arXiv subjects

Vance Blankers

Publications and source records attributed to Vance Blankers.

9 recordsLinked to original sources

arXiv2026

Sign-reversing involutions in moduli spaces of curves

We use sign-reversing involutions to solve two computational problems that arise naturally in the geometry of moduli spaces of curves. In particular, we give an explicit combinatorial formula for arbitrary $ψ$ class intersection products on the genus zero multicolored spaces $\overline{M}_{0,[r_1,\ldots,r_m]}$ using a novel sign reversing involution on decorated diagrams. As an application, we give a necessary and sufficient condition for when these intersection products are nonzero in terms of matchings on graphs. We also calculate the analog of the tropical Euler characteristic for the graphical moduli spaces $\overline{M}_{0,Γ}$ for graphs with two dominant vertices $P, Q$, by constructing two new sign-reversing involutions to simplify the sum. We show that (up to sign) it is the number of acyclic orientations of $Γ\smallsetminus \{P, Q\}$.

math.CO↗

arXiv2025

Moduli Theory of the $r$-Braid Arrangement

We describe a family of hyperplane arrangements depending on a positive integer parameter $r$, which we refer to as the $r$-braid arrangements, and which can be viewed as a generalization of the classical braid arrangement. The wonderful compactification of the braid arrangement (with respect to its minimal building set) is well-known to yield the moduli space $\overline{\mathcal{M}}_{0,n}$, and, in this work, we generalize this result, constructing a moduli space $\overline{\mathcal{M}}^r_{n}$ of certain genus-zero curves with an order-$r$ involution that we identify with the corresponding wonderful compactification of the $r$-braid arrangement. The resulting space is a variant of the previously studied moduli space $\overline{\mathcal{L}}^r_n$ [arXiv:2104.06526], related via a change of weights on the markings.

math.AG↗

arXiv2022

On compactifications of $\mathcal{M}_{g,n}$ with colliding markings

In this paper, we study all ways of constructing modular compactifications of the moduli space $\mathcal{M}_{g,n}$ of $n$-pointed smooth algebraic curves of genus $g$ by allowing markings to collide. We find that for any such compactification, collisions of markings are controlled by a simplicial complex which we call the collision complex. Conversely, we identify modular compactifications of $\mathcal{M}_{g,n}$ with essentially arbitrary collision complexes, including complexes not associated to any space of weighted pointed stable curves. These moduli spaces classify the modular compactifications of $\mathcal{M}_{g,n}$ by nodal curves with smooth markings as well as the modular compactifications of $\mathcal{M}_{1,n}$ with Gorenstein curves and smooth markings. These compactifications generalize previous constructions given by Hassett, Smyth, and Bozlee--Kuo--Neff.

math.AG↗

arXiv2021

Extremality of Rational Tails Boundary Strata in $\overline{\mathcal{M}}_{g,n}$

We review and develop some techniques used to investigate the effective cones of higher codimension classes. Our results show that a large collection of boundary strata of rational tails type are extremal in their effective cones on $\overline{\mathcal{M}}_{g,n}$ and provide evidence for the conjecture that all boundary strata of $\overline{\mathcal{M}}_{g,n}$ are extremal. As a corollary, we show that all boundary strata are extremal in genus zero.

math.AG↗

arXiv2020

Hyperelliptic classes are rigid and extremal in genus two

We show that the class of the locus of hyperelliptic curves with $\ell$ marked Weierstrass points, $m$ marked conjugate pairs of points, and $n$ free marked points is rigid and extremal in the cone of effective codimension-($\ell + m$) classes on $\overline{\mathcal{M}}_{2,\ell+2m+n}$. This generalizes work of Chen and Tarasca and establishes an infinite family of rigid and extremal classes in arbitrarily-high codimension.

math.AG↗

arXiv2019

Wall-crossings for Hassett descendant potentials

This paper solves the combinatorics relating the intersection theory of $ψ$-classes of Hassett spaces to that of $\overline{\mathcal{M}}_{g,n}$. A generating function for intersection numbers of $ψ$ classes on all Hassett spaces is obtained from the Gromov-Witten potential of a point via a non-invertible transformation of variables. When restricting to diagonal weights, the changes of variables are invertible and explicitly described as polynomial functions. Finally, the comparison of potentials is extended to the level of cycles: the pinwheel cycle potential, a generating function for tautological classes of rational tail type on $\overline{\mathcal{M}}_{g,n}$ is the right instrument to describe the pull-back to $\overline{\mathcal{M}}_{g,n}$ of all monomials of $ψ$ classes on Hassett spaces.

math.AG↗

arXiv2019

Julia and Mandelbrot sets for dynamics over the hyperbolic numbers

Julia and Mandelbrot sets, which characterize bounded orbits in dynamical systems over the complex numbers, are classic examples of fractal sets. We investigate the analogs of these sets for dynamical systems over the hyperbolic numbers. Hyperbolic numbers, which have the form $x+τy$ for $x,y \in \mathbb{R}$, and $τ^2 = 1$ but $τ\neq \pm 1$, are the natural number system in which to encode geometric properties of the Minkowski space $\mathbb{R}^{1,1}$. We show that the hyperbolic analog of the Mandelbrot set parameterizes connectedness of hyperbolic Julia sets. We give a wall-and-chamber decomposition of the hyperbolic plane in terms of these Julia sets.

math.DS↗

arXiv2018

Witten's conjecture and recursions for $κ$ classes

We construct a countable number of differential operators $\hat{L}_n$ that annihilate a generating function for intersection numbers of $κ$ classes on $\Moduli_g$ (the $κ$-potential). This produces recursions among intersection numbers of $κ$ classes which determine all such numbers from a single initial condition. The starting point of the work is a combinatorial formula relating intersecion numbers of $ψ$ and $κ$ classes. Such a formula produces an exponential differential operator acting on the Gromov-Witten potential to produce the $κ$-potential; after restricting to a hyperplane, we have an explicit change of variables relating the two generating functions, and we conjugate the "classical" Virasoro operators to obtain the operators $\hat{L}_n$.

math.AG↗

arXiv2017

Intersections of $ω$ classes in $\overline{\mathcal{M}}_{g,n}$

We provide a graph formula which describes an arbitrary monomial in ω classes (also referred to as stable ψ classes) in terms of a simple family of dual graphs (pinwheel graphs) with edges decorated by rational functions in ψ classes. We deduce some numerical consequences and in particular a combinatorial formula expressing top intersections of \k{appa} classes on Mg in terms of top intersections of ψ classes.

math.AG↗