution of $\mathcal{O}_\Delta$ on $X_\Sigma\times X_\Sigma$. We also discuss how a deformation parameter $\epsilon$ should lead to a broader family of resolutions.

math.AG↗

arXiv2023

A Resolution of the Diagonal for Toric Deligne-Mumford Stacks

Beilinson's resolution of the diagonal for complex projective space was generalized by Bayer-Popescu-Sturmfels for any unimodular toric variety. Here, we give a resolution of the diagonal for any smooth toric variety (viewed as a toric Deligne-Mumford stack) in families by deformation of the cellular complex of Bayer-Popescu-Sturmfels and show that the cokernel of this resolution gives the diagonal, modulo torsion from the irrelevant ideal. Furthermore, we give a resolution of the diagonal for a toric Deligne-Mumford stack associated to the global quotient of a smooth toric variety by a finite abelian group.

math.AG↗