arXiv subjects

Ronno Das

Publications and source records attributed to Ronno Das.

11 recordsLinked to original sources

arXiv2025

Homological stability and Manin's conjecture for rational curves on quartic del Pezzo surfaces

We prove a version of Manin's conjecture (over $\mathbb{F}_{q}$ for $q$ large) and the Cohen--Jones--Segal conjecture (over $\mathbb{C}$) for maps from rational curves to split quartic del Pezzo surfaces. The proofs share a common method which builds upon prior work of the first and fourth authors. The main ingredients of this method are (i) the construction of bar complexes formalizing the inclusion-exclusion principle and its point counting estimates, (ii) dimension estimates for spaces of rational curves using conic bundle structures, (iii) estimates of error terms using arguments of Sawin--Shusterman based on Katz's results, and (iv) a certain virtual height zeta function revealing the compatibility of bar complexes and Peyre's constant. Our argument substantiates the heuristic approach to Manin's conjecture over global function fields given by Batyrev and Ellenberg--Venkatesh in this case.

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arXiv2024

Homology of spaces of curves on blowups

We consider the space of holomorphic maps from a compact Riemann surface to a projective space blown up at finitely many points. We show that the homology of this mapping space equals that of the space of continuous maps that intersect the exceptional divisors positively, once the degree of the maps is sufficiently positive compared to the degree of homology. The proof uses a version of Vassiliev's method of simplicial resolution. As a consequence, we obtain a homological stability result for rational curves on the degree $5$ del Pezzo surface, which is analogous to a case of the Batyrev--Manin conjectures on rational point counts.

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arXiv2024

The Mumford conjecture (after Bianchi)

We give a self-contained and streamlined rendition of Andrea Bianchi's recent proof of the Mumford conjecture using moduli spaces of branched covers.

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arXiv2023

Homological stability for the space of hypersurfaces with marked points

We study the space of smooth marked hypersurfaces in a given linear system. Specifically, we prove a homology h-principle to compare it with a space of sections of an appropriate jet bundle. Using rational models, we compute its rational cohomology in a range of degrees, and deduce a homological stability result for hypersurfaces of increasing degree. We also describe the Hodge weights on the stable cohomology, and thereby connect our topological result to motivic results of Howe.

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arXiv2022

Cohomological and motivic inclusion-exclusion

We categorify the inclusion-exclusion principle for partially ordered topological spaces and schemes to a filtration on the derived category of sheaves. As a consequence, we obtain functorial spectral sequences that generalize the two spectral sequences of a stratified space and certain Vassiliev-type spectral sequences; we also obtain Euler characteristic analogs in the Grothendieck ring of varieties. As an application, we give an algebro-geometric proof of Vakil and Wood's homological stability conjecture for the space of smooth hypersurface sections of a smooth projective variety. In characteristic zero this conjecture was previously established by Aumonier via topological methods.

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arXiv2020

Zeta statistics and Hadamard functions

We introduce the Hadamard topology on the Witt ring of rational functions, giving a simultaneous refinement of the weight and point-counting topologies. Zeta functions of algebraic varieties over finite fields are elements of the rational Witt ring, and the Hadamard topology allows for a conjectural unification of results in arithmetic and motivic statistics: The completion of the Witt ring for the Hadamard topology can be identified with a space of meromorphic functions which we call Hadamard functions, and we make the meta-conjecture that any "natural" sequence of zeta functions which converges to a Hadamard function in both the weight and point-counting topologies converges also in the Hadamard topology. For statistics arising from Bertini problems, zero-cycles or the Batyrev-Manin conjecture, this yields an explicit conjectural unification of existing results in motivic and arithmetic statistics that were previously connected only by analogy. As evidence for our conjectures, we show that Hadamard convergence holds for many natural statistics arising from zero-cycles, as well as for the motivic height zeta function associated to the motivic Batyrev-Manin problem for split toric varieties.

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arXiv2020

Arithmetic statistics on cubic surfaces

In this paper we compute the distributions of various markings on smooth cubic surfaces defined over the finite field $\mathbb{F}_q$, for example the distribution of pairs of points, `tritangents' or `double sixes'. We also compute the (rational) cohomology of certain associated bundles and covers over complex numbers.

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arXiv2019

Configurations of noncollinear points in the projective plane

We consider the space $F_n$ of configurations of $n$ points in $P^2$ satisfying the condition that no three of the points lie on a line. For $n = 4, 5, 6$, we compute $H^*(F_n; \mathbb{Q})$ as an $S_n$-representation. The cases $n = 5, 6$ are computed via the Grothendieck--Lefschetz trace formula in \'etale cohomology and certain "twisted" point counts for analogous spaces over $\mathbb{F}_q$.

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arXiv2019

Cohomology of the universal smooth cubic surface

We compute the rational cohomology of the universal family of smooth cubic surfaces using Vassiliev's method of simplicial resolution. Modulo embedding, the universal family has cohomology isomorphic to that of $\mathbb{P}^2$. A consequence of our theorem is that over the finite field $\mathbb{F}_q$, away from finitely many characteristics, the average number of points on a smooth cubic surface is $q^2 + q + 1$.

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arXiv2018

The space of cubic surfaces equipped with a line

The Cayley--Salmon theorem implies the existence of a 27-sheeted covering space specifying lines contained in smooth cubic surfaces over $\mathbb{C}$. In this paper we compute the rational cohomology of the total space of this cover, using the spectral sequence in the method of simplicial resolution developed by Vassiliev. The covering map is an isomorphism in cohomology (in fact of mixed Hodge structures) and the cohomology ring is isomorphic to that of $PGL(4,\mathbb{C})$. We derive as a consequence of our theorem that over the finite field $\mathbb{F}_q$ the average number of lines on a cubic surface equals 1 (away from finitely many characteristics); this average is $1 + O(q^{-1/2})$ by a standard application of the Weil conjectures.

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arXiv2014

Coxeter transformation groups and reflection arrangements in smooth manifolds

Artin groups are a natural generalization of braid groups and are well-understood in certain cases. Artin groups are closely related to Coxeter groups. There is a faithful representation of a Coxeter group $W$ as a linear reflection group on a real vector space $V$. The group acts properly and fixes a union of hyperplanes. The $W$-action extends as the covering space action to the complexified complement of these hyperplanes. The fundamental groups of the complement and the orbit space are the pure Artin group and the Artin group respectively. For the Artin groups of finite type Deligne proved that the associated complement is aspherical. Using the Coxeter group data Salvetti gave a construction of a cell complex which is a $W$-equivariant deformation retract of the complement. This construction was independently generalized by Charney and Davis to the Artin groups of infinite type. A lot of algebraic properties of these groups were discovered using combinatorial and topological properties of this cell complex. In this paper we represent a Coxeter group as a subgroup of diffeomorphisms of a smooth manifold. These so-called Coxeter transformation groups fix a union of codimension-$1$ (reflecting) submanifolds and permute the connected components of the complement. Their action naturally extends to the tangent bundle of the ambient manifold and fixes the union of tangent bundles of these reflecting submanifolds. Fundamental group of the tangent bundle complement and that of its quotient serve as the analogue of pure Artin group and Artin group respectively. The main aim of this paper is to prove Salvetti's theorems in this context. We show that the combinatorial data of the Coxeter transformation group can be used to construct a cell complex homotopy equivalent to the tangent bundle complement and that this homotopy equivalence is equivariant.

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