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Olivier Martin

Publications and source records attributed to Olivier Martin.

16 recordsLinked to original sources
arXiv2026

Low degree subvarieties of universal hypersurfaces

We study irreducible subvarieties of the universal hypersurface $\mathcal{X}/B$ of degree $d$ and dimension $n$. We prove that when $d$ is sufficiently large, a degree $kd$ subvariety $Z$ which dominates $B$ comes from intersection with a family of degree $k$ projective varieties parametrized by $B$. This answers a question raised independently by Farb and Ma. Our main tools consist of a Grassmannian technique due to Riedl and Yang, a theorem of Mumford-Roitman on rational equivalence of zero-cycles, and an analysis of Cayley-Bacharach conditions in the presence of a Galois action. We also show that the large degree assumption is necessary; for $d=3$, rational points are dense in $\text{Sym}^dX_{k(B)}$, and in particular are not collinear.

math.AG↗
arXiv2025

Spin-Split Dispersion of Leaky Surface plasmons in Inversion- Symmetric System

Spin-dependent dispersion and Rashba effect are manifestations of universal spin orbit interaction associated with the breaking of the spatial inversion symmetry in condensed matter and in optical systems. In sharp contrast to this, we report a spin-split dispersion effect of leaky surface plasmons in an inversion-symmetric one dimensional plasmonic grating system. In our system, the signature of spin-momentum locking and the resulting spin-polarization dependent splitting of dispersion of the surface plasmons are observed through the leakage radiation detected in a Fourier (momentum) domain optical arrangement. The setup enables single-shot recording of the full polarization-resolved dispersion (frequency vs transverse momentum (k)) of the leaky surface plasmons. Momentum domain polarization analysis identified a transverse momentum (k) dependent linear birefringence-linear dichroism effect (referred to as the geometric LB-LD effect) responsible for the observed spin-split dispersion. This unconventional SOI effect is reminiscent of the recently reported LB-LD effect resulting in giant chirality in centrosymmetric crystal, albeit with geometric origin. It is demonstrated that the interplay of the geometrical polarization transformation in focused polarized light and subsequent interaction of the structured field polarization with the plasmonic grating leads to the evolution of strong geometrical phase gradient or spin(circular polarization)-dependent transverse momentum of light resulting in spin-split dispersion. Our study offers a new paradigm of spin-based dispersion engineering and spin-enabled nano-optical functionalities in simple symmetric metasurfaces using geometric LB-LD effect.

physics.optics↗
arXiv2025

A primer on measures of irrationality

Measures of irrationality are a numerical way of quantifying how far a given variety is from being rational (or rationally connected, uniruled, etc.). In the last two decades, there has been renewed interest in the study of these invariants. The goal of this expository survey is to summarize known results from the point of view of the Kodaira--Enriques classification of surfaces, highlight recent progress, and discuss a number of open problems and questions.

math.AG↗
arXiv2024

Isotrivial Lagrangian fibrations of compact hyper-Kähler manifolds

This article initiates the study of isotrivial Lagrangian fibrations of compact hyper-Kähler manifolds. We present four foundational results that extend well-known facts about isotrivial elliptic fibrations of K3 surfaces. First, we prove that smooth fibers of an isotrivial Lagrangian fibration are isogenous to a power of an elliptic curve. Second, we exhibit a dichotomy between two types of isotrivial Lagrangian fibrations, which we call A and B. Third, we give a classification result for type A isotrivial Lagrangian fibrations. Namely, if a type A isotrivial Lagrangian fibration admits a rational section, then it is birational to one of two straightforward examples of isotrivial fibrations of hyper-Kähler manifolds of $\text{K3}^{[n]}$-type and $\text{Kum}_n$-type. Finally, we prove that a genericity assumption on the smooth fiber of an isotrivial Lagrangian fibration ensures that the fibration is of type A.

math.AG↗
arXiv2024

Effective zero-cycles and the Bloch-Beilinson filtration

A conjecture of Voisin states that two points on a smooth projective complex variety whose algebra of holomorphic forms is generated in degree 2 are rationally equivalent to each other if and only if their difference lies in the third step of the Bloch-Beilinson filtration. In this note, we formulate a generalization that allows for rational equivalence of effective zero-cycles of higher degree, at the expense of looking deeper in the Bloch-Beilinson filtration. In the first half, we provide evidence in support of this conjecture in the case of abelian varieties and projective hyper-Kähler manifolds. Notably, we give explicit criteria for rational equivalence of effective zero-cycles on moduli spaces of semistable sheaves on K3 surfaces, generalizing that of Marian-Zhao. In the second half, in an effort to explain our main conjecture, we formulate a second conjecture predicting when the diagonal of a smooth projective variety belongs to a subalgebra of the ring of correspondences generated in low degree.

math.AG↗
arXiv2021

Measures of association between algebraic varieties

This paper begins the exploration of what we call measures of association between two irreducible complex projective varieties of the same dimension. The idea is to study from various points of view the minimal complexity of correspondences between them. We extend to this setting results about degrees of irrationality for hypersurfaces, and we study joint covering invariants for pairs of curves and hypersurfaces. We also propose a number of conjectures and open problems.

math.AG↗
arXiv2021

Rational maps from products of curves to surfaces with $p_g = q = 0$

We study dominant rational maps from a product of two curves to surfaces with $p_{g} = q = 0$. Given two curves which satisfy a mild genericity assumption and have large genus relative to their gonality, we show that the degree of irrationality of their product is equal to the product of their gonalities. Moreover, we prove that the degree of irrationality of a product of two hyperelliptic curves is 4.

math.AG↗
arXiv2020

On a conjecture of Voisin on the gonality of very general abelian varieties

We study orbits for rational equivalence of zero-cycles on very general abelian varieties by adapting a method of Voisin to powers of abelian varieties. We deduce that, for $k$ at least $3$, a very general abelian variety of dimension at least $2k-2$ has covering gonality greater than $k$. This settles a conjecture of Voisin. We also discuss how upper bounds for the dimension of orbits for rational equivalence can be used to provide new lower bounds on other measures of irrationality. In particular, we obtain a strengthening of the Alzati-Pirola bound on the degree of irrationality of abelian varieties.

math.AG↗
arXiv2020

Equilibrium and sensitivity analysis of a spatio-temporal host-vector epidemic model

Insect-borne diseases are diseases carried by insects affecting humans, animals or plants. They have the potential to generate massive outbreaks such as the Zika epidemic in 2015-2016 mostly distributed in the Americas, the Pacific and Southeast Asia, and the multi-foci outbreak caused by the bacterium {\it Xylella fastidiosa} in Europe in the 2010s. In this article, we propose and analyze the behavior of a spatially-explicit compartmental model adapted to pathosystems with fixed hosts and mobile vectors disseminating the disease. The behavior of this model based on a system of partial differential equations is complementarily characterized via a theoretical study of its equilibrium states and a numerical study of its transitive phase using global sensitivity analysis. The results are discussed in terms of implications concerning the surveillance and control of the disease over a medium-to-long temporal horizon.

math.AP↗
arXiv2019

The degree of irrationality of most abelian surfaces is 4

The degree of irrationality of a smooth projective variety $X$ is the minimal degree of a dominant rational map $X\dashrightarrow \mathbb{P}^{\dim X}$. We show that if an abelian surface $A$ over $\mathbb{C}$ is such that the image of the intersection pairing $\text{Sym}^2NS(A)\to \mathbb{Z}$ does not contain $12$, then it has degree of irrationality $4$. In particular, a very general $(1,d)$-polarized abelian surface has degree of irrationality $4$ provided that $d\nmid 6$. This answers two questions of Yoshihara by providing the first examples of abelian surfaces with degree of irrationality greater than $3$ and showing that the degree of irrationality is not isogeny-invariant for abelian surfaces.

math.AG↗
arXiv2019

Degree of irrationality of a very general abelian variety

Consider a very general abelian variety $A$ of dimension at least $3$ and an integer $0<d\leq \dim A$. We show that if the map $A^k\to CH_0(A)$ has a $d$-dimensional fiber then $k\geq d+(\dim A+1)/2$. This extends results of the second-named author which covered the cases $d=1,2$. As a geometric application, we obtain that any dominant rational map from a very general abelian $g$-fold to $\mathbb{P}^g$ has degree at least $(3\dim A+1)/2$ for $g\geq 3$. This improves results of Alzati and the last-named author in the case of a very general abelian variety.

math.AG↗
arXiv2015

Short relaxation times but long transient times in both simple and complex reaction networks

When relaxation towards an equilibrium or steady state is exponential at large times, one usually considers that the associated relaxation time $τ$, i.e., the inverse of that decay rate, is the longest characteristic time in the system. However that need not be true, and in particular other times such as the lifetime of an infinitesimal perturbation can be much longer. In the present work we demonstrate that this paradoxical property can arise even in quite simple systems such as a chain of reactions obeying mass action kinetics. By mathematical analysis of simple reaction networks, we pin-point the reason why the standard relaxation time does not provide relevant information on the potentially long transient times of typical infinitesimal perturbations. Overall, we consider four characteristic times and study their behavior in both simple chains and in more complex reaction networks taken from the publicly available database "Biomodels." In all these systems involving mass action rates, Michaelis-Menten reversible kinetics, or phenomenological laws for reaction rates, we find that the characteristic times corresponding to lifetimes of tracers and of concentration perturbations can be much longer than $τ$.

q-bio.MN↗
arXiv2014

Analysis of on-sky MOAO performance of CANARY using natural guide stars

The first on-sky results obtained by CANARY, the Multi-Object Adaptive Optics (MOAO) demonstrator, are analysed. The data were recorded at the William Herschel Telescope, at the end of September 2010. We describe the command and calibrations algorithms used during the run and present the observing conditions. The processed data are MOAO-loop engaged or disengaged slope buffers, comprising the synchronised measurements of the four Natural Guide Stars (NGS) wavefront sensors running in parallel, and near Infra-Red (IR) images. We describe the method we use to establish the error budget of CANARY. We are able to evaluate the to- mographic and the open loop errors, having median values around 216 nm and 110 nm respectively. In addition, we identify an unexpected residual quasi-static field aberration term of mean value 110 nm. We present the detailed error budget analysed for three sets of data for three different asterisms. We compare the experimental budgets with the numerically simulated ones and demonstrate a good agreement. We find also a good agreement between the computed error budget from the slope buffers and the measured Strehl ratio on the IR images, ranging between 10% and 20% at 1 530 nm. These results make us confident in our ability to establish the error budget of future MOAO instruments.

astro-ph.IM↗
arXiv2012

First faint dual-field phase-referenced observations on the Keck interferometer

Ground-based long baseline interferometers have long been limited in sensitivity by the short integration periods imposed by atmospheric turbulence. The first observation fainter than this limit was performed on January 22, 2011 when the Keck Interferometer observed a K=11.5 target, about one magnitude fainter than its K=10.3 limit. This observation was made possible by the Dual Field Phase Referencing instrument of the ASTRA project: simultaneously measuring the real-time effects of the atmosphere on a nearby bright guide star, and correcting for it on the faint target, integration time longer than the turbulence time scale are made possible. As a prelude to this demonstration, we first present the implementation of Dual Field Phase Referencing on the interferometer. We then detail its on-sky performance focusing on the accuracy of the turbulence correction, and on the resulting fringe contrast stability. We conclude with a presentation of early results obtained with Laser Guide Star AO and the interferometer.

astro-ph.IM↗
arXiv2010

Spatial and topological organization of DNA chains induced by gene co-localization

Transcriptional activity has been shown to relate to the organization of chromosomes in the eukaryotic nucleus and in the bacterial nucleoid. In particular, highly transcribed genes, RNA polymerases and transcription factors gather into discrete spatial foci called transcription factories. However, the mechanisms underlying the formation of these foci and the resulting topological order of the chromosome remain to be elucidated. Here we consider a thermodynamic framework based on a worm-like chain model of chromosomes where sparse designated sites along the DNA are able to interact whenever they are spatially close-by. This is motivated by recurrent evidence that there exists physical interactions between genes that operate together. Three important results come out of this simple framework. First, the resulting formation of transcription foci can be viewed as a micro-phase separation of the interacting sites from the rest of the DNA. In this respect, a thermodynamic analysis suggests transcription factors to be appropriate candidates for mediating the physical interactions between genes. Next, numerical simulations of the polymer reveal a rich variety of phases that are associated with different topological orderings, each providing a way to increase the local concentrations of the interacting sites. Finally, the numerical results show that both one-dimensional clustering and periodic location of the binding sites along the DNA, which have been observed in several organisms, make the spatial co-localization of multiple families of genes particularly efficient.

physics.bio-ph↗