arXiv subjects

Navid Nabijou

Publications and source records attributed to Navid Nabijou.

At least 19 recordsLinked to original sources

arXiv2026

Counting point configurations in projective space

We investigate the enumerative geometry of point configurations in projective space. We define "projective configuration counts": these enumerate configurations of points in projective space such that certain specified subsets are in fixed relative positions. The $\mathbb{P}^1$ case recovers cross-ratio degrees, which arise naturally in numerous contexts. We establish two main results. The first is a combinatorial upper bound given by the number of weighted transversals of a bipartite graph. The second is a recursion that relates counts associated to projective spaces of different dimensions, by projecting away from a given point. Key inputs include the Gelfand-MacPherson correspondence, the Jacobi-Trudi and Thom-Porteous formulae, and the notion of surplus from matching theory of bipartite graphs.

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arXiv2025

Euler characteristics of higher rank double ramification loci in genus one

Double ramification loci parametrise marked curves where a weighted sum of the markings is linearly trivial; higher-rank loci are obtained by imposing several such conditions simultaneously. We obtain closed formulae for the orbifold Euler characteristics of double ramification loci, and their higher-rank generalisations, in genus one. The rank-one formula is a polynomial, while the higher-rank formula involves greatest common divisors of matrix minors. The proof is based on a recurrence relation, which allows for induction on the rank and number of markings.

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arXiv2024

Chow theory of toric variety bundles

We describe the Chow homology and cohomology of toric variety bundles, with no restrictions on the singularities of the fibre. We present the ordinary and equivariant homologies as modules over the cohomology of the base, identify the ordinary cohomology with homology-valued Minkowski weights, and identify the equivariant cohomology with cohomology-weighted piecewise polynomial functions. We describe the product structure on Minkowski weights via a fan displacement rule, and the non-equivariant limit via equivariant multiplicities. Along the way we establish relative analogues of the K\"unneth property and Kronecker duality. Applications include the balancing condition in logarithmic enumerative geometry.

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arXiv2024

Toric configuration spaces: the bipermutahedron and beyond

We establish faithful tropicalisation for point configurations on algebraic tori. Building on ideas from enumerative geometry, we introduce tropical scaffolds and use them to construct a system of modular fan structures on the tropical configuration spaces. The corresponding toric varieties provide modular compactifications of the algebraic configuration spaces, with boundary parametrising transverse configurations on tropical expansions. The rubber torus, used to identify equivalent configurations, plays a key role. As an application, we obtain a modular interpretation for the bipermutahedral variety.

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arXiv2024

Divisors and curves on logarithmic mapping spaces

We determine the rational class and Picard groups of the moduli space of stable logarithmic maps in genus zero, with target projective space relative a hyperplane. For the class group we exhibit an explicit basis consisting of boundary divisors. For the Picard group we exhibit a spanning set indexed by piecewise-linear functions on the tropicalisation. In both cases a complete set of boundary relations is obtained by pulling back the WDVV relations from the space of stable curves. Our proofs hinge on a controlled technique for manufacturing test curves in logarithmic mapping spaces, opening up the topology of these spaces to further study.

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arXiv2024

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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arXiv2023

Rubber tori in the boundary of expanded stable maps

We investigate torus actions on logarithmic expansions in the context of enumerative geometry. Our main result is an intrinsic and coordinate-free description of the higher-rank rubber torus appearing in the boundary of the space of expanded stable maps. The rubber torus is constructed canonically from the tropical moduli space, and its action on each stratum of the expanded target is encoded in a linear tropical position map. The presence of 2-morphisms in the universal target forces expanded stable maps differing by the rubber action to be identified. This provides the first step towards a recursive description of the boundary of the expanded moduli space, with future applications including localisation and rubber calculus.

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arXiv2023

Gromov-Witten theory of bicyclic pairs

A bicyclic pair is a smooth surface equipped with a pair of smooth divisors intersecting in two reduced points. Resolutions of self-nodal curves constitute an important special case. We investigate the logarithmic Gromov-Witten theory of bicyclic pairs. We establish correspondences with local Gromov-Witten theory and open Gromov-Witten theory in all genera, a correspondence with orbifold Gromov-Witten theory in genus zero, and correspondences between all-genus refined Gopakumar-Vafa invariants and refined quiver Donaldson-Thomas invariants. For self-nodal curves in $\mathbb{P}(1,1,r)$ we obtain closed formulae for the genus zero invariants and relate these to the invariants of local curves. We also establish a conceptual relationship between invariants relative a self-nodal plane cubic and invariants relative a smooth plane cubic. The technical heart of the paper is a qualitatively new analysis of the degeneration formula for stable logarithmic maps, involving a tight intertwining of tropical and intersection-theoretic vanishing arguments.

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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

Universality for tropical and logarithmic maps

We prove that every toric monoid appears in a space of maps from tropical curves to an orthant. It follows that spaces of logarithmic maps to Artin fans exhibit arbitrary toric singularities: a virtual universality theorem for logarithmic maps to pairs. The target rank depends on the chosen singularity: we show that the cone over the 7-gon never appears in a space of maps to a rank 1 target. We obtain similar results for tropical maps to affine space.

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arXiv2022

Tangent curves to degenerating hypersurfaces

We study the behaviour of rational curves tangent to a hypersurface under degenerations of the hypersurface. Working within the framework of logarithmic Gromov-Witten theory, we extend the degeneration formula to the logarithmically singular setting, producing a virtual class on the space of maps to the degenerate fibre. We then employ logarithmic deformation theory to express this class as an obstruction bundle integral over the moduli space of ordinary stable maps. This produces new refinements of the logarithmic Gromov-Witten invariants, encoding the degeneration behaviour of tangent curves. In the example of a smooth plane cubic degenerating to the toric boundary we employ localisation and tropical techniques to compute these refinements. Finally, we leverage these calculations to describe how embedded curves tangent to a smooth cubic degenerate as the cubic does; the results obtained are of a classical nature, but the proofs make essential use of logarithmic Gromov-Witten theory.

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arXiv2022

Tropical expansions and toric variety bundles

A tropical expansion is a degeneration of a toroidal embedding, induced by a polyhedral subdivision of its tropicalisation. Each irreducible component of a tropical expansion admits a collapsing map down to a stratum of the original variety. We study the relative geometry of this map. We give a polyhedral criterion for the map to have the structure of a toric variety bundle, and prove that this structure always exists over the interior of the codomain. We give examples demonstrating that this is the strongest statement one can hope for in general. In addition, we provide a combinatorial recipe for constructing the toric variety bundle as a fibrewise GIT quotient of an explicit split vector bundle. Our proofs make systematic use of Artin fans as a language for globalising local toric models.

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arXiv2022

The local-orbifold correspondence for simple normal crossings pairs

For $X$ a smooth projective variety and $D=D_1+\ldots+D_n$ a simple normal crossings divisor, we establish a precise cycle-level correspondence between the genus zero local Gromov-Witten theory of the bundle $\oplus_{i=1}^n \mathcal{O}_X(-D_i)$ and the maximal contact Gromov-Witten theory of the multi-root stack $X_{D,\vec{r}}$. The proof is an implementation of the rank reduction strategy. We use this point of view to clarify the relationship between logarithmic and orbifold invariants.

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arXiv2021

Gromov-Witten theory with maximal contacts

We propose an intersection-theoretic method to reduce questions in genus zero logarithmic Gromov-Witten theory to questions in the Gromov-Witten theory of smooth pairs, in the presence of positivity. The method is applied to the enumerative geometry of rational curves with maximal contact orders along a simple normal crossings divisor and to recent questions about its relationship to local curve counting. Three results are established. We produce counterexamples to the local/logarithmic conjectures of van Garrel-Graber-Ruddat and Tseng-You. We prove that a weak form of the conjecture holds for product geometries. Finally, we explicitly determine the difference between local and logarithmic theories, in terms of relative invariants for which efficient algorithms are known. The polyhedral geometry of the tropical moduli of maps plays an essential and intricate role in the analysis.

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arXiv2021

GV and GW invariants via the enhanced movable cone

Given any smooth germ of a threefold flopping contraction, we first give a combinatorial characterisation of which Gopakumar-Vafa (GV) invariants are non-zero, by prescribing multiplicities to the walls in the movable cone. On the Gromov-Witten (GW) side, this allows us to describe, and even draw, the critical locus of the associated quantum potential. We prove that the critical locus is the infinite hyperplane arrangement of Iyama and the second author, and moreover that the quantum potential can be reconstructed from a finite fundamental domain. We then iterate, obtaining a combinatorial description of the matrix which controls the transformation of the non-zero GV invariants under a flop. There are three main ingredients and applications: (1) a construction of flops from simultaneous resolution via cosets, which describes how the dual graph changes, (2) a closed formula which describes the change in dimension of the contraction algebra under flop, and (3) a direct and explicit isomorphism between quantum cohomologies of different crepant resolutions, giving a Coxeter-style, visual proof of the Crepant Transformation Conjecture for isolated cDV singularities.

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arXiv2021

Curve counting in genus one: elliptic singularities & relative geometry

We construct and study the reduced, relative, genus one Gromov--Witten theory of very ample pairs. These invariants form the principal component contribution to relative Gromov--Witten theory in genus one and are relative versions of Zinger's reduced Gromov--Witten invariants. We relate the relative and absolute theories by degeneration of the tangency conditions, and the resulting formulas generalise a well-known recursive calculation scheme put forward by Gathmann in genus zero. The geometric input is a desingularisation of the principal component of the moduli space of genus one logarithmic stable maps to a very ample pair, using the geometry of elliptic singularities. Our study passes through general techniques for calculating integrals on logarithmic blowups of moduli spaces of stable maps, which may be of independent interest.

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arXiv2020

Relative quasimaps and mirror formulae

We construct and study the theory of relative quasimaps in genus zero, in the spirit of Gathmann. When $X$ is a smooth toric variety and $Y$ is a smooth very ample hypersurface in $X$, we produce a virtual class on the moduli space of relative quasimaps to $(X,Y)$, which we use to define relative quasimap invariants. We obtain a recursion formula which expresses each relative invariant in terms of invariants of lower tangency, and apply this formula to derive a quantum Lefschetz theorem for quasimaps, expressing the restricted quasimap invariants of $Y$ in terms of those of $X$. Finally, we show that the relative $I$-function of Fan-Tseng-You coincides with a natural generating function for relative quasimap invariants, providing mirror-symmetric motivation for the theory.

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arXiv2018

The fundamental solution matrix and relative stable maps

Givental's Lagrangian cone $\mathcal{L}_X$ is a Lagrangian submanifold of a symplectic vector space which encodes the genus-zero Gromov-Witten invariants of $X$. Building on work of Braverman, Coates has obtained the Lagrangian cone as the push-forward of a certain class on the moduli space of stable maps to $X \times \mathbb{P}^1$. This provides a conceptual description for an otherwise mysterious change of variables called the dilaton shift. In this note we recast this construction in its natural context, namely the moduli space of stable maps to $X \times \mathbb{P}^1$ relative the divisor $X \times \infty$. We find that the resulting push-forward is another familiar object, namely the transform of the Lagrangian cone under the action of the fundamental solution matrix. This hints at a generalisation of Givental's quantisation formalism to the setting of relative invariants. Finally, we use a hidden polynomiality property implied by our construction to obtain a sequence of universal relations for the Gromov-Witten invariants, as well as new proofs of several foundational results concerning both the Lagrangian cone and the fundamental solution matrix.

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