arXiv subjects

Rohini Ramadas

Publications and source records attributed to Rohini Ramadas.

14 recordsLinked to original sources

arXiv2026

Degenerations in tropical compactifications and tropical intersection theory of $\overline{M}_{0,n}$

The main result of this paper is a formula for the limit cycle of a 1-parameter family of subvarieties of a tropical compactification, expressed in terms of tropical intersections. Our theorem generalizes results of Dickenstein-Feichtner-Sturmfels and Katz to the case of tropical compactifications. In the second part of the paper, we apply our formula to the moduli space $\overline{M}_{0, n}$ of stable marked rational curves. We describe the tropicalization of the Kapranov maps $\overline{M}_{0, n}\to\mathbb{P}^{n-3}$, whose hyperplane pullbacks are the $\psi$-classes, with respect to a suitable choice of torus. We introduce tropical $\psi$-hypersurfaces (in genus zero). These are different from the standard definition of Mikhalkin and Kerber-Markwig, and may be of independent interest. We demonstrate our main result by giving a "firework algorithm" that computes limits of intersections of $\psi$-hypersurfaces.

math.AG↗

arXiv2025

Handlebodies, Outer space, and tropical geometry

The moduli space of graphs $M_{g,n}^{\mathrm{trop}}$ is a polyhedral object that mimics the behavior of the moduli spaces $M_{g,n}$, $\overline{M}_{g,n}$ of (stable) Riemann surfaces; this relationship has been made precise in several different ways, which collectively identify $M_{g,n}^{\mathrm{trop}}$ as the "tropicalization" of $M_{g,n}$. We describe how this relationship lifts to some objects that live over $M_{g,n}$ (like Teichm\"uller space) and that live over $M_{g,n}^{\mathrm{trop}}$ (like the Culler-Vogtmann space $CV_{g,n}^*$). We introduce the notion of a stable complex handlebody, and show that $CV_{g,n}^*$ can be viewed as the tropicalization of a certain complex manifold $hT(V_{g,n})$ that parametrizes complex handlebodies. An important ingredient is our construction of a partial compactification $\overline{hT}(V_{g,n})\supset hT(V_{g,n})$, which we prove is a simply connected complex manifold with simple normal crossings boundary. When $n=0$, $hT(V_{g,n})$ coincides with the moduli space of Schottky groups, $\overline{hT}(V_{g,n})$ coincides with Gerritzen-Herrlich's extended Schottky space, and $CV_{g,0}^*$ is the simplicial completion of the original Outer space. The resulting picture fits together many familiar objects from geometric group theory and surface topology, including Harvey's curve complex, mapping class groups of surfaces and handlebodies, and augmented Teichm\"uller space. Many of the relationships between the objects that we see in this picture already exist in the literature, but we add some new ones, and generalize several existing relationships to include a number $n>0$ of punctures/leaves.

math.GT↗

arXiv2024

Thurston obstructions and tropical geometry

We describe an application of tropical moduli spaces to complex dynamics. A post-critically finite branched covering $\varphi$ of $S^2$ induces a pullback map on the Teichm\"uller space of complex structures of $S^2$; this descends to an algebraic correspondence on the moduli space of point-configurations of $\mathbb{C}\mathbb{P}^1$. We make a case for studying the action of the tropical moduli space correspondence by making explicit the connections between objects that have come up in one guise in tropical geometry and in another guise in complex dynamics. For example, a Thurston obstruction for $\varphi$ corresponds to a ray that is fixed by the tropical moduli space correspndence, and scaled by a factor $\ge 1$. This article is intended to be accessible to algebraic and tropical geometers as well as to complex dynamicists.

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arXiv2022

Wonderful compactifications and rational curves with cyclic action

We prove that the moduli space of rational curves with cyclic action, constructed in our previous work, is realizable as a wonderful compactification of the complement of a hyperplane arrangement in a product of projective spaces. By proving a general result on such wonderful compactifications, we conclude that this moduli space is Chow-equivalent to an explicit toric variety (whose fan can be understood as a tropical version of the moduli space), from which a computation of its Chow ring follows.

math.AG↗

arXiv2022

Moduli spaces of quadratic maps: arithmetic and geometry

We establish an implication between two long-standing open problems in complex dynamics. The roots of the $n$-th Gleason polynomial $G_n\in\mathbb{Q}[c]$ comprise the $0$-dimensional moduli space of quadratic polynomials with an $n$-periodic critical point. $\mathrm{Per}_n(0)$ is the $1$-dimensional moduli space of quadratic rational maps on $\mathbb{P}^1$ with an $n$-periodic critical point. We show that if $G_n$ is irreducible over $\mathbb{Q}$, then $\mathrm{Per}_n(0)$ is irreducible over $\mathbb{C}$. To do this, we exhibit a $\mathbb{Q}$-rational smooth point on a projective completion of $\mathrm{Per}_n(0)$, using the admissible covers completion of a Hurwitz space. In contrast, the Uniform Boundedness Conjecture in arithmetic dynamics would imply that for sufficiently large $n$, $\mathrm{Per}_n(0)$ itself has no $\mathbb{Q}$-rational points.

math.DS↗

arXiv2021

Permutohedral complexes and rational curves with cyclic action

We define a moduli space of rational curves with finite-order automorphism and weighted orbits, and we prove that the combinatorics of its boundary strata are encoded by a particular polytopal complex that also captures the algebraic structure of a complex reflection group acting on the moduli space. This generalizes the situation for Losev-Manin's moduli space of curves (whose boundary strata are encoded by the permutohedron and related to the symmetric group) as well as the situation for Batyrev-Blume's moduli space of curves with involution, and it extends that work beyond the toric context.

math.AG↗

arXiv2021

Equations at infinity for critical-orbit-relation families of rational maps

We develop techniques for using compactifications of Hurwitz spaces to study families of rational maps $\mathbb{P}^1\to\mathbb{P}^1$ defined by critical orbit relations. We apply these techniques in two settings: We show that the parameter space $\mathrm{Per}_{d,n}$ of degree-$d$ bicritical maps with a marked 4-periodic critical point is a $d^2$-punctured Riemann surface of genus $\frac{(d-1)(d-2)}{2}$. We also show that the parameter space $\mathrm{Per}_{2,5}$ of degree-2 rational maps with a marked 5-periodic critical point is a 10-punctured elliptic curve, and we identify its isomorphism class over $\mathbb{Q}$. We carry out an experimental study of the interaction between dynamically defined points of $\mathrm{Per}_{2,5}$ (such as PCF points or punctures) and the group structure of the underlying elliptic curve.

math.AG↗

arXiv2020

Pullbacks of $κ$ classes on $\overline{\mathcal{M}}_{0,n}$

The moduli space $\overline{\mathcal{M}}_{0,n}$ carries a codimension-$d$ cycle class $κ_{d}$. We consider the subspace $\mathcal{K}^{d}_{n}$ of $A^d(\overline{\mathcal{M}}_{0,n},\mathbb{Q})$ spanned by pullbacks of $κ_d$ via forgetful maps. We find a permutation basis for $\mathcal{K}^{d}_{n}$, and describe its annihilator under the intersection pairing in terms of $d$-dimensional boundary strata. As an application, we give a new permutation basis of the divisor class group of $\overline{\mathcal{M}}_{0,n}$.

math.AG↗

arXiv2020

Algebraic stability of meromorphic maps descended from Thurston's pullback maps

Let $ϕ:S^2 \to S^2$ be an orientation-preserving branched covering whose post-critical set has finite cardinality $n$. If $ϕ$ has a fully ramified periodic point $p_{\infty}$ and satisfies certain additional conditions, then, by work of Koch, $ϕ$ induces a meromorphic self-map $R_ϕ$ on the moduli space $\mathcal{M}_{0,n}$; $R_ϕ$ descends from Thurston's pullback map on Teichmüller space. Here, we relate the dynamics of $R_ϕ$ on $\mathcal{M}_{0,n}$ to the dynamics of $ϕ$ on $S^2$. Let $\ell$ be the length of the periodic cycle in which the fully ramified point $p_{\infty}$ lies; we show that $R_ϕ$ is algebraically stable on the heavy-light Hassett space corresponding to $\ell$ heavy marked points and $(n-\ell)$ light points.

math.AG↗

arXiv2020

Two-dimensional cycle classes on $\overline{\mathcal{M}_{0,n}}$

For each $n\ge5$, we give an $S_n$-equivariant basis for $H_4(\overline{\mathcal{M}_{0,n}},\mathbb{Q})$, as well as for $H_{2(n-5)}(\overline{\mathcal{M}_{0,n}},\mathbb{Q})$. Such a basis exists for $H_2(\overline{\mathcal{M}_{0,n}},\mathbb{Q})$ and for $H_{2(n-4)}(\overline{\mathcal{M}_{0,n}},\mathbb{Q})$, but it is not known whether one exists for $H_{2k}(\overline{\mathcal{M}_{0,n}},\mathbb{Q})$ when $3\le k\le n-6$.

math.AG↗

arXiv2020

Dynamical invariants of monomial correspondences

We focus on various dynamical invariants associated to toric correspondences, using algebraic geometry or arithmetic. We find a formula for the dynamical degrees, relate the exponential growth of the degree sequences with a strict log-concavity condition on the dynamical degrees and compute the asymptotic ratio of the growth of heights of points of such correspondences.

math.DS↗

arXiv2019

Post-Critically Finite Maps on $\mathbb{P}^n$ for $n\ge2$ are Sparse

Let $f:{\mathbb P}^n\to{\mathbb P}^n$ be a morphism of degree $d\ge2$. The map $f$ is said to be post-critically finite (PCF) if there exist integers $k\ge1$ and $\ell\ge0$ such that the critical locus $\operatorname{Crit}_f$ satisfies $f^{k+\ell}(\operatorname{Crit}_f)\subseteq{f^\ell(\operatorname{Crit}_f)}$. The smallest such $\ell$ is called the tail-length. We prove that for $d\ge3$ and $n\ge2$, the set of PCF maps $f$ with tail-length at most $2$ is not Zariski dense in the the parameter space of all such maps. In particular, maps with periodic critical loci, i.e., with $\ell=0$, are not Zariski dense.

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arXiv2017

Dynamical degrees of Hurwitz correspondences

Let $ϕ$ be a post-critically finite branched covering of a two-sphere. By work of Koch, the Thurston pullback map induced by $ϕ$ on Teichmüller space descends to a multi-valued self-map --- a Hurwitz correspondence $\mathcal{H}_ϕ$ --- of the moduli space $\mathcal{M}_{0,P}$. We study the dynamics of Hurwitz correspondences via numerical invariants called dynamical degrees. We show that the sequence of dynamical degrees of $\mathcal{H}_ϕ$ is always non-increasing, and the behavior of this sequence is constrained by the behavior of $ϕ$ at and near points of its post-critical set.

math.AG↗

arXiv2016

Hurwitz correspondences on compactifications of $\mathcal{M}_{0,N}$

Hurwitz correspondences are certain multivalued self-maps of the moduli space $\mathcal{M}_{0,N}$. They arise in the study of Thurston's topological characterization of rational functions. We consider the dynamics of Hurwitz correspondences and ask: On which compactifications of $\mathcal{M}_{0,N}$ should they be studied? We compare a Hurwitz correspondence $\mathcal{H}$ across various modular compactifications of $\mathcal{M}_{0,N}$, and find a weighted stable curves compactification $X_N^\dagger$ that is optimal for its dynamics. We use $X_N^\dagger$ to show that the $k$th dynamical degree of $\mathcal{H}$ is the absolute value of the dominant eigenvalue of the pushforward induced by $\mathcal{H}$ on a natural quotient of $H_{2k}(\overline{\mathcal{M}}_{0,N})$.

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