arXiv subjects

Dhruv Ranganathan

Publications and source records attributed to Dhruv Ranganathan.

At least 19 recordsLinked to original sources

arXiv2026

Combinatorics of Hurwitz degenerations and tropical realizability

We investigate the realizability of balanced functions on tropical curves, establishing new sufficient criteria for superabundant functions on genus two curves, analogous to the well-spacedness condition in genus one. We find that realizability is sensitive to the precise locations of conjugate and Weierstrass points on the tropical curve. The key input is a combinatorial comparison of semistable limit theorems for maps of curves. Amini-Baker-Brugallé-Rabinoff previously showed that realizability of functions is equivalent to modifiability to a tropical admissible cover. While the resulting criteria are typically inexplicit, we develop combinatorial techniques to derive explicit, verifiable conditions. We further introduce a dimensional reduction technique to deduce statements about maps to $\mathbb{R}^r$ from corresponding statements about maps to $\mathbb{R}$. By proving directly that modifiability and well-spacedness are equivalent in genus one, we obtain a new proof that well-spaced maps are realizable. Along the way, we explain how the modifiability criterion can be interpreted as a comparison result for properness statements in moduli spaces of relative maps and admissible covers.

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arXiv2026

The GW/PT conjectures for toric pairs

We prove the conjectural correspondence between logarithmic Gromov-Witten theory and logarithmic Donaldson/Pandharipande-Thomas theory for pairs $(Y|\partial Y)$ consisting of a toric threefold $Y$ and any torus invariant divisor $\partial Y$, with primary insertions. The results are the first verifications of this conjecture when $\partial Y$ is singular, i.e., the ``fully logarithmic'' setting, and the first proof of the equivariant toric correspondence for pairs when $\partial Y$ is nonempty. When $\partial Y$ is empty, we get a new proof of the known toric correspondence, but our methods also lead to stronger conclusions. In particular, we show the PT series is a Laurent polynomial in the presence of sufficient positivity and prove a 2008 conjecture of Oblomkov, Okounkov, Pandharipande, and the first author stating the capped vertex is a Laurent polynomial. The methods also verify the logarithmic DT/PT conjecture for toric threefold pairs. Using the constraints of the logarithmic theory, the complete evaluation of toric pairs is determined by a single calculation -- the degree $1$ series of $\mathbb{P}^3$.

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arXiv2026

Logarithmic negative tangency and root stacks

We study stable maps to normal crossings pairs with possibly negative tangency orders. There are two independent models: punctured Gromov-Witten theory of pairs and orbifold Gromov-Witten theory of root stacks with extremal ages. Exploiting the tropical structure of the punctured mapping space, we define and study a new virtual class for the punctured theory. This arises as a refined intersection product on the Artin fan, and produces a distinguished sector of punctured Gromov-Witten invariants. Restricting to genus zero, we show that these invariants coincide with the orbifold invariants, first for smooth pairs, and then for normal crossings pairs after passing to a sufficiently refined blowup. This builds on previous work to provide a complete picture of the logarithmic-orbifold comparison in genus zero, which is compatible with splitting and thus allows for the wholesale importation of orbifold techniques, including boundary recursion and torus localisation. Contemporaneous work of Johnston uses the comparison to give a new proof of the associativity of the Gross-Siebert intrinsic mirror ring.

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arXiv2026

An invitation to the enumerative geometry of degenerations

This expository article is an introduction to logarithmic Gromov--Witten (GW) theory. We discuss how to study the GW theory of a smooth projective variety via simple normal crossings degenerations. We survey several approaches to constructing well-behaved, virtually smooth moduli spaces of stable maps to such degenerations. Each irreducible component of the special fiber of a degeneration determines a pair consisting of a variety and a normal crossings divisor, and these pairs carry their own logarithmic GW theory. We explain how the GW theory of the general fiber can be expressed in terms of the logarithmic GW theory of these pairs. Finally, we discuss applications to tautological classes on the moduli space of curves.

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arXiv2025

Gromov-Witten theory, degenerations, and the tautological ring

Gromov-Witten (GW) theory produces Chow and cohomology classes on the moduli of curves, and there are several conjectures/speculations about their relation to the tautological ring. We develop new degeneration techniques to address these. In Chow, we show that GW cycles of complete intersections in products of projective spaces (and more generally a broad class of toric varieties) with restricted insertions are tautological. This gives significant evidence for a 2010 speculation of Pandharipande that GW cycles of varieties over the algebraic numbers are tautological. In particular, the 0-cycle for curves on the quintic threefold is proportional to a zero stratum in the moduli space of stable curves. In cohomology, we show that in normal crossings degenerations, GW classes of the general fiber lie in the span of absolute GW classes of the special fiber strata. This confirms a 2006 conjecture of Levine-Pandharipande for targets that degenerate into elementary pieces, including complete intersections in products of projective spaces and many toric varieties. Our proofs rely on several reconstruction theorems in logarithmic GW theory, which make the logarithmic degeneration formula an inductive tool to compute GW cycles via snc degenerations. We prove a folklore conjecture that logarithmic GW cycles of a pair are determined by absolute invariants of the strata. We prove a conjecture of Urundolil Kumaran and the second author that GW cycles of toric pairs are tautological, and analogous results for broken toric bundles. We also develop tools to study GW cycles with vanishing cohomology and strengthen the logarithmic degeneration formula to allow iteration.

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arXiv2025

Logarithmic Quot spaces, boundedness, and K-tropicalizations

Logarithmic Hilbert and Quot spaces are generalizations of their traditional versions adapted to study pairs and degenerations. The logarithmic Quot spaces of $(X,D)$ parameterize "algebraically transverse" (logarithmically flat) quotient sheaves on degenerations of $X$. We prove boundedness and deduce properness of logarithmic Quot spaces. The results complete the basic foundations of logarithmic Quot spaces and specialize to work of Li-Wu and Maulik and the second author in special cases. Boundedness relies on two results of independent interest. First, we show that for a simple normal crossing pair $(X, D)$ and a subscheme $Z$, there is a smallest logarithmic space ${X}^\flat$ modifying $X$ such that the strict transform of $Z$ is algebraically transverse. Precisely, given $Z$ in $X$, there is a canonical logarithmic space $X^\flat$ over $X$ with the following universal property - an snc logarithmic blowup $X'\to X$ makes the strict transform of $Z$ algebraically transverse if and only if $X'$ is a modification of $X^\flat$. Parallel results hold for arbitrary coherent sheaves. This proves boundedness for logarithmic quotients with fixed tropicalization. A logarithmic quotient sheaf defines a K-tropicalization, an enhancement of tropicalization that is sensitive to scheme structures. The K-tropicalization has the same relationship to K-theory as traditional tropicalization has to Chow, and is related to Gröbner theory and convex geometry via state and secondary polytopes. Using the K-theory of toric bundles, we derive a balancing condition for K-tropicalizations that imposes strong finiteness properties. The second key result is that K-tropicalizations with fixed numerics are parametrized by a finite-dimensional polyhedral complex.

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arXiv2025

Pluricanonical cycles and tropical covers

We extract a system of numerical invariants from logarithmic intersection theory on pluricanonical double ramification cycles, and show that these invariants exhibit a number of properties that are enjoyed by double Hurwitz numbers. Among their properties are (i) the numbers can be efficiently calculated by counts of tropical curves with a modified balancing condition, (ii) they are piecewise polynomial in the entries of the ramification vector, and (iii) they are matrix elements of operators on the Fock space. The numbers are extracted from the logarithmic double ramification cycle, which is a lift of the standard double ramification cycle to a blowup of the moduli space of curves. The blowup is determined by tropical geometry. We show that the traditional double Hurwitz numbers are intersections of the refined cycle with the cohomology class of a piecewise polynomial function on the tropical moduli space of curves. This perspective then admits a natural, combinatorially motivated, generalization to the pluricanonical setting. Tropical correspondence results for the new invariants lead immediately to the structural results for these numbers.

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arXiv2025

Logarithmic tautological rings of the moduli spaces of curves

We define the logarithmic tautological rings of the moduli spaces of Deligne-Mumford stable curves (together with a set of additive generators lifting the decorated strata classes of the standard tautological rings). While these algebras are infinite dimensional, a connection to polyhedral combinatorics via a new theory of homological piecewise polynomials allows an effective study. A complete calculation is given in genus 0 via the algebra of piecewise polynomials on the cone stack of the associated Artin fan (lifting Keel's presentation of the Chow ring of $\overline{\mathcal{M}}_{0,n}$). Counterexamples to the simplest generalizations in genus 1 are presented. We show, however, that the structure of the log tautological rings is determined by the complete knowledge of all relations in the standard tautological rings of the moduli spaces of curves. In particular, Pixton's conjecture concerning relations in the standard tautological rings lifts to a complete conjecture for relations in the log tautological rings of the moduli spaces of curves. Several open questions are discussed. We develop the entire theory of logarithmic tautological classes in the context of arbitrary smooth normal crossings pairs $(X,D)$ with explicit formulas for intersection products. As a special case, we give an explicit set of additive generators of the full logarithmic Chow ring of $(X,D)$ in terms of Chow classes on the strata of $X$ and piecewise polynomials on the cone stack.

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arXiv2025

Logarithmic enumerative geometry for curves and sheaves

We propose a logarithmic enhancement of the Gromov-Witten/Donaldson-Thomas correspondence, with descendants, and study its behavior under simple normal crossings degenerations. The formulation of the logarithmic correspondence requires a matching of tangency conditions with relative insertions. This is achieved via a version of the Nakajima basis for the cohomology of the Hilbert schemes of points on a logarithmic surface. Next, we establish a strong form of the degeneration formula in logarithmic DT theory - the numerical DT invariants of the general fiber of a degeneration are determined by the numerical DT invariants attached to strata of the special fiber. The GW version of this result, which we prove in all target dimensions, strengthens currently known formulas. A key role is played by a certain exotic class of insertions, introduced here, that impose non-local incidence conditions coupled across multiple boundary strata of the target geometry. Finally, we prove compatibility of the new logarithmic GW/DT correspondence with degenerations. In particular, the logarithmic conjecture for all strata of the special fiber of a degeneration implies the traditional GW/DT conjecture on the general fiber. Compatibility is a strong constraint, and can be used to calculate logarithmic DT invariants. Several examples are included to illustrate the nature and utility of the formula.

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arXiv2024

The many faces of a logarithmic scheme

A standard assumption in the study of logarithmic structures is "fineness", but this assumption is not preserved by intersections, fiber products, and more general limits. We explain how a coherent logarithmic scheme $X$ has a natural decomposition into "fine faces" -- a collection of fine logarithmic subschemes. This allows for importation of the techniques of tropical geometry into the study of more general logarithmic schemes. We explain the relevance of the construction in enumerative geometry and moduli via a range of examples. Techniques developed in the study of binomial schemes lead to effective algorithms for computing the fine faces.

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arXiv2024

Logarithmic Donaldson-Thomas theory

Let $X$ be a smooth threefold with a simple normal crossings divisor $D$. We construct the Donaldson-Thomas theory of the pair $(X|D)$ enumerating ideal sheaves on $X$ relative to $D$. These moduli spaces are compactified by studying subschemes in expansions of the target geometry, and the moduli space carries a virtual fundamental class leading to numerical invariants with expected properties. We formulate punctual evaluation, rationality and wall-crossing conjectures, in parallel with the standard theory. Our formalism specializes to the Li-Wu theory of relative ideal sheaves when the divisor is smooth, and is parallel to recent work on logarithmic Gromov-Witten theory with expansions.

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arXiv2023

Logarithmic Gromov-Witten theory and double ramification cycles

We examine the logarithmic Gromov-Witten cycles of a toric variety relative to its full toric boundary. The cycles are expressed as products of double ramification cycles and natural tautological classes in the logarithmic Chow ring of the moduli space of curves. We introduce a simple new technique that relates the Gromov-Witten cycles of rigid and rubber geometries; the technique is based on a study of maps to the logarithmic algebraic torus. By combining this with recent work on logarithmic double ramification cycles, we deduce that all logarithmic Gromov-Witten pushforwards, for maps to a toric variety relative to its full toric boundary, lie in the tautological ring of the moduli space of curves. A feature of the approach is that it avoids the as yet undeveloped logarithmic virtual localization formula, instead relying directly on piecewise polynomial functions to capture the structure that would be provided by such a formula. The results give a common generalization of work of Faber-Pandharipande, and more recent work of Holmes-Schwarz and Molcho-Ranganathan. The proof passes through general structure results on the space of stable maps to the logarithmic algebraic torus, which may be of independent interest.

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arXiv2023

A case study of intersections on blowups of the moduli of curves

We explain how logarithmic structures select principal components in an intersection of schemes. These manifest in Chow homology and can be understood using strict transforms under logarithmic blowups. Our motivation comes from Gromov--Witten theory. The \textit{toric contact cycles} in the moduli space of curves parameterize curves that admit a map to a fixed toric variety with prescribed contact orders. We show that they are intersections of virtual strict transforms of double ramification cycles in blowups of the moduli space of curves. We supply a calculation scheme for the virtual strict transforms, and deduce that toric contact cycles lie in the tautological ring of the moduli space of curves. This is a higher dimensional analogue of a result of Faber--Pandharipande. The operational Chow rings of Artin fans play a basic role, and are shown to be isomorphic to rings of piecewise polynomials on associated cone complexes. The ingredients in our analysis are Fulton's blowup formula, Aluffi's formulas for Segre classes of monomial schemes, piecewise polynomials, and degeneration methods. A model calculation in toric intersection theory is treated without logarithmic methods and may be read independently.

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arXiv2023

Gromov-Witten theory via roots and logarithms

Orbifold and logarithmic structures provide independent routes to the virtual enumeration of curves with tangency orders for a simple normal crossings pair $(X|D)$. The theories do not coincide and their relationship has remained mysterious. We prove that the genus zero orbifold theories of multi-root stacks of strata blowups of $(X|D)$ converge to the corresponding logarithmic theory of $(X|D)$. With fixed numerical data, there is an explicit combinatorial criterion that guarantees when a blowup is sufficiently refined for the theories to coincide. There are two key ideas in the proof. The first is the construction of a naive Gromov-Witten theory, which serves as an intermediary between roots and logarithms. The second is a smoothing theorem for tropical stable maps; the geometric theorem then follows via virtual intersection theory relative to the universal target. The results import new computational tools into logarithmic Gromov-Witten theory. As an application, we show that the genus zero logarithmic Gromov-Witten theory of a pair is determined by the absolute Gromov-Witten theories of its strata.

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arXiv2022

Counting tropical curves in $\mathbb{P}^1\times\mathbb{P}^1$: computation & polynomiality properties

Counts of curves in $\mathbb{P}^1\times\mathbb{P}^1$ with fixed contact order with the toric boundary and satisfying point conditions can be determined with tropical methods by Mikhalkin. If we require that our curves intersect the zero- and infinity-section only in points of contact order $1$, but allow arbitrary contact order for the zero- and infinity-fiber, the corresponding numbers reveal beautiful structural properties such as piecewise polynomiality, similar to the case of double Hurwitz numbers counting covers of $\mathbb{P}^1$ with special ramification profiles over zero and infinity by Ardila and Brugallé. This result was obtained using the floor diagram method to count tropical curves. Here, we expand the tropical tools to determine counts of curves in $\mathbb{P}^1\times\mathbb{P}^1$. We provide a computational tool (building on Polymake by Gawrilow and Joswig) that determines such numbers of tropical curves for any genus and any contact orders via a straightforward generalization of Mikhalkin's lattice path algorithm. The tool can also be used for other toric surfaces. To enable efficient computations also by hand, we introduce a new counting tool (for the case of rational curves with transverse contacts with the infinity section) which can be seen as a combination of the floor diagram and the lattice path approach: subfloor diagrams. We use both our computational tool and the subfloor diagrams for experiments revealing structural properties of these counts. We obtain first results on the (piecewise) polynomial structure of counts of rational curves in $\mathbb{P}^1\times\mathbb{P}^1$ with arbitrary contact orders on the zero- and infinity-fiber and restricted choices for the contact orders on the zero- and infinity-section.

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arXiv2022

A note on the cycle of curves in a product of pairs

We prove that the cycle-valued logarithmic Gromov--Witten theory of a product of simple normal crossings pairs $X\times Y$ decomposes into a product of pieces coming from $X$ and $Y$, provided that the decomposition is considered over a blowup of the moduli space of curves. The result is established by applying the toroidal semistable reduction theorem to the stabilization morphism on the stack of maps to the Artin fans. Since products of smooth pairs are naturally simple normal crossings pairs, the result provides a direct avenue of access to questions in logarithmic Gromov--Witten theory via relative Gromov--Witten theory.

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arXiv2022

Logarithmic Gromov-Witten theory with expansions

We construct relative Gromov--Witten theory with expanded degenerations in the normal crossings setting and establish a degeneration formula for the resulting invariants. Given a simple normal crossings pair $(X,D)$, we show that there exist proper moduli spaces of curves in $X$ with prescribed boundary conditions along $D$, equipped with virtual classes. Each point in such a moduli space parameterizes a map from a nodal curve to an expanded degeneration of $X$ that is dimensionally transverse to the strata. In the context of maps to a simple normal crossings degeneration, the virtual fundamental class is known to decompose as a sum over tropical maps. We use the expanded formalism to prove the degeneration formula -- we reconstruct the virtual class attached to a tropical map in terms of spaces of maps to expansions attached to the vertices.

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arXiv2022

Gromov-Witten theory and invariants of matroids

We use techniques from Gromov-Witten theory to construct new invariants of matroids taking value in the Chow groups of spaces of rational curves in the permutohedral toric variety. When the matroid is realizable by a complex hyperplane arrangement, our invariants coincide with virtual fundamental classes used to define the logarithmic Gromov-Witten theory of wonderful models of arrangement complements, for any logarithmic structure supported on the wonderful boundary. When the boundary is empty, this implies that the quantum cohomology ring of a hyperplane arrangement's wonderful model is a combinatorial invariant, i.e., it depends only on the matroid. When the boundary divisor is maximal, we use toric intersection theory to convert the virtual fundamental class into a balanced weighted fan in a vector space, having the expected dimension. We explain how the associated Gromov-Witten theory is completely encoded by intersections with this weighted fan. We include a number of questions whose positive answers would lead to a well-defined Gromov-Witten theory of non-realizable matroids.

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